Sequence and Series
262 JEE Maths previous year questions on Sequence and Series — options free on every question; 26 include the answer & explanation free, the rest unlock with PYQ Pass.
If \(\alpha\) and \(\beta\) are the roots of the equation \(3 x^2-p x+q=0\) Let an A.P whose common diff. is \(\frac{3}{2}\) and \(10^{\text {th }}\) term is \(\alpha\) and \(11^{\text {th }}\) term is \(\beta\) and \(S_{11}=88\). Find the value of \(q-2 p\).
474
\(\begin{array}{rlrl}d=3 / 2 & a_{10}=\alpha =a+9 d \\a_{11} & =\beta=a+10 d \\3_{11}=\frac{11}{2}(2 a+10 d) & =88 \\(a+5 d) & =8 \\a+\frac{15}{2} & =8 \Rightarrow a=1 / 2\end{array}\)
\(\begin{gathered}\alpha=\frac{1}{2}+9 \times \frac{3}{2}=14 \beta=\frac{1}{2}+10 \times \frac{3}{2}=\frac{31}{2} 3 x^2-p x+q=0 \alpha+\beta=\frac{p}{3} \frac{14+81}{2}=\frac{p}{3} \Rightarrow \frac{28+31}{2}=\frac{p}{3} \frac{59}{2}=\frac{p}{3} \Rightarrow p=\frac{59 \times 3}{2}=\frac{177}{2}\end{gathered}\)
\(\begin{aligned}& \alpha \beta=q / 3 \\& 14 \times \frac{31}{2}=\frac{q}{3} \\& 217 \times 3=q \\& q=651 \\& q-2 p \\& 651-2\left(\frac{177}{2}\right)=651-177=474\end{aligned}\)
\(\left(\frac{1}{3}+\frac{4}{7}\right)+\left(\frac{1}{{3}^{2}}+\frac{1}{3}\times \frac{4}{7}+\frac{{4}^{2}}{{7}^{2}}\right)+\)\(\left(\frac{1}{{3}^{3}}+\frac{1}{{3}^{2}}\times \frac{4}{7}+\frac{1}{3}\times \frac{{4}^{2}}{{7}^{2}}+\frac{{4}^{3}}{{7}^{3}}\right)+...\)upto infinite terms, is equal to
\(\frac{5}{2}\)
Let \(a=\frac{4}{7},b=\frac{1}{3}\)
Therefore we get
\(S=\left(a+b\right)+\left({a}^{2}+ab+{b}^{2}\right)+\left({a}^{3}+{a}^{2}b+a{b}^{2}+{b}^{3}\right)+...\infty\)
Multiply numerator and denominator by \((a – b)\) \(=\frac{4}{7}−\frac{1}{3}=\frac{5}{21}\)
\(S=\frac{1}{a−b}\left[\left({a}^{2}−{b}^{2}\right)+\left({a}^{3}−{b}^{3}\right)+\left({a}^{4}−{b}^{4}\right)+\ldots .∞\right]\)
\(S=\frac{1}{a−b}\left[\frac{{a}^{2}}{1−a}−\frac{{b}^{2}}{1−b}\right]\)
\(=\frac{21}{5}\left[\frac{\frac{16}{49}}{1−\frac{4}{7}}−\frac{\frac{1}{9}}{1−\frac{1}{3}}\right]\)
\(=\frac{21}{5}\left[\frac{16}{21}−\frac{1}{6}\right]=\frac{21}{5}\left[\frac{96−21}{21\cdot 6}\right]\)
\(=\frac{75}{5\cdot 6}=\frac{15}{6}=\frac{5}{2}\)
If the arithmetic mean of two distinct positive real numbers \(a\) and \(b\) (where \(a>b\) ) is twice their geometric mean, then \(a: b\) is:
\(2+\sqrt{3}: 2-\sqrt{3}\)
From the given condition, we can say that,
\(AM=2\times GM\\ \frac{a+b}{2}=2\sqrt{ab}\\ a+b=4\sqrt{ab}\)
\(\text{we know,}\\ {\left(a-b\right)}^{2}={\left(a+b\right)}^{2}-4ab\\ =16ab-4ab=12ab\\ a-b=2\sqrt{3}\sqrt{ab}\)
\(\text{ (Taking }+ve\text{ sign only as }a>b\text{ ) }\\ ∴\frac{a+b}{a-b}=\frac{4\sqrt{ab}}{2\sqrt{3}\sqrt{ab}}=\frac{2}{\sqrt{3}}\\ \text{ By componendo and dividendo, }\\ \frac{a}{b}=\frac{2+\sqrt{3}}{2-\sqrt{3}}\)
The sum \(\frac{{1}^{3}}{1}+\frac{{1}^{3}+{2}^{3}}{1+3}+\frac{{1}^{3}+{2}^{3}+{3}^{3}}{1+3+5}+⋯\) up to \(8\) terms, is:
[JEE Main 2026, 2 Apr (Shift 2)]
\(71\)
\( \mathrm{T}_{\mathrm{r}}=\frac{1^3+2^3+3^3+\ldots +\ldots r^3}{1+3+5+\ldots .+(2 \mathrm{r}-1)}=\frac{\left(\frac{\mathrm{r}(\mathrm{r}+1)}{2}\right)^2}{\mathrm{r}^2} \)
\( =\frac{\mathrm{r}^2+2 \mathrm{r}+1}{4} \)
\( \mathrm{~S}_{\mathrm{n}}=\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{~T}_{\mathrm{r}} \)
\( \mathrm{~S}_{\mathrm{n}}=\frac{1}{4} \sum_{\mathrm{r}=1}^{\mathrm{n}} \left(\mathrm{r}^2+2 \mathrm{r}+1\right) \)
\( =\frac{1}{4}\left[\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}+2 \frac{\mathrm{n}(\mathrm{n}+1)}{2}+\mathrm{n}\right] \)
\( \mathrm{S}_8=\frac{1}{4}\left[\frac{8 \times 9 \times 17}{6}+8 \times 9+8\right]\)
\( =\frac{1}{4}[204+72+8]=71\)
If \(2{a}_{n+2}=5{a}_{n+1}-3{a}_{n}\), where \(n=0,1,2,\ldots\). If \({a}_{0}=\)3 and \({a}_{1}=4\), then the value of \(\sum _{k=1}^{100}{a}_{k}\) is equal to (28 Jan, Shift I, Memory Based)
\(3{a}_{100}+91\)
\(2{a}_{n+2}=5{a}_{n+1}-3{a}_{n}\\ \Rightarrow 2{t}^{2}-5t+3=0\\ \Rightarrow t=1,\frac{3}{2}\\ {a}_{n}=A\cdot (1{)}^{n}+B\cdot {\left(\frac{3}{2}\right)}^{n}\\ {a}_{0}=3,{a}_{1}=4\\ thenA=1\&B=2\\ Now,{a}_{n}=1+2{\left(\frac{3}{2}\right)}^{n}\\ {a}_{100}=1+2{\left(\frac{3}{2}\right)}^{100}\\ {\left(\frac{3}{2}\right)}^{100}=\frac{{a}_{100}-1}{2}\\ Now\sum _{k=1}^{100}{a}_{k}={S}_{100}=100+6\left({\left(\frac{3}{2}\right)}^{100}-1\right)\\ =100+6\left(\frac{{a}_{100}-1}{2}-1\right)\\ =3{a}_{100}+91\)
Let the sum of the first \(n\) terms of an A.P. be \(3 n^2+5 n\). Then the sum of squares of the first \(10\) terms of the A.P. is:
[JEE Main 2026, 5 Apr (Shift 1)]
\(15220\)
\(S_n=3 n^2+5 n\)
\(T_n=S_n-S_{n-1}\)
\(=3\left(n^2-(n-1)^2\right)+5(n-(n-1))\)
\(=3(2 n-1)+5\)
\(=6 n+2\)
\(\sum_{n=1}^{10}(6 n+2)^2=\sum 36 n^2+\sum 4+\sum 24 n\)
\(=36 \times \frac{10 \times 11 \times 21}{6}+4 \times 10+24 \frac{10 \times 11}{2}\)
\(=15220\)
Let \(A B C\) be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle \(A B C\) and the same process is repeated infinitely many times. If \(\mathrm{P}\) is the sum of perimeters and \(\mathrm{Q}\) is be the sum of areas of all the triangles formed in this process, then :
[JEE Main 2024, 6 Apr (Shift 2)]
\({\mathrm{P}}^{2}=36\sqrt{3}\mathrm{Q}\)
Let the side of the original equilateral triangle be \(a\).
When the middle points of the sides are joined, the side of the new equilateral triangle becomes half of the previous side.
So, the sides of the triangles formed are \(\frac{a}{2},\frac{a}{4},\frac{a}{8},\ldots\)
Now, perimeter of an equilateral triangle of side \(s\) is \(3s\).
Therefore, \(P=3\cdot\frac{a}{2}+3\cdot\frac{a}{4}+3\cdot\frac{a}{8}+\cdots\)
\(P=3a\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\right)\)
\(P=3a\cdot 1\)
\(P=3a\)
Area of an equilateral triangle of side \(s\) is \(\frac{\sqrt{3}}{4}s^2\).
Therefore, \(Q=\frac{\sqrt{3}}{4}\left(\frac{a}{2}\right)^2+\frac{\sqrt{3}}{4}\left(\frac{a}{4}\right)^2+\frac{\sqrt{3}}{4}\left(\frac{a}{8}\right)^2+\cdots\)
\(Q=\frac{\sqrt{3}}{4}a^2\left(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots\right)\)
\(Q=\frac{\sqrt{3}}{4}a^2\cdot \frac{\frac{1}{4}}{1-\frac{1}{4}}\)
\(Q=\frac{\sqrt{3}}{4}a^2\cdot \frac{1}{3}\)
\(Q=\frac{\sqrt{3}a^2}{12}\)
Now, \(P^2=(3a)^2=9a^2\)
Also, \(36\sqrt{3}Q=36\sqrt{3}\cdot \frac{\sqrt{3}a^2}{12}\)
\(=36\cdot \frac{3a^2}{12}\)
\(=9a^2\)
Thus, \(P^2=36\sqrt{3}Q\)
Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m,\) \({\mathrm{T}}_{\mathrm{m}}=\frac{1}{25},{\mathrm{T}}_{25}=\frac{1}{20}\text{ and }20\sum _{\mathrm{r}=1}^{25}{\mathrm{T}}_{\mathrm{r}}=13,\) then \(5\mathrm{m}\sum _{\mathrm{r}=\mathrm{m}}^{2\mathrm{m}}{\mathrm{T}}_{\mathrm{r}}\) is equal to:
[JEE Main 2025, 28 Jan (Shift 1)]
126
\({\text{Given, T}}_{r}\text{represents terms of A.P.}\\ {\mathrm{T}}_{\mathrm{m}}=\frac{1}{25},{\mathrm{T}}_{25}=\frac{1}{20},20\sum _{\mathrm{r}=1}^{25}{\mathrm{T}}_{\mathrm{r}}=13\\ 20\sum _{r=1}^{25}{T}_{r}=20\left[\frac{25}{2}\left[a+\frac{1}{20}\right]\right]=13\\ \Rightarrow a=\frac{1}{20\times 25}=\frac{1}{500}\\ ∵{T}_{25}=a+24d=\frac{1}{20}\\ \Rightarrow d=\frac{1}{20\times 25}=\frac{1}{500}\\ \text{Now},\\ {T}_{m}=a+(m-1)d=\frac{1}{25}\\ =\frac{1}{500}m=\frac{1}{25}\\ \Rightarrow m=20\\ \text{ Now, }5m\sum _{r=m}^{2m}{T}_{r}=5\times 20\left[\sum _{r=20}^{40}{T}_{r}\right]\\ =100\left[\frac{40}{2}(2a+39d)-\frac{19}{2}(2a+18d)\right]\\ \text{but a=d (we already find above)}\\ 5m\sum _{r=m}^{2m}{T}_{r}=100\left[\frac{40}{2}\times 41d-\frac{19}{2}\times 20d\right]\\ =126\)
If the sum of the second, fourth and sixth terms of a G.P. of positive terms is \(21\) and the sum of its eighth, tenth and twelfth terms is \(15309\), then the sum of its first nine terms is :
[JEE Main 2025, 7 Apr (Shift 2)]
\(757\)
Let the first term of the G.P. be \(a\) and the common ratio be \(r\).
Given
\(a r+a r^3+a r^5=21 \Rightarrow a r\left(1+r^2+r^4\right)=21 \ldots (1)\)
and \(a r^7+a r^9+a r^{11}=15309 \Rightarrow a r^7\left(1+r^2+r^4\right)=15309 \ldots (2)\)
Divide Equation 2 by Equation 1
\(\frac{a r^7\left(1+r^2+r^4\right)}{a r\left(1+r^2+r^4\right)}=\frac{15309}{21}\)
\(r^6=729\)
\(r=3\)
from eq (1)
\(\begin{aligned} & 3 a(1+9+81)=21 \\ & 3 a(91)=21 \Rightarrow 273 a=21 \\ & a=\frac{21}{273}=\frac{1}{13}\end{aligned}\)
Now
\(\begin{aligned} & S_9=\frac{\frac{1}{13}\left(3^9-1\right)}{3-1} \\ & S_9=\frac{1}{13} \cdot \frac{19683-1}{2} \\ & S_9=\frac{19682}{26} \\ & S_9=757\end{aligned}\)
\(\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}\)is equal to:
[JEE Main 2026, 28 Jan (Shift 2)]
\({2}^{26}\)
\(S=\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}\\ =\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[1+6+{6}^{2}+...+{6}^{24}\right]\)
\(=\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[\frac{{\left(6\right)}^{25}−1}{6−1}\right]\)
\(=\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[\frac{{6}^{25}−1}{5}\right]\)
\(=\frac{2}{{3}^{25}}+2\left[{2}^{25}−\frac{1}{{3}^{25}}\right]\)
\(={2}^{26}\)
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
[JEE Main 2024, 29 Jan (Shift 1)]
6
Given the sum of all the terms is 7 times the sum of the odd terms of the G.P.
\(a+a r+a r^2+a r^3+\ldots+a r^{63}=7\left(a+a r^2+a r^4 \ldots+a r^{62}\right)\)
\(\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7a\left(1-\left(r^2\right)^{32}\right)}{1-r^2}\)
\(\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7 a\left(1-r^{64}\right)}{1-r^2}\)
\(\Rightarrow 1+r=7\\ \Rightarrow r=6\)
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
[JEE Main 2024, 29 Jan (Shift 1)]
6
Given the sum of all the terms is 7 times the sum of the odd terms of the G.P.
\(a+a r+a r^2+a r^3+\ldots+a r^{63}=7\left(a+a r^2+a r^4 \ldots+a r^{62}\right)\)
\(\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7a\left(1-\left(r^2\right)^{32}\right)}{1-r^2}\)
\(\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7 a\left(1-r^{64}\right)}{1-r^2}\)
\(\Rightarrow 1+r=7\\ \Rightarrow r=6\)
Let \(\alpha, \beta\) be the roots of the equation \(x^2-x+p=0\) and \(\gamma, \delta\) be the roots of the equation \(x^2-4 x+q=0, \mathrm{p}, \mathrm{q} \in \mathrm{Z}\). If \(\alpha, \beta, \gamma, \delta\) are in G.P., then \(|p+q|\) equals:
[JEE Main 2026, 5 Apr (Shift 2)]
\(34\)
Let \(\alpha, \beta, \gamma, \delta\) be \(a, a r, a r^2, a r^3\), respectively.
For \(x^2-x+p=0\),
sum of roots \(=1\),
so \(a+a r=a(1+r)=1\)
For \(x^2-4 x+q=0\),
sum of roots \(=4\),
so \(a r^2+a r^3=a r^2(1+r)=4\)
Dividing,
\(\frac{a r^2(1+r)}{a(1+r)}=\frac{4}{1}\)
\(r^2=4\)
So, \(r=2\) or \(r=-2\).
If \(r=2\), then \(a(3)=1 \Rightarrow a=\frac{1}{3}\),
so \(p=\alpha \beta=a \cdot a r=a^2 r=\frac{2}{9}\), not an integer.
Hence \(r=-2\).
Now \(a(1-2)=1 \Rightarrow-a=1 \Rightarrow a=-1\).
So,
\(p=\alpha \beta=a \cdot a r=a^2 r=-2\)
Also,
\(q=\gamma \delta=a r^2 \cdot a r^3=a^2 r^5=(-2)^5=-32\)
\(|p+q|=|-2-32|=34\)
Let \(a_1, a_2, a_3, a_4\) be an A.P. of four terms such that each term of the A.P. and its common difference \(l\) are integers. If \(a_1+a_2+a_3+ a_4=48\) and \(a_1 a_2 a_3 a_4+l^4=361\), then the largest term of the A.P. is equal to
[JEE Main 2026, 24 Jan (Shift 2)]
\(27\)
Let four terms are \(a−3d,a−d,a+d,a+3d\)
Where \(d=\frac{l}{2}\)
\(∵{a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}=48\)
\(\Rightarrow 4a=48\Rightarrow a=12\)
\({a}_{1}{a}_{2}{a}_{3}{a}_{4}+{l}^{4}=361\)
\(\Rightarrow \left({a}^{2}−9{d}^{2}\right)\left({a}^{2}−{d}^{2}\right)+16{d}^{4}=361\)
\(\Rightarrow \left(144−9{d}^{2}\right)\left(144−{d}^{2}\right)+16{d}^{4}=361\)
\(\Rightarrow 25{d}^{4}−1440{d}^{2}+{(144)}^{2}=361\)
\(\Rightarrow {\left(5{d}^{2}−144\right)}^{2}={19}^{2}\)
\(\Rightarrow 5{d}^{2}−144=19,−19\)
\(\Rightarrow {d}^{2}=\frac{163}{5},25\)
\(\Rightarrow d=\sqrt{\frac{163}{5}},5\)
\(∴l=2\sqrt{\frac{163}{5}},10\)
Common difference is an integer therefore \(l=10\)
Largest term \(= 12 + 15 = 27\)
Consider the quadratic equation \(\left({n}^{2}-2n+2\right){x}^{2}-3x+{\left({n}^{2}-2n+2\right)}^{2}=0,n\in R\). Let \(\alpha\) be the minimum value of the product of its roots and \(\beta\) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \(\alpha\) and the common ratio is \(\frac{\alpha }{\beta }\), is:
[JEE Main 2026, 6 Apr (Shift 2)]
\(\frac{364}{243}\)
We have \(\left(n^2-2 n+2\right) x^2-3 x+\left(n^2-2 n+2\right)^2=0\)
Since \(n^2-2 n+2=(n-1)^2+1\)
Sum of roots \(=\frac{3}{(n-1)^2+1}\)
Since the denominator is minimum when \((n-1)^2=0\)
Therefore \(\beta=3\)
Product of roots \(=(n-1)^2+1\)
So the minimum value of the product is clearly
\(\alpha=1\)
Now, first term \(a=\alpha=1\)
Common ratio \(r=\frac{\alpha}{\beta}=\frac{1}{3}\)
Now, \(S_6=1 \cdot \frac{1-\left(\frac{1}{3}\right)^6}{1-\frac{1}{3}}\)
\(S_6=\frac{1092}{729}=\frac{364}{243}\)
Let \(S_n\) denote the sum of first \(n\) terms of an arithmetic progression. If \(S_{20}=790\) and \(S_{10}=145\), then \(S_{15}-S_5\) is :
[JEE Main 2024, 30 Jan (Shift 1)]
395
Given, \(S_{20}=\frac{20}{2}[2 a+19 d]=790\)
\(2 a+19 d=79\)\(\ldots(1)\)
and, \(S_{10}=\frac{10}{2}[2 a+9 d]=145\)
\(2 a+9 d=29\)\(\ldots(2)\)
From (1) and (2)
\(a=-8, d=5\)
Now, \(S_{15}-S_5=\frac{15}{2}[2 a+14 d]-\frac{5}{2}[2 a+4 d]\)
\(=\frac{15}{2}[-16+70]-\frac{5}{2}[-16+20]\)
\(=405-10\)
\(=395\)
Let \(S_n\) denote the sum of first \(n\) terms of an arithmetic progression. If \(S_{20}=790\) and \(S_{10}=145\), then \(S_{15}-S_5\) is :
[JEE Main 2024, 30 Jan (Shift 1)]
395
Given, \(S_{20}=\frac{20}{2}[2 a+19 d]=790\)
\(2 a+19 d=79\)\(\ldots(1)\)
and, \(S_{10}=\frac{10}{2}[2 a+9 d]=145\)
\(2 a+9 d=29\)\(\ldots(2)\)
From (1) and (2)
\(a=-8, d=5\)
Now, \(S_{15}-S_5=\frac{15}{2}[2 a+14 d]-\frac{5}{2}[2 a+4 d]\)
\(=\frac{15}{2}[-16+70]-\frac{5}{2}[-16+20]\)
\(=405-10\)
\(=395\)
If \(7=5+\frac{1}{7}(5+\alpha )+\frac{1}{{7}^{2}}(5+2\alpha )\)\(+\frac{1}{{7}^{3}}(5+3\alpha )+\ldots \ldots \ldots ...\infty\) then the value of \(\alpha\) is :
[JEE Main 2025, 24 Jan (Shift 2)]
6
Let
\(S=5+\frac{1}{7}(5+\alpha)+\frac{1}{72}(5+2 \alpha)+\ldots \infty \quad \ldots(i)\)
On multiply by \(\frac{1}{7}\), we get
\(\frac{1}{7} S=\frac{1}{7}(5)+\frac{1}{7^2}(5+\alpha)+\ldots \infty \ldots(i i)\)
Subtracting equation (ii) from (i), we get
\(\begin{aligned} & \frac{6}{7}(S)=5+\frac{1}{7} \alpha\left(\frac{1}{1-\frac{1}{7}}\right) \\ & 6=5+\frac{\alpha}{6} \\ & \Rightarrow \alpha=6\end{aligned}\)
\(\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}\)is equal to:
[JEE Main 2026, 28 Jan (Shift 2)]
\({2}^{26}\)
\(S=\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}\\ =\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[1+6+{6}^{2}+...+{6}^{24}\right]\)
\(=\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[\frac{{\left(6\right)}^{25}−1}{6−1}\right]\)
\(=\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[\frac{{6}^{25}−1}{5}\right]\)
\(=\frac{2}{{3}^{25}}+2\left[{2}^{25}−\frac{1}{{3}^{25}}\right]\)
\(={2}^{26}\)
If \(\log _e a , \log _e b , \log _e c\) are in an A.P. and \(\log _e a -\log _e 2 b , \log _e 2 b -\log _e 3 c , \log _e 3 c\) \(-\log _e a\) are also in an A.P, then \(a : b : c\) is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
\(9: 6: 4\)
\( \log _e a, \log _e b, \log _e c \) are in A.P.
\(\therefore \mathrm{b}^2=\mathrm{ac} \quad \ldots .(i)\)
Also \(\log _e\left(\frac{a}{2 b}\right), \log _c\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right)\) are in A.P.
\( \left(\frac{2 \mathrm{~b}}{3 \mathrm{c}}\right)^2=\frac{\mathrm{a}}{2 \mathrm{~b}} \times \frac{3 \mathrm{c}}{\mathrm{a}} \)
\( \frac{\mathrm{~b}}{\mathrm{c}}=\frac{3}{2}\)
Putting in eq. (i)
\( b^2=a \times \frac{2 b}{3} \)
\(\frac{a}{b}=\frac{3}{2} \)
\( a: b: c=9: 6: 4\)
If \(\log _e a , \log _e b , \log _e c\) are in an A.P. and \(\log _e a -\log _e 2 b , \log _e 2 b -\log _e 3 c , \log _e 3 c\) \(-\log _e a\) are also in an A.P, then \(a : b : c\) is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
\(9: 6: 4\)
\( \log _e a, \log _e b, \log _e c \) are in A.P.
\(\therefore \mathrm{b}^2=\mathrm{ac} \quad \ldots .(i)\)
Also \(\log _e\left(\frac{a}{2 b}\right), \log _c\left(\frac{2 b}{3 c}\right), \log _e\left(\frac{3 c}{a}\right)\) are in A.P.
\( \left(\frac{2 \mathrm{~b}}{3 \mathrm{c}}\right)^2=\frac{\mathrm{a}}{2 \mathrm{~b}} \times \frac{3 \mathrm{c}}{\mathrm{a}} \)
\( \frac{\mathrm{~b}}{\mathrm{c}}=\frac{3}{2}\)
Putting in eq. (i)
\( b^2=a \times \frac{2 b}{3} \)
\(\frac{a}{b}=\frac{3}{2} \)
\( a: b: c=9: 6: 4\)
The value of \({1}^{3}-{2}^{3}+{3}^{3}-\ldots +{15}^{3}\) is:
[JEE Main 2026, 6 Apr (Shift 1)]
\(1856\)
\(S=1^3-2^3+3^3-\ldots+15^3\)
\(=\left(1^3+2^3+\ldots+15^3\right)-2\left(2^3+4^3+\ldots+14^3\right)\)
\(=\left(1^3+2^3+\ldots+15^3\right)-2^4\left(1^3+2^3+\ldots+7^3\right)\)
\(=\left[\frac{15 \times 16}{2}\right]^2-16\left[\frac{7 \times 8}{2}\right]^2\)
\(=14400-16\times 784\)
\(=14400-12544\)
\(=1856\)
\(\text { If } s_n=\sum_{r=0}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64} \text { then find } \operatorname{Lim}_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{T_r}=\) (22 Jan, Shift I, Memory Based)
\(\frac{2}{3}\)
\(\begin{aligned}& T_n=S_{n-} S_{n-1} \\& =\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)-(2 n-3)(2 n-1)(2 n+1)(2 n+3)}{64} \\& T_n=\frac{(2 n-1)(2 n+1)(2 n+3)}{8} \\& \frac{1}{T_n}=\frac{8}{(2 n-1)(2 n+1)(2 n+3)} \\& \frac{1}{T_n}=2\left(\frac{1}{(2 n-1)(2 n+1)}-\frac{1}{(2 n-1)(2 n+3)}\right) \\& \sum_{r=1}^n \frac{1}{T_r}=2\left(\frac{1}{1 \times 3}-\frac{1}{(2 n-1)(2 n+3)}\right) \\& \operatorname{Lim}_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{T_r}=\frac{2}{3}\end{aligned}\)
Let \(a,ar,a{r}^{2},\ldots ..\). be an infinite G.P. If \(\sum _{n=0}^{\infty }a{r}^{n}=57\) and \(\sum _{n=0}^{\infty }{a}^{3}{r}^{3n}=9747\), then \(a + 18r\) is equal to
[JEE Main 2024, 9 Apr (Shift 2)]
31
\(\sum_{n=0}^{\infty} a r^n=57\)
\(\frac{a}{1-r}=57\) \(\ldots(i)\)
\(\sum_{n=0}^{\infty} a^3 r^{3 n}=9747\)
\(\frac{\mathrm{a}^3}{1-\mathrm{r}^3}=9747\) \(\ldots (ii)\)
from (i) and (ii)
\(\frac{\frac{a^3}{(1-r)^3}}{\frac{a^3}{1-r^3}}=\frac{57^3}{9747}=19\)
On solving, \(r=\frac{2}{3}\) and \(r=\frac{3}{2}\) (rejected)
\(a=19\)
\(\therefore a+18 r=19+18 \times \frac{2}{3}=31\)
If the arithmetic mean of two distinct positive real numbers \(a\) and \(b\) (where \(a>b\) ) is twice their geometric mean, then \(a: b\) is:
\(2+\sqrt{3}: 2-\sqrt{3}\)
From the given condition, we can say that,
\(AM=2\times GM\\ \frac{a+b}{2}=2\sqrt{ab}\\ a+b=4\sqrt{ab}\)
\(\text{we know,}\\ {\left(a-b\right)}^{2}={\left(a+b\right)}^{2}-4ab\\ =16ab-4ab=12ab\\ a-b=2\sqrt{3}\sqrt{ab}\)
\(\text{ (Taking }+ve\text{ sign only as }a>b\text{ ) }\\ ∴\frac{a+b}{a-b}=\frac{4\sqrt{ab}}{2\sqrt{3}\sqrt{ab}}=\frac{2}{\sqrt{3}}\\ \text{ By componendo and dividendo, }\\ \frac{a}{b}=\frac{2+\sqrt{3}}{2-\sqrt{3}}\)
Let \(A\) be the set of first \(101\) terms of an A.P., whose first term is \(1\) and the common difference is \(5\) and let \(B\) be the set of first \(71\) terms of an A.P., whose first term is \(9\) and the common difference is \(7\). Then the number of elements in \(A\cap B\), which are divisible by \(3\), is:
[JEE Main 2026, 2 Apr (Shift 1)]
\(5\)
First A.P.
Set \(\mathrm{A}=\{1,6,11,16 \ldots .101\) terms \(\}\)
Second A.P.
Set \(\mathrm{B}=\{9,16 \ldots . .71\) terms \(\}\)
\(\mathrm{D}=\mathrm{L} . \mathrm{C} . \mathrm{M}\left\{\mathrm{d}_1, \mathrm{~d}_2\right\}=35\)
\(1^{\text {st }}\) Common term is \(16\)
\(16+(\mathrm{n}-1) 35 \leq 499\)
\(\mathrm{n} \leq 14.8\)
\(\Rightarrow \mathrm{n}=14\)
\(A \cap B=\{16,51,86,121,156,191,226,261,296,331,366,401,436,471\}\)
Terms divisible by \(3=\{51,156,261,366,471\}\)
\(=5\) terms
In an arithmetic progression, if \(\mathrm{S}_{40}=1030\) and \(\mathrm{S}_{12}=57\), then \(\mathrm{S}_{30}-\mathrm{S}_{10}\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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Let \({a}_{1},{a}_{2},{a}_{3},\ldots \ldots\) be an A.P. and \({g}_{1}={a}_{1},{g}_{2},{g}_{3}\ldots \ldots\) be an increasing G.P. If \({a}_{1}={a}_{2}+{g}_{2}=1\) and \({a}_{3}+{g}_{3}=4\), then \({a}_{10}+{g}_{5}\) is equal to:
[JEE Main 2026, 2 Apr (Shift 2)]
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Let \({2}^{\text{nd }},{8}^{\text{th }}\) and \({44}^{\text{th }}\) terms of a non-constant A. P. be respectively the \({1}^{\text{st }},{2}^{\text{nd }}\) and\({3}^{\text{rd }}\) terms of a G. P. If the first term of the A. P. is 1 , then the sum of its first 20 terms is equal to -
[JEE Main 2024, 31 Jan (Shift 2)]
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The first term of an A.P. of \(30\) non-negative terms is \(\frac{10}{3}\). If the sum of this A.P. is the cube of its last term, then its common difference is:
[JEE Main 2026, 4 Apr (Shift 1)]
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The sum \(1+\frac{1}{2}\left({1}^{2}+{2}^{2}\right)+\frac{1}{3}\left({1}^{2}+{2}^{2}+{3}^{2}\right)+\ldots\) upto \(10\) terms is equal to:
[JEE Main 2026, 6 Apr (Shift 2)]
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If \(\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}=\frac{(2\mathrm{n}-1)(2\mathrm{n}+1)(2\mathrm{n}+3)(2\mathrm{n}+5)}{64}\), then \(\lim _{n\to \infty }\sum _{r=1}^{n}\left(\frac{1}{{T}_{r}}\right)\) is equal to :
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Number of terms in an arithmetic progression is \(2 n\). Sum of terms occurring at even places is 40 and sum of terms occurring at odd places is 55 . If the first term exceeds the last term by 27 , then \(n\) equals to (22 Jan, Shift II, Memory Based)
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Let \(f\) and \(g\) be functions satisfying \(f(x+y)=\)\(f(x)f(y),\text{ }f(1)=7\) and \(g(x+y)=g(xy)\), \(g(1)=1,\) for all \(x,\text{ }y\in N.\) If \(\sum _{x=1}^{n}\left(\frac{f(x)}{g(x)}\right)=19607,\) then \(n\) is equal to:
[JEE Main 2026, 22 Jan (Shift 2)]
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If for an arithmetic progression, if first term is 3 and sum of first four terms is equal to \(\frac{1}{5}\) of the sum of next four terms, then the sum of first 20 terms is
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The product of all real solutions of the equation \({e}^{5{\left({\log }_{e}x\right)}^{2}+3}={x}^{8},x>0,\) is:
[JEE Main 2025, 22 Jan (Shift 1)]
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The interior angle of a polygon with n side are in A.P with common difference of \(6^\circ\). If the biggest interior angle of polygon is \(219^\circ\) then n is equal to
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The common difference of the A.P.: \({a}_{1},{a}_{2},\ldots ,{a}_{m}\) is \(13\) more than the common difference of the A.P.: \({b}_{1},{b}_{2},\ldots ,{b}_{n}\). If \({b}_{31}=−277,{b}_{43}=−385\) and \({a}_{78}=327\), then \({a}_{1}\) is equal to
[JEE Main 2026, 28 Jan (Shift 1)]
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Consider an A.P.: \({a}_{1},{a}_{2},\ldots ,{a}_{\text{n}};{a}_{1}>0.\) If \({a}_{2}−{a}_{1}=\frac{−3}{4},{a}_{\text{n}}=\frac{1}{4}{a}_{1}\) , and \({\sum }_{\text{i}=1}^{\text{n}}{a}_{\text{i}}=\frac{525}{2},\) then \({\sum }_{\text{i}=1}^{17}{a}_{\text{i}}\) is equal to
[JEE Main 2026, 24 Jan (Shift 1)]
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Let \(\alpha\) and \(\beta\) be the roots of the equation \(p x^2+ q x- r =0\), where \(p \neq 0\). If \(p , q\) and \(r\) be the consecutive terms of a non constant G.P. and \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}\), then the value of \((\alpha-\beta)^2\) is :
[JEE Main 2024, 1 Feb (Shift 2)]
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The sum of the infinite series \(1+\frac{5}{6}+\frac{12}{{6}^{2}}+\frac{22}{{6}^{3}}+\frac{35}{{6}^{4}}+............\)is equal to:
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Let \(S_n\) denote the sum of the first \(n\) terms of an arithmetic progression. If \(S_{10}=390\) and the ratio of the tenth and the fifth terms is \(15: 7\), then \(S_{15}-S_5\) is equal to:EndFragment
[JEE Main 2024, 01 Feb (Shift 2)]
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Let \(S_n\) denote the sum of the first \(n\) terms of an arithmetic progression. If \(S_{10}=390\) and the ratio of the tenth and the fifth terms is \(15: 7\), then \(S_{15}-S_5\) is equal to:EndFragment
[JEE Main 2024, 01 Feb (Shift 2)]
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Let \({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3}\ldots\) be in an A.P. such that \(\sum _{\mathrm{k}=1}^{12}{\mathrm{a}}_{2\mathrm{k}-1}=-\frac{72}{5}{\mathrm{a}}_{1},{\mathrm{a}}_{1}\neq 0\). If \(\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{k}}=0\), then n is:
[JEE Main 2025, 2 Apr (Shift 1)]
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Let \(3, a , b , c\) be in A.P. and \(3, a -1, b +1, c +9\) be in G.P. Then, the arithmetic mean of \(a, b\) and \(c\) is :
[JEE Main 2024, 1 Feb (Shift 1)]
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Let \(
[JEE Main 2025, 28 Jan (Shift 1)]
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If the sum of the first \(20\) terms of the series \(\frac{4.1}{4+3.{1}^{2}+{1}^{4}}+\frac{4.2}{4+3.{2}^{2}+{2}^{4}}+\frac{4.3}{4+3.{3}^{2}+{3}^{4}}+\frac{4.4}{4+3.{4}^{2}+{4}^{4}}+\ldots\) is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to
[JEE Main 2025, 4 Apr (Shift 2)]
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\({a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{2024}\text{ are in A.P. }{a}_{1}+\left({a}_{5}+{a}_{10}+{a}_{15}+⋯+{a}_{2020}\right)+{a}_{2024}=2233,\\ then{a}_{1}+{a}_{2}+{a}_{3}+⋯+{a}_{2024}=\text{ ? }\)
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Sum of first three terms of an AP with integral common difference is 54 and sum of first twenty terms lies between 1600 to 1800 , find \({a}_{11}\)
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In an A.P., the sixth term \(a_{6}=2\). If the product \(a_{1} a_{4} a_{5}\) is the greatest, then the common difference of the A.P. is equal to
[JEE Main 2024, 29 Jan (Shift 1)]
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A software company sets up \(m\) number of computer systems to finish an assignment in \(17\) days. If \(4\) computer systems crashed on the start of the second day, \(4\) more computer systems crashed on the start of the third day and so on, then it took \(8\) more days to finish the assignment. The value of \(\mathrm{m}\) is equal to :
[JEE Main 2024, 6 Apr (Shift 2)]
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If the sum of the series\(\text{ }\frac{1}{1\cdot (1+d)}+\frac{1}{(1+d)(1+2d)}+\ldots +\frac{1}{(1+9d)(1+10d)}\text{ , }\)is equal to \(5\) then \(50d\) is equal to :
[JEE Main 2024, 9 Apr (Shift 1)]
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\(\text { If } s_n=\sum_{r=0}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64} \text { then find } \operatorname{Lim}_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{T_r}=\) (22 Jan, Shift I, Memory Based)
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Let \({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3}\ldots\) be in an A.P. such that \(\sum _{\mathrm{k}=1}^{12}{\mathrm{a}}_{2\mathrm{k}-1}=-\frac{72}{5}{\mathrm{a}}_{1},{\mathrm{a}}_{1}\neq 0\). If \(\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{k}}=0\), then \(n\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
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\(\sum _{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)\) is equal to:
[JEE Main 2026, 5 Apr (Shift 1)]
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Let \(a_1, a_2, a_3, a_4\) be an A.P. of four terms such that each term of the A.P. and its common difference \(l\) are integers. If \(a_1+a_2+a_3+ a_4=48\) and \(a_1 a_2 a_3 a_4+l^4=361\), then the largest term of the A.P. is equal to
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If \(7=5+\frac{1}{7}(5+\alpha )+\frac{1}{{7}^{2}}(5+2\alpha )\)\(+\frac{1}{{7}^{3}}(5+3\alpha )+\ldots \ldots \ldots ...\infty\)
then the value of \(\alpha\) is :
[JEE Main 2025, 24 Jan (Shift 2)]
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Let \({a}_{n}\) be the \({\mathrm{n}}^{\text{th }}\) term of an A. P. If \({\mathrm{S}}_{\mathrm{n}}={\mathrm{a}}_{1}+{\mathrm{a}}_{2}+{\mathrm{a}}_{3}+\ldots +{\mathrm{a}}_{\mathrm{n}}=700,{\mathrm{a}}_{6}=7\) and \({S}_{7}=7\), then \({a}_{n}\) is equal to :
[JEE Main 2025, 7 Apr (Shift 2)]
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Let \(f\) and \(g\) be functions satisfying \(f(x+y)=\)\(f(x)f(y),\text{ }f(1)=7\) and \(g(x+y)=g(xy)\), \(g(1)=1,\) for all \(x,\text{ }y\in N.\) If \(\sum _{x=1}^{n}\left(\frac{f(x)}{g(x)}\right)=19607,\) then \(n\) is equal to:
[JEE Main 2026, 22 Jan (Shift 2)]
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If \(\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}=\frac{(2\mathrm{n}-1)(2\mathrm{n}+1)(2\mathrm{n}+3)(2\mathrm{n}+5)}{64}\) then \(\lim _{n\to \infty }\sum _{r=1}^{n}\left(\frac{1}{{T}_{r}}\right)\) is equal to :
[JEE Main 2025, 22 Jan (Shift 1)]
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If \(7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\ldots \infty\) terms, then \(\alpha\) is equal to (24 Jan, Shift II, Memory Based)
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Let \({S}_{n}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots\)upto n terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2026{\mathrm{S}}_{2025}},\) then the absolute difference between \({20}^{\text{th }}\text{ and }{15}^{\text{th }}\) terms of the A.P. is
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\(1+3+{5}^{2}+7+{9}^{2}+\ldots\) up to \(40\) terms is equal to
[JEE Main 2025, 4 Apr (Shift 1)]
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Let the first three terms \(2, p\) and \(q\), with \(q\neq 2\), of a G.P. be respectively the \({7}^{\text{th }},{8}^{\text{th }}\) and \({13}^{\text{th }}\) terms of an A.P. If the \({5}^{\text{th }}\) term of the G.P. is the \({n}^{\text{th }}\) term of the A.P., then \(n\) is equal to :
[JEE Main 2024, 4 Apr (Shift 1)]
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The product of all solutions of the equation \({e}^{5{\left({\log }_{e}x\right)}^{2}+3}={x}^{8},x>0,\) is:
[JEE Main 2025, 22 Jan (Shift 1)]
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The sum of the infinite series \(1+\frac{5}{6}+\frac{12}{{6}^{2}}+\frac{22}{{6}^{3}}+\frac{35}{{6}^{4}}+............\)is equal to:
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\(\left(\frac{1}{3}+\frac{4}{7}\right)+\left(\frac{1}{{3}^{2}}+\frac{1}{3}\times \frac{4}{7}+\frac{{4}^{2}}{{7}^{2}}\right)+\)\(\left(\frac{1}{{3}^{3}}+\frac{1}{{3}^{2}}\times \frac{4}{7}+\frac{1}{3}\times \frac{{4}^{2}}{{7}^{2}}+\frac{{4}^{3}}{{7}^{3}}\right)+...\)upto infinite terms, is equal to
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The common difference of the A.P.: \({a}_{1},{a}_{2},\ldots ,{a}_{m}\) is \(13\) more than the common difference of the A.P.: \({b}_{1},{b}_{2},\ldots ,{b}_{n}\). If \({b}_{31}=−277,{b}_{43}=−385\) and \({a}_{78}=327\), then \({a}_{1}\) is equal to
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The number of terms of an A.P. is even; the sum of all the odd terms is \(24\) , the sum of all the even terms is \(30\) and the last term exceeds the first by \(\frac{21}{2}\). Then the number of terms which are integers in the A.P. is :
[JEE Main 2025, 2 Apr (Shift 2)]
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The value of \(\sum _{k=1}^{∞}{(−1)}^{k+1}\left(\frac{k\left(k+1\right)}{k!}\right)\) is
[JEE Main 2026, 28 Jan (Shift 1)]
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Consider an \(A.P.\) of positive integers, whose sum of the first three terms is \(54\) and the sum of the first twenty terms lies between \(1600\) and \(1800.\) Then its \(11^{th}\) term is :
[JEE Main 2025, 29 Jan (Shift 1)]
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If each term of a geometric progression \(a_1, a_2, a_3, \ldots\) with \(a_1=\frac{1}{8}\) and \(a_2 \neq a_1\), is the arithmetic mean of the next two terms and \(S_n=a_1+a_2+\ldots . .+a_n\), then \(S_{20}-S_{18}\) is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
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Let \(\alpha =3+4+8+9+13+14+\ldots\) upto \(40\) terms. If \((\tan \beta {)}^{\frac{\alpha }{1020}}\) is a root of the equation \({x}^{2}+x-2=0,\beta \in \left(0,\frac{\pi }{2}\right)\), then \({\sin }^{2}\beta +3{\cos }^{2}\beta\) is equal to:
[JEE Main 2026, 8 Apr (Shift 2)]
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If the sum of the first four terms of an A.P. is \(6\) and the sum of its first six terms is \(4\), then the sum of its first twelve terms is
[JEE Main 2026, 22 Jan (Shift 1)]
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The sum of the series \(\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots\) up to \(10\) terms is
[JEE Main 2024, 31 Jan (Shift 1)]
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\(\text{ If }\log y=x\log \frac{2}{5},x\in \mathrm{N}\cup {0}\text{.}\\ \text{Then sum of all values of }y\text{ equals to }\)
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The sum \(1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots\) up to \(\infty\) terms, is equal to
[JEE Main 2025, 3 Apr (Shift 2)]
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If for an arithmetic progression, if first term is 3 and sum of first four terms is equal to \(\frac{1}{5}\) of the sum of next four terms, then the sum of first 20 terms is
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If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
[JEE Main 2025, 23 Jan (Shift 1)]
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\(\lim _{n\to \infty }\sum _{k=1}^{n}\frac{{k}^{3}+6{k}^{2}+11k+5}{(k+3)!}=\)
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If \(\alpha\) and \(\beta\) are the roots of the equation \(3 x^2-p x+q=0\) Let an A.P whose common diff. is \(\frac{3}{2}\) and \(10^{\text {th }}\) term is \(\alpha\) and \(11^{\text {th }}\) term is \(\beta\) and \(S_{11}=88\). Find the value of \(q-2 p\).
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If \(\frac{1}{{1}^{4}}+\frac{1}{{2}^{4}}+\frac{1}{{3}^{4}}+\ldots ..\infty =\frac{{\pi }^{4}}{90}\), \(\frac{1}{{1}^{4}}+\frac{1}{{3}^{4}}+\frac{1}{{5}^{4}}+\ldots ..\infty =\alpha ,\) \(\frac{1}{{2}^{4}}+\frac{1}{{4}^{4}}+\frac{1}{{6}^{4}}+\ldots ..\infty =\beta ,\) then \(\frac{\alpha }{\beta }\) is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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Let \(a\) and \(b\) be two distinct positive real numbers. Let \(11^{\text {th }}\) term of a GP, whose first term is \(a\) and third term is \(b\), is equal to \(p^{\text {th }}\) term of another GP, whose first term is \(a\) and fifth term is \(b\). Then \(p\) is equal to
[JEE Main 2024, 30 Jan (Shift 2)]
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Let \(a\) and \(b\) be two distinct positive real numbers. Let \(11^{\text {th }}\) term of a GP, whose first term is \(a\) and third term is \(b\), is equal to \(p^{\text {th }}\) term of another GP, whose first term is \(a\) and fifth term is \(b\). Then \(p\) is equal to
[JEE Main 2024, 30 Jan (Shift 2)]
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\(\lim _{n\to \infty }\sum _{k=1}^{n}\frac{{k}^{3}+6{k}^{2}+11k+5}{(k+3)!}=\)
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If \(\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}=\frac{(2\mathrm{n}-1)(2\mathrm{n}+1)(2\mathrm{n}+3)(2\mathrm{n}+5)}{64}\), then \(\lim _{n\to \infty }\sum _{r=1}^{n}\left(\frac{1}{{T}_{r}}\right)\) is equal to :
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If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
[JEE Main 2025, 23 Jan (Shift 1)]
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Let \(\alpha =\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \infty\) and \(\beta =\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots \infty\). Then the value of \((0.2{)}^{{\log }_{\sqrt{5}}(\alpha )}+(0.04{)}^{{\log }_{5}(\beta )}\) is equal to:
[JEE Main 2026, 4 Apr (Shift 2)]
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If the sum of the first \(20\) terms of the series \(\frac{4.1}{4+3.{1}^{2}+{1}^{4}}+\frac{4.2}{4+3.{2}^{2}+{2}^{4}}+\frac{4.3}{4+3.{3}^{2}+{3}^{4}}+\frac{4.4}{4+3.{4}^{2}+{4}^{4}}+\ldots\) is \(\frac{m}{n}\), where m and n are coprime, then m + n is equal to :-
[JEE Main 2025, 4 Apr (Shift 1)]
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Sum of first three terms of an AP with integral common difference is 54 and sum of first twenty terms lies between 1600 to 1800 , find \({a}_{11}\)
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Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m,\) \({\mathrm{T}}_{\mathrm{m}}=\frac{1}{25},{\mathrm{T}}_{25}=\frac{1}{20}\text{ and }20\sum _{\mathrm{r}=1}^{25}{\mathrm{T}}_{\mathrm{r}}=13,\) then \(5\mathrm{m}\sum _{\mathrm{r}=\mathrm{m}}^{2\mathrm{m}}{\mathrm{T}}_{\mathrm{r}}\) is equal to:
[JEE Main 2025, 28 Jan (Shift 1)]
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The number of common terms in the progressions \(4,9,14,19, \ldots\), up to \(25^{\text {th }}\) term and \(3,6,9,12, \ldots \ldots\), up to \(37^{\text {th }}\) term is :
[JEE Main 2024, 27 Jan (Shift 1)]
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The number of common terms in the progressions \(4,9,14,19, \ldots\), up to \(25^{\text {th }}\) term and \(3,6,9,12, \ldots \ldots\), up to \(37^{\text {th }}\) term is :
[JEE Main 2024, 27 Jan (Shift 1)]
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If \(8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}\left(3+p^2\right)+\ldots \infty\) then the value of \(p\) is (22 Jan, Shift I, Memory Based)
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For \(x⩾0\), the least value of \(K\) for which \({4}^{1+x}+{4}^{1-x},\frac{K}{2},{16}^{x}+{16}^{-x}\text{ }\) are three consecutive terms of an A.P., is equal to:
[JEE Main 2024, 5 Apr (Shift 2)]
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Consider two sets \(A\)and \(B\), each containing three numbers in A.P. Let the sum and the product of the elements of \(A\) be \(36\)and \(p\) respectively and the sum and the product of the elements of \(B\)be \(36\) and \(q\)respectively. Let \(d\) and \(D\)be the common differences of AP's in \(A\) and \(B\)respectively such that \(\mathrm{D}=\mathrm{d}+3,\mathrm{d}>0\). If \(\frac{\mathrm{p}+\mathrm{q}}{\mathrm{p}-\mathrm{q}}=\frac{19}{5}\), then \(\mathrm{p}-\mathrm{q}\) is equal to
[JEE Main 2025, 4 Apr (Shift 1)]
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\(\text{ If }\log y=x\log \frac{2}{5},x\in \mathrm{N}\cup {0}\text{.}\\ \text{Then sum of all values of }y\text{ equals to }\)
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Let \(
[JEE Main 2025, 28 Jan (Shift 1)]
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If \(\sum _{\mathrm{r}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{r}}=\frac{(2\mathrm{n}-1)(2\mathrm{n}+1)(2\mathrm{n}+3)(2\mathrm{n}+5)+15}{64}\) then \(\lim _{n\to \infty }\sum _{r=1}^{n}\left(\frac{1}{{T}_{r}}\right)\) is equal to :
[JEE Main 2025, 22 Jan (Shift 1)]
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Let \({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3}\ldots .\)be a G.P. of increasing positive terms. If \({a}_{1}{a}_{5}=28\text{ and }{a}_{2}+{a}_{4}=29\text{, the }{a}_{6}\) is equal to
[JEE Main 2025, 22 Jan (Shift 1)]
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Let \(A_1, A_2, A_3, \ldots \ldots \ldots \ldots . ., A_{39}\) be \(39\) arithmetic means between the numbers \(59\) and \(159\) . Then the mean of \(\mathrm{A}_{25}, \mathrm{~A}_{28}, \mathrm{~A}_{31}\), and \(\mathrm{A}_{36}\) is equal to:
[JEE Main 2026, 5 Apr (Shift 2)]
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If \(\frac{1}{{1}^{4}}+\frac{1}{{2}^{4}}+\frac{1}{{3}^{4}}+\ldots ..\infty =\frac{{\pi }^{4}}{90}\), \(\frac{1}{{1}^{4}}+\frac{1}{{3}^{4}}+\frac{1}{{5}^{4}}+\ldots ..\infty =\alpha ,\) \(\frac{1}{{2}^{4}}+\frac{1}{{4}^{4}}+\frac{1}{{6}^{4}}+\ldots ..\infty =\beta ,\) then \(\frac{\alpha }{\beta }\) is equal to
[JEE Main 2025, 8 Apr (Shift 1)]
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If \(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots +\frac{1}{\sqrt{99}+\sqrt{100}}=m\) and \(\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\ldots +\frac{1}{99\cdot 100}=n\), then the point \((m, n)\) lies on the line
[JEE Main 2024, 5 Apr (Shift 1)]
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Let \(\sum _{k=1}^{n}{a}_{k}=\alpha {n}^{2}+\beta n\). If \({a}_{10}=59\) and \({a}_{6}=7{a}_{1}\), then \(\alpha+\beta\) is equal to
[JEE Main 2026, 23 Jan (Shift 2)]
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Suppose that the number of terms in an A.P. is \(2 k\), \(\mathrm{k} \in \mathrm{N}\). If the sum of all odd terms of the A.P. is \(40,\) the sum of all even terms is \(55\) and the last term of the A.P. exceeds the first term by \(27,\) then \(k\) is equal to
[JEE Main 2025, 22 Jan (Shift 2)]
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Let \({x}_{1},{x}_{2},{x}_{3},{x}_{4}\) be in a geometric progression. If \(2,7,9,5\)are subtracted respectively from \({x}_{1},{x}_{2},{x}_{3}\), \({\mathrm{x}}_{4}\) then the resulting numbers are in an arithmetic progression. Then the value of \(\frac{1}{24}\left({\mathrm{x}}_{1}{\mathrm{x}}_{2}{\mathrm{x}}_{3}{\mathrm{x}}_{4}\right)\) is :
[JEE Main 2025, 7 Apr (Shift 1)]
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Let \({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3}\ldots .\)be a G.P. of increasing positive terms. If \({a}_{1}{a}_{5}=28\text{ and }{a}_{2}+{a}_{4}=29\text{, the }{a}_{6}\) is equal to
[JEE Main 2025, 22 Jan (Shift 1)]
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Let the range of the function \(f\left(x\right)=\frac{1}{2+\sin 3x+\cos 3x},x\in R\) be \([a,b]\). If \(\alpha\) and \(\beta\) are respectively the A.M. and the G.M. of a and b, then \(\frac{\alpha }{\beta }\) is equal to
[JEE Main 2024, 9 Apr (Shift 2)]
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The value of \(\frac{1\times {2}^{2}+2\times {3}^{2}+\ldots +100\times (101{)}^{2}}{{1}^{2}\times 2+{2}^{2}\times 3+\ldots +{100}^{2}\times 101}\)
[JEE Main 2024, 4 Apr (Shift 2)]
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The product of all solutions of the equation \({e}^{5{\left({\log }_{e}x\right)}^{2}+3}={x}^{8},x>0,\) is:
[JEE Main 2025, 22 Jan (Shift 1)]
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The value of \(\sum _{k=1}^{∞}{(−1)}^{k+1}\left(\frac{k\left(k+1\right)}{k!}\right)\) is
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If \(2{a}_{n+2}=5{a}_{n+1}-3{a}_{n}\), where \(n=0,1,2,\ldots\). If \({a}_{0}=\)3 and \({a}_{1}=4\), then the value of \(\sum _{k=1}^{100}{a}_{k}\) is equal to (28 Jan, Shift I, Memory Based)
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For positive integers \(n\), if \(4 a_n=\left(n^2+5 n+6\right)\) and \(S_n=\sum_{k=1}^n\left(\frac{1}{a_k}\right)\), then the value of \(507 \mathrm{~S}_{2025}\) is:
[JEE Main 2025, 28 Jan (Shift 2)]
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The sum \(1+3+11+25+45+71+.\). upto \(20\) terms, is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
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Let \({\mathrm{a}}_{1},{\mathrm{a}}_{2},{\mathrm{a}}_{3},\ldots\) be a G. P. of increasing positive numbers. If \({\mathrm{a}}_{3}{\mathrm{a}}_{5}=729\) and \({\mathrm{a}}_{2}+{\mathrm{a}}_{4}=\frac{111}{4}\), then \(24\left({a}_{1}+{a}_{2}+{a}_{3}\right)\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
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Let \(\sum _{k=1}^{n}{a}_{k}=\alpha {n}^{2}+\beta n\). If \({a}_{10}=59\) and \({a}_{6}=7{a}_{1}\), then \(\alpha+\beta\) is equal to
[JEE Main 2026, 23 Jan (Shift 2)]
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Let \({x}_{1},{x}_{2},{x}_{3},{x}_{4}\) be in a geometric progression. If \(2,7,9,5\) are subtracted respectively from \({x}_{1},{x}_{2},{x}_{3},{x}_{4}\) then the resulting numbers are in an arithmetic progression. Then the value of \(\frac{1}{24}\left({x}_{1},{x}_{2},{x}_{3},{x}_{4}\right)\) is :
[JEE Main 2025, 7 Apr (Shift 1)]
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The interior angle of a polygon with n side are in A.P with common difference of \(6^\circ\). If the biggest interior angle of polygon is \(219^\circ\) then n is equal to
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If \(7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\ldots \infty\) terms, then \(\alpha\) is equal to (24 Jan, Shift II, Memory Based)
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If the sum of the first 10 terms of the series
\(\frac{1}{1+1^4 \times 4}+\frac{2}{1+2^4 \times 4}+\frac{3}{1+3^4 \times 4}+\frac{4}{1+4^4 \times 4}+\ldots \ldots\)
is \(\frac{m}{n}, \operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to:
[JEE Main 2026, 5 Apr (Shift 2)]
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Suppose that the number of terms in an A.P. is \(2 k\), \(\mathrm{k} \in \mathrm{N}\). If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then \(k\) is equal to
[JEE Main 2025, 22 Jan (Shift 2)]
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In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \(\frac{70}{3}\) and the product of the third and fifth terms is 49 . Then the sum of the \({4}^{\text{th }},{6}^{\text{th }}\)and \({8}^{\text{th }}\) terms is equal to :
[JEE Main 2024, 8 Apr (Shift 2)]
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Let \({S}_{n}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots\)up to n terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2026{\mathrm{S}}_{2025}},\) then the absolute difference between \({20}^{\text{th }}\text{ and }{15}^{\text{th }}\) terms of the A.P. is
[JEE Main 2025, 24 Jan (Shift 1)]
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Let three real numbers \(a,b,c\) be in arithmetic progression and \(a+1,b,c+3\) be in geometric progression. If \(a>10\) and the arithmetic mean of \(a,b\) and \(c\) is \(8\) , then the cube of the geometric mean of \(a, b\) and \(c\) is
[JEE Main 2024, 4 Apr (Shift 2)]
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Let \({a}_{n}\) be the \({\mathrm{n}}^{\text{th }}\) term of an A. P. If \({\mathrm{S}}_{\mathrm{n}}={\mathrm{a}}_{1}+{\mathrm{a}}_{2}+{\mathrm{a}}_{3}+\ldots +{\mathrm{a}}_{\mathrm{n}}=700,{\mathrm{a}}_{6}=7\) and \({S}_{7}=7\), then \({a}_{n}\) is equal to :
[JEE Main 2025, 7 Apr (Shift 2)]
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The sum \(1+3+11+25+45+71+...\) up to \(20\) terms, is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
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\({a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{2024}\text{ are in A.P. }{a}_{1}+\left({a}_{5}+{a}_{10}+{a}_{15}+⋯+{a}_{2020}\right)+{a}_{2024}=2233,\\ then{a}_{1}+{a}_{2}+{a}_{3}+⋯+{a}_{2024}=\text{ ? }\)
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Consider two sets \(A\)and \(B\), each containing three numbers in A.P. Let the sum and the product of the elements of \(A\) be \(36\)and \(p\) respectively and the sum and the product of the elements of \(B\)be \(36\) and \(q\)respectively. Let \(d\) and \(D\)be the common differences of AP's in \(A\) and \(B\)respectively such that \(\mathrm{D}=\mathrm{d}+3,\mathrm{d}>0\). If \(\frac{\mathrm{p}+\mathrm{q}}{\mathrm{p}-\mathrm{q}}=\frac{19}{5}\), then \(\mathrm{p}-\mathrm{q}\) is equal to
[JEE Main 2025, 4 Apr (Shift 1)]
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Number of terms in an arithmetic progression is \(2 n\). Sum of terms occurring at even places is 40 and sum of terms occurring at odd places is 55 . If the first term exceeds the last term by 27 , then \(n\) equals to (22 Jan, Shift II, Memory Based)
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For positive integer \(n,4{a}_{n}={n}^{2}+5n+6\) and \({S}_{n}=\sum _{k=1}^{n}\left(\frac{1}{{a}_{k}}\right)\), then the value of \(507{S}_{2025}\)
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The number of terms of an A.P. is even; the sum of all the odd terms is \(24\) , the sum of all the even terms is \(30\) and the last term exceeds the first by \(\frac{21}{2}\). Then the number of terms which are integers in the A.P. is :
[JEE Main 2025, 2 Apr (Shift 2)]
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For positive integers \(n\), if \(4 a_n=\left(n^2+5 n+6\right)\) and \(S_n=\sum_{k=1}^n\left(\frac{1}{a_k}\right)\), then the value of \(507 \mathrm{~S}_{2025}\) is:
[JEE Main 2025, 28 Jan (Shift 2)]
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If the sum of the first four terms of an A.P. is \(6\) and the sum of its first six terms is \(4\), then the sum of its first twelve terms is
[JEE Main 2026, 22 Jan (Shift 1)]
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Let \(729,81,9,1, \ldots\) be a sequence and \(P_n\) denote the product of the first \(n\) terms of this sequence.
If \(2\sum _{n=1}^{40}{\left({P}_{n}\right)}^{\frac{1}{n}}=\frac{{3}^{\alpha }−1}{{3}^{\beta }}\) and \(\operatorname{gcd}(\alpha, \beta)=1\), then \(\alpha+\beta\) is equal to
[JEE Main 2026, 24 Jan (Shift 1)]
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Consider an \(A.P.\) of positive integers, whose sum of the first three terms is \(54\) and the sum of the first twenty terms lies between \(1600\) and \(1800.\) Then its \(11^{th}\) term is :
[JEE Main 2025, 29 Jan (Shift 1)]
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The \(20^{\text {th }}\) term from the end of the progression \(20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}\) is :
[JEE Main 2024, 27 Jan (Shift 2)]
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The \(20^{\text {th }}\) term from the end of the progression \(20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}\) is :
[JEE Main 2024, 27 Jan (Shift 2)]
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In an arithmetic progression, if \(\mathrm{S}_{40}=1030\) and \(\mathrm{S}_{12}=57\), then \(\mathrm{S}_{30}-\mathrm{S}_{10}\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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The sum of the first ten terms of an A.P. is \(160\) and the sum of the first two terms of a G.P. is \(8\) . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:
[JEE Main 2026, 6 Apr (Shift 1)]
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If the sum of an infinite GP \(a, a r, a r^2, a r^3, \ldots \ldots\) is 15 and the sum of the squares of its each term is 150 , then the sum of \(a r^2, a r^4, a r^6, \ldots \ldots\) is:
[JEE Main 2021, 26 Aug (Shift 1)]
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Let a, b, c and d be positive real numbers such that \(a+b +c+d=11.\) If the maximum value of \({a}^{5}{b}^{3}{c}^{2}d\) is \(3750\beta\), then the value of \(\beta\) is
[JEE Main 2023, 11 Apr (Shift 2)]
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Let \(0 [JEE Main 2023, 8 Apr (Shift 2)]
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If the sum of the second, third and fourth terms of a positive term G.P is 3 and the sum of its sixth , seventh and eighth terms is 243 , then the sum of the first 50 terms of this G.P. is
[JEE Main 2020, 5 Sep (Shift 2)]
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Three numbers are in an increasing geometric progression with common ratio \(r\). If the middle number is doubled, then the new numbers are in arithmetic progression with common difference \(d\). If the fourth term of GP is \(3r^{2}\) then \(r^{2} - d\) is equal to ?
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If sum of the first 21 terms of the series \({\log }_{{9}^{1/2}}\text{x}+{\log }_{{9}^{1/3}}\text{x}+{\log }_{{9}^{1/4}}\text{x}+\ldots \ldots ,\) where \(x>0\) is 504 , then \(x\) is equal to:
[JEE Main 2021, 20 Jul (Shift 2)]
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The sum \( \sum_{\mathrm{k}=1}^{20}(1+2+3+\ldots \ldots .+\mathrm{k}) \) is
[JEE Main 2020, 8 Jan (Shift 1)]
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Let \(x_1, x_2 \ldots, x_{100}\) be in an arithmetic progression, with \(x_1=2\) and their mean equal to 200. If \(y_{ i }=i\left(x_{ i }-i\right), 1 \leq i \leq\) 100 , then the mean of \(y_1, y_2, \ldots, y_{100}\) is
[JEE Main 2023, 11 Apr (Shift 1)]
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Let \(a_n\) be the \(n^{\text {th }}\) term of the series \(5+8+14+23+35+\) \(50+\ldots\) and \(S_n=\sum_{k=1}^n a_k\). Then \(S_{30}-a_{40}\) equal to
[JEE Main 2023, 8 Apr (Shift 2)]
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A student read common difference of an A. P. as \(-2\) instead of 2 and got the sum of first 5 terms as \(-5\). Actual sum of first five terms is
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Let \(S_K=\frac{1+2+\ldots+K}{K}\) and \(\sum_{j=1}^n S_j^2=\frac{n}{A}\left(B n^2+C n+D\right)\) where \(A, B, C, D \in N\) and \(A\) has least value. Then
[JEE Main 2023, 8 Apr (Shift 1)]
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The \(\operatorname{sum} \sum_{n=1}^{\infty} \frac{2 n^2+3 n+4}{(2 n) !}\) is equal to:
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Let \({S}_{n}=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\ldots +(n-1)\cdot 1,n\geq 4\). The sum \(\sum _{n=4}^{\infty }\left(\frac{2{S}_{n}}{n!}-\frac{1}{(n-2)!}\right)\) is equal to
[JEE Main 2021, 1 Sep (Shift 2)]
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Let \(a_1=b_1=1\) and \(a_n=a_{n-1}+(n-1), b_n=b_{n-1}+\) \(a_{n-1}, \forall n \geq 2\). If \(S=\sum_{n=1}^{10} \frac{b_n}{2^n}\) and \(T=\sum_{n=1}^8 \frac{n}{2^{n-1}}\), then \(2^7(2 S-T)\) is equal to ______ .
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Let \(S_n\) denote the sum of the first \(n\)-terms of an arithmetic progression. If \(S_{10}=530, S_5=140\), then \(S_{20}-S_6\) is equal to :
[JEE Main 2021, 22 Jul (Shift 2)]
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The greatest positive integer \( \mathrm{k} \), for which \( 49^{\mathrm{k}}+1 \) is a factor of the sum \( 49^{125}+49^{124}+\ldots 49^{2}+49+1 \), is :
[JEE Main 2020, 7 Jan (Shift 1)]
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If the sum of an infinite GP \(a, a r, a r^2, a r^3, \ldots . .\). is 15 and the sum of the squares of its each term is 150 , then the sum of \(a r^2, a r^4, a r^6, \ldots . .\). is:
[JEE Main 2021, 26 Aug (Shift 1)]
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Let a1, a2, a3, … be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the product of its 3rd and 5th terms be \(\frac{1}{9}\) . Then 6(a2 + a4)(a4 + a6) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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If \(a, c, b\) are in G.P., then the area of the triangle formed by the lines \(a x+b y+c=0\) with the coordinates axes is equal to
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If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is
[JEE Main 2023, 31 Jan (Shift 1)]
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Let \(a, b, c,>1, a^3, b^3\) and \(c^3\) be in A.P., and \(\log _a b, \log _c a\) and \(\log _b c\) be in G.P. If the sum of first 20 terms of an A.P., whose first term is \(\frac{a+4 b+c}{3}\) and the common difference is \(\frac{a-8 b+c}{10}\) is -444 , then \(a b c\) is equal to
[JEE Main 2023, 30 Jan (Shift 2)]
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Let \(x_1, x_2 \ldots, x_{100}\) be in an arithmetic progression, with \(x_1=2\) and their mean equal to 200 . If \(y_{ i }=i\left(x_{ i }-i\right), 1 \leq i \leq\) 100 , then the mean of \(y_1, y_2, \ldots, y_{100}\) is
[JEE Main 2023, 11 Apr (Shift 1)]
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If the first term of an A.P. is \(3\) and the sum of its first \(25\) terms is equal to the sum of its next \(15\) terms, then the common difference of this A.P. is:
[JEE Main 2020, 3 Sep (Shift 1)]
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Let \(n\) denote the number of solutions of the equation \(z^2+3 \bar{z}=0\), where \(z\) is a complex number. Then the value of \(\sum^{\infty} \frac{1}{n^k}\) is equal to:
[JEE Main 2021, 22 Jul (Shift 2)]
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If sum of the first \(21\) terms of the series \(\log _{9^{\frac{1}{2}}} x+\log _{9^{\frac{1}{3}}} x+\) \(\log _{9^{\frac{1}{4}}} x+\ldots\), where \(x>0\) is \(504,\) then \(x\) is equal to
[JEE Main 2021, 20 Jul (Shift 2)]
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If \(0 [JEE Main 2021, 27 Aug (Shift 2)]
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Let \(a_n\) be a sequence such that \(a_1+a_2+\ldots+a_n\) \(=\frac{n^2+3 n}{(n+1)(n+2)}\).
If \(28 \sum_{k=1}^{10} \frac{1}{a_k}=p_1 p_2 p_3 \ldots p_m\), where \(p_1, p_2, \ldots, p_m\) are the first \(m\) prime numbers, then \(m\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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If \(a_n=\frac{-2}{4 n^2-16 n+15}\), then \(a_1+a_2+\ldots+a_{25}\) is equal to:
[JEE Main 2023, 30 Jan (Shift 1)]
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If the arithmetic mean and geometric mean of the \(p^{\text {th }}\) and \(q^{\text {th }}\) terms of the sequence \(-16,8,-4,2 \ldots\) satisfy the equation \(4 x^2-9 x+5=0\), then \(p+q\) is equal to......
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If \(a,b,c\) are positive numbers, then least value of \((a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\) is
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If the sum of an infinite GP \(\mathrm{a}, \mathrm{ar}, \mathrm{ar}^2, \mathrm{ar}^3, \ldots\) is 15 and the sum of the squares of its each term is 150 , then the sum of \(\operatorname{ar}^2,\operatorname{ar}^4, \operatorname{ar}^6, \ldots\) is :
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The sum of the first 20 terms of the series \(5+11+19+29\) \(+41+\ldots\) is
[JEE Main 2023, 6 Apr (Shift 1)]
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If \(S_n=4+11+21+34+50+\ldots\). to \(n\) terms, then \(\frac{1}{60}\left(S_{29}-S_9\right)\)
[JEE Main 2023, 10 Apr (Shift 2)]
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Let the first term \(a\) and the common ratio \(r\) of a geometric progression be positive integers. If the sum of its squares of first three terms is 33033 , then the sum of these three terms is equal to
[JEE Main 2023, 10 Apr (Shift 1)]
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The sum to 10 terms of the series \(\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\ldots\) is
[JEE Main 2023, 1 Feb (Shift 1)]
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\(\frac{1}{3^2-1}+\frac{1}{5^2-1}+\frac{1}{7^2-1}+\ldots \ldots +\frac{1}{(201)^2-1}\) is equal to :
[JEE Main 2021, 18 Mar (Shift 1)]
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Let \(S_n(x)=\log _{a^{\frac{1}{2}}} x+\log _{a^{1 / 3}} x+\log _{a^{\frac{1}{6}}} x+\) \(\log _{a^{1 / 11}} x+\log _{a^{1 / 18}} x+\log _{a^{1 / 27}} x+\ldots\) up to \(n\)-terms. Where \(a>1\). If \(S _{24}(x)=1093\) and \(S _{12}(2x)=265\), then the value of \(a\) is equal to ___
[JEE Main 2021, 16 Mar (Shift 2)]
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The minimum value of \(2^{\sin x}+2^{\cos x}\) is:
[JEE Main 2020, 4 Sep (Shift 2)]
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The sum of the series \(\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\ldots+\frac{2^{100}}{x^{2^{100}}+1}\) when \(x=2\) is:
[JEE Main 2021, 26 Aug (Shift 1)]
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For three positive integers \(p, q, r, x^{p q^2}=y^{q r}=z^{p^{2}r}\) and \(r=p q+1\) such that \(3,3 \log _y x, 3 \log _z y, 7 \log _x z\) are in A.P. with common difference \(\frac{1}{2}\). Then \(r-p-q\) is equal to
[JEE Main 2023, 24 Jan (Shift 1)]
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Let \(\left\{a_k\right\}\) and \(\left\{b_k\right\}, k \in N\), be two G.P.s with common ratio \(r_1\) and \(r_2\) respectively such that \(a_1=b_1=4\) and \(r_1
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Let \(a_1, a_2, a_3, \ldots\) be a G.P. of increasing positive numbers. Let the sum of its \(6^{\text {th }}\) and \(8^{\text {th }}\) terms be 2 and the product of its \(3^{\text {rd }}\) and \(5^{\text {th }}\) terms be \(\frac{1}{9}\). Then \(6\left(a_2+a_4\right)\left(a_4+a_6\right)\) is equal to
[JEE Main 2023, 13 Apr (Shift 2)]
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Let \(a, b, c, d\) and \(p\) be any non zero distinct real numbers such that \(\left(a^2+b^2+c^2\right) p^2-2(a b+b c+c d) p+\left(b^2+c^2+d^2\right)=0\). Then :
[JEE Main 2020, 6 Sep (Shift 1)]
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Let \( \mathrm{S} \) be the sum of the first 9 terms of the series : \( \{x+k a\}+\left\{x^{2}+(k+2) a\right\}+\left\{x^{3}+(k+4) a\right\}+\left\{x^{4}+\right. \) \( (\mathrm{k}+6) \mathrm{a}\}+\ldots \) where \( \mathrm{a} \neq 0 \) and \( \mathrm{x} \neq 1 \). If \( S=\frac{x^{10}-x+45 a(x-1)}{x-1} \), then \( k \) is equal to
[JEE Main 2020, 2 Sep (Shift 2)]
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Let \({S}_{n}=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\ldots +(n-1)\cdot 1,n\geq 4\). The sum \(\sum _{n=4}^{\infty }\left(\frac{2{S}_{n}}{n!}-\frac{1}{(n-2)!}\right)\) is equal
[JEE Main 2021, 1 Sep (Shift 2)]
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If the sum of a certain number of terms of the A.P. \(\ 25,22,19, \ldots \) is 116 then the last term is
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If \( \tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right) \) are in A.P. and \( \tan \left(\frac{\pi}{9}\right) , y, \tan \left(\frac{5 \pi}{18}\right) \) are also in A.P. Then, \( |x-2 y|= \)
[JEE Main 2021, 27 Jul (Shift 2)]
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A student read common difference of an A. P. as -2 instead of 2 and got the sum of first 5 terms as -5 . Actual sum of first five terms is
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The sum of 10 terms of the series \(\frac{3}{1^{2} \times 2^{2}} + \frac{5}{2^{2} \times 3^{2}} + \frac{7}{3^{2} \times 4^{2}} + ......\) is?
[JEE Main 2021, 31 Aug (Shift 1)]
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Consider an infinite geometric series with first term a and common ratio \(\ r \). If its sum is 4 and the second term is \(\ \frac{3}{4} \), then :
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The product \( 2^{\frac{1}{4}}, 4^{\frac{1}{16}}, 8^{\frac{1}{48}}, 16^{\frac{1}{128}} \ldots \ldots \) to \( \infty \) is equal to
[JEE Main 2020, 9 Jan (Shift 1)]
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Let \(s _1, s _2, s _3, \ldots . ., s _{10}\) respectively be the sum to 12 terms of 10 A.P.s whose first terms are 1, 2, 3, ....., 10 and the common differences are \(1,3,5, \ldots . ., 19\) respectively. Then \(\sum_{i=10}^{10} s _i\) is equal to
[JEE Main 2023, 13 Apr (Shift 1)]
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The sum of 10 terms of the series
\(\frac{3}{1^2 \times 2^2}+\frac{5}{2^2 \times 3^2}+\frac{7}{3^2 \times 4^2}+\ldots\) is
[JEE Main 2021, 31 Aug (Shift 1)]
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There are four distinct numbers of which the first three are in G.P. and the last three are in A.P., whose common difference is 6 . If the first and the last numbers are equal then two other numbers are
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If \(a>0, b>0, c>0\) and \(a, b, c\) are distinct, then \((a+b) (b+c)(c+a)\) is greater than
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If \( \log _{3} 2, \log _{3}\left(2^{x}-5\right), \log _{3}\left(2^{x}-\frac{7} { 2}\right) \) are in A.P., then \( x \) is equal to
[JEE Main 2021, 27 Jul (Shift 1)]
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The sum of the first three terms of a G.P. is \(S\) and their product is 27 . Then all such \(S\) lie in:
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If \(0 [JEE Main 2021, 27 Aug (Shift 1)]
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Let \( \mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} \) be such that for all \( \mathrm{x} \in \mathrm{R}\left(2^{1+\mathrm{x}}+2^{1-\mathrm{x}}\right), \mathrm{f}(\mathrm{x}) \) and \( \left(3^{x}+3^{-x}\right) \) are in A.P., then the minimum value of \( f(x) \) is
[JEE Main 2020, 8 Jan (Shift 2)]
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Let \(a_1, a_2, \ldots \ldots, a_{21}\) be an A.P. such that \(\sum_{n=1}^{20} \frac{1}{a_n a_{n+1}}=\frac{4}{9}\). If the sum of this A.P. is 189, then \(a_6 a_{16}\) is equal to:
[JEE Main 2021, 1 Sep (Shift 2)]
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Let \(a_{1},a_{2},a_{3},.....\) be an A.P. If \(\frac{a_{1} + a_{2} + .... + a_{10}}{a_{1} + a_{2} + .... + a_{p}} = \frac{100}{p^{2}},p \neq 10\), then \(\frac{a_{11}}{a_{10}}\) is equal to:
[JEE Main 2021, 31 Aug (Shift 2)]
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Let \(a_1, a_2, \ldots a_n\) be a given A.P. whose common difference is an integer and \(S_n=a_1+a_2+\ldots+a_n\). If \(a_1=1\), \(a_n=300\) and \(15 \leq n \leq 50\), then the ordered pair \(\left(S_{n-4}, a_{n-4}\right)\) is equal to
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If for \(x, y \in R, x>0, y=\log _{10} x+\log _{10} x^{1 / 3}+\log _{10} x^{1 / 9}+\ldots\) upto \(\infty\) terms and \(\frac{2+4+6+\ldots+2 y}{3+6+9+\ldots+3 y}=\frac{4}{\log _{10} x}\), then the ordered pair \((x, y)\) is equal to.
[JEE Main 2021, 27 Aug (Shift 1)]
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If the arithmetic mean and geometric mean of the \(p^{\text {th }}\) and \(q^{\text {th }}\) terms of the sequence \(-16,8,-4,2 \ldots\) satisfy the equation \(4 x^2-9 x+5=0\), then \(p+q\) is equal to......
[JEE Main 2021, 26 Feb (Shift 2)]
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Let \(A_1\) and \(A_2\) be two arithmetic means and \(G_1, G_2, G_3\) be three geometric means of two distinct positive numbers. The \(G_1{ }^4+G_2{ }^4+G_3{ }^4+G_1{ }^2 G_3{ }^2\) is equal to
[JEE Main 2023, 15 Apr (Shift 1)]
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The parabolas : \(a x^2+2 b x+c y=0\) and \(d x^2+2 e x+f y=0\) intersect on the line \(y=1\). If \(a, b, c, d, e, f\) are positive real numbers and \(a, b, c\) are in G.P., then
[JEE Main 2023, 30 Jan (Shift 2)]
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Let \(a_1, a_2, a_3, \ldots .\). be an A.P. If \(a_7=3\), the product \(a_1 a_4\) is minimum and the sum of its first \(n\) terms is zero, then \(n !-4 a_{n(n+2)}\) is equal to:
[JEE Main 2023, 31 Jan (Shift 2)]
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Let \(S_1\) be the sum of first \(2 n\) terms of an arithmetic progression. Let \(S_2\) be the sum of first \(4 n\) terms of the same arithmetic progression. If \(\left(S_2-S_1\right)\) is 1000 , then the sum of the first \(6 n\) terms of the arithmetic progression is equal to
[JEE Main 2021, 18 Mar (Shift 2)]
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Let \(S_n\) denote the sum of the first \(n\)-terms of an arithmetic progression. If \(S_{10}=530, S_5=140\), then \(S_{20}-S_6\) is equal to:
[JEE Main 2021, 22 Jul (Shift 2)]
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The common difference of the A.P. \(b_1, b_2, \ldots ., b_m\) is 2 more than the common difference of A.P. \(a_1, a_2, \ldots ., a_n\). If \(a_{40}=-159, a_{100}=-399\) and \(b_{100}=a_{70}\), then \(b_1\) is equal to:
[JEE Main 2020, 6 Sep (Shift 2)]
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The sum of the infinite series
\( 1+\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\frac{22}{3^{5}}+\ldots \cdots \) is equal to:
[JEE Main 2021, 26 Feb (Shift 1)]
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If \( 1+\left(1-2^{2} \cdot 1\right)+\left(1-4^{2} \cdot 3\right)+\left(1-6^{2} \cdot 5\right)+\ldots \ldots \ldots \ldots \ldots+(1 \left.-20^{2} \cdot 19\right)=\alpha-220 \beta \), then an ordered pair \( (\alpha, \beta) \) is equal to :
[JEE Main 2020, 4 Sep (Shift 1)]
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Let \(a_1, a_2, \ldots \ldots, a_{21}\) be an AP such that \(\sum_{n=1}^{20} \frac{1}{a_n a_{n+1}}=\frac{4}{9}\). If the sum of this AP is 189 , then \(a_6 a_{16}\) is equal to:
[JEE Main 2021, 1 Sep (Shift 2)]
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Let \(a_1, a_2, \ldots a_n\) be a given A.P. whose common difference is an integer and \(S_n=a_1+a_2+\ldots+a_n\). If \(a_1=1\), \(a_n=300\) and \(15 \leq n \leq 50\), then the ordered pair \(\left(S_{n-4}, a_{n-4}\right)\) is equal to
[JEE Main 2020, 4 Sep (Shift 2)]
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If \(a_n=\frac{-2}{4 n^2-16 n+15}\), then \(a_1+a_2+\ldots+a_{25}\) is equal to:
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For three positive integers \(p, q, r, x^{p q^2}=y^{q r}=z^{p^{2}r}\) and \(r=p q+1\) such that \(3,3 \log _y x, 3 \log _z y, 7 \log _x z\) are in \(A.P.\) with common difference \(\frac{1}{2}\). Then \(r-p-q\) is equal to
[JEE Main 2023, 24 Jan (Shift 1)]
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The sum of the series \( \sum_{n=1}^{\infty} \frac{n^{2}+6 n+10}{(2 n+1) !} \) is equal to :
[JEE Main 2021, 26 Feb (Shift 2)]
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If the sum of first 11 terms of an A.P., \( a_{1}, a_{2}, a_{3} \ldots \) is 0 \( \left(a_{1} \neq 0\right) \), then the sum of the A.P., , \( a_{1}, a_{3}, a_{5} \ldots, a_{23} \) is \(k{a}_{1}\) , where \( k \) is equal to :
[JEE Main 2020, 2 Sep (Shift 2)]
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If the 10th term of an A.P. is \( \frac{1}{20} \) and its 20th term is \( \frac{1}{10} \), then the sum of its first 200 term is
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If \( 2^{10}+2^{9} \cdot 3^{1}+2^{8} \cdot 3^{2}+\ldots .+2 \cdot 3^{9}+3^{10}=S-2^{11} \), then \( \mathrm{S} \) is equal to :
[JEE Main 2020, 5 Sep (Shift 1)]
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Let \(a_1, a_2, \ldots ., a_{10}\) be an A.P. with common difference \(-3\) and \(b_1, b_2, \ldots \ldots, b_{10}\) be a \(GP\) with common ratio \(2.\) Let \(c_k=a_k+b_k, k=1,2, \ldots ., 10\). If \(c_2=12\) and \(c_3=13\), then \(\sum_{k=1}^{10} c_k\) is equal to _______ .
[JEE Main 2021, 26 Aug (Shift 2)]
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If \( |x|<1,|y|<1 \) and \( x \neq y \), then the sum to infinity of the following series \( (x+y)+\left(x^{2}+x y+y^{2}\right)+ \) \( \left(x^{3}+x^{2} y+x y^{2}+y^{3}\right)+\ldots . . \) is :
[JEE Main 2020, 2 Sep (Shift 1)]
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Let \(S=109+\frac{108}{5}+\frac{107}{5^2}+\ldots \ldots \ldots+\frac{2}{5^{107}}+\frac{1}{5^{108}}\). Then the value of \(\left(16 S-(25)^{-54}\right)\) is equal to ____
[JEE Main 2023, 11 Apr (Shift 1)]
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Let \(a_1, a_2, a_3 \ldots a_n\) be \(n\) positive consecutive terms of an arithmetic progression. If \(d>0\) is its common difference, then \( \lim _{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots \frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right) \)
[JEE Main 2023, 6 Apr (Shift 1)]
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If the sum of the first \(20\) terms of the series \( \log _{\left(7^{1 / 2}\right)} x+\log _{\left(7^{1 / 3}\right)} x+\log _{\left(7^{1 / 4}\right)} x+\ldots \) is \(460\) , then \(x\) is equal to
[JEE Main 2020, 5 Sep (Shift 2)]
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Let \(S_n\) be the sum of the first \(n\) terms of an arithmetic progression. If \(S_{3 n}=3 S_{2 n}\), then the value of \(\frac{S_{4 n}}{S_{2 n}}\) is:
[JEE Main 2021, 25 Jul (Shift 1)]
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If the sum of the first 40 terms of the series, \( 3+4+8+9+13+14+18+19+\ldots \) is \((102)m,\) then \(m \) is equal to :
[JEE Main 2020, 7 Jan (Shift 2)]
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If the harmonic mean between a and \(\ b \) be \(\ H \), then the value of \(\ \frac{1}{H-a}+\frac{1}{H-b} \) is
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If \( 3^{2 \sin 2 \alpha-1}, 14 \) and \( 3^{4-2 \sin 2 \alpha} \) are the first three terms of an A.P. for some \( \alpha \), then the sixth term of this A.P is :
[JEE Main 2020, 5 Sep (Shift 1)]
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If \(S = \frac{7}{5} + \frac{9}{5^{2}} + \frac{13}{5^{3}} + \frac{19}{5^{4}} + .....\) , then \(160S\) is equal to
[JEE Main 2021, 31 Aug (Shift 2)]
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If \( 0<\theta, \phi<\frac{\pi}{2}, x=\sum_{n=0}^{\infty} \cos ^{2 n} \theta, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi \) and \( z=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \cdot \sin ^{2 n} \phi \), then:
[JEE Main 2021, 25 Feb (Shift 1)]
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If \( \alpha, \beta \) are natural numbers such that \(\begin{array}{r}100^{\alpha}-199 \beta=(100)(100)+(99)(101) +(98)(102)+\ldots \ldots+(1)(199)\end{array}\) ,then the slope of the line passing through \( (\alpha, \beta) \) and origin is :
[JEE Main 2021, 18 Mar (Shift 1)]
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Let the first term \(a\) and the common ratio \(r\) of a geometric progression be positive integers. If the sum of its squares of first three terms is 33033, then the sum of these three terms is equal to
[JEE Main 2023, 10 Apr (Shift 1)]
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If \(0 [JEE Main 2021, 27 Aug (Shift 2)]
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The sum of all odd numbers between 1 and 1000 which are divisible by 3 is
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If \(\operatorname{gcd}(m, n)=1\) and \(1^2-2^2+3^2-4^2+\ldots \ldots .+(2021)^2-\) \((2022)^2+(2023)^2=1012 m^2 n\), then \(m^2-n^2\) is equal to
[JEE Main 2023, 6 Apr (Shift 2)]
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Let \(a, b, c\) and \(d\) be positive real numbers such that \(a+b\) \(+c+d=11\). If the maximum value of \(a^5 b^3 c^2 d\) is \(3750 \beta\), then the value of \(\beta\) is
[JEE Main 2023, 11 Apr (Shift 2)]
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Let \( a_{n} \) be the \( n^{\text {th }} \) term of G.P. of positive terms. If \( \sum_{n=1}^{100} a_{2 n+1}=200 \) and \( \sum_{n=1}^{100} a_{2 n}=100 \), then \( \sum_{n=1}^{200} a_{n} \) is equal to:
[JEE Main 2020, 9 Jan (Shift 2)]
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Let \( a, b, c>1, a^{3}, b^{3} \) and \( c^{3} \) be in A.P., and
\( \log _{a} b \), \( \log _{c} a \) and \( \log _{b} c \) be in G.P. If the sum
of first \( 20 \) terms of an A.P., whose first term is
\( \frac{a+4 b+c}{3} \) and the common difference is
\( \frac{a-8 b+c}{10} \) is \( -444 \), then \( a b c \) is equal to
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In an increasing geometric series, the sum of the second and the sixth term is \( \frac{25}{2} \) and the product of the third and fifth term is 25 . Then, the sum of \( 4^{\text {th }}, 6^{\text {th }} \) and \( 8^{\text {th }} \) terms is equal to:
[JEE Main 2021, 26 Feb (Shift 1)]
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If \( |x|<1,|y|<1 \) and \( x \neq y \), then the sum to infinity of the following series \( (x+y)+\left(x^{2}+x y+y^{2}\right)+ \) \( \left(x^{3}+x^{2} y+x y^{2}+y^{3}\right)+\ldots . . \) is :
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Let \( \alpha \) and \( \beta \) be the roots of \( x^{2}-3 x+p=0 \) and \( \gamma \) and \( \delta \) be the roots of \( x^{2}-6 x+q=0 \). If \( \alpha, \beta, \gamma, \delta \) form a geometric progression. Then ratio \( (2 q+p):(2 q-p) \) is:
[JEE Main 2020, 4 Sep (Shift 1)]
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Let \(s _1, s _2, s _3, \ldots ., s _{10}\) respectively be the sum to 12 terms of 10 A.P.s whose first terms are \(1,2,3, \ldots . ., 10\) and the common differences are \(1,3,5, \ldots . ., 19\) respectively. Then \(\sum_{i=1}^{10} s _i\) is equal to
[JEE Main 2023, 13 Apr (Shift 1)]
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Three numbers are in an increasing geometric progression with common ratio \(r\). If the middle number is doubled, then the new numbers are in arithmetic progression with common difference \(d\). If the fourth term of GP is \(3r^{2}\) then \(r^{2} - d\) is equal to ?
[JEE Main 2021, 31 Aug (Shift 1)]
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Let \( a_{1}, a_{2}, a_{3}, \ldots \) be a G.P. such that \( a_{1}<0, a_{1}+a_{2}=4 \) and \( a_{3}+a_{4}=16 \). If \( \sum_{\mathrm{i}=1}^{9} a_{i}=4 \lambda \), then \( \lambda \) is equal to :
[JEE Main 2020, 7 Jan (Shift 2)]
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The sum of the first three terms of a G.P. is \(S\) and their product is 27 . Then all such \(S\) lie in:
[JEE Main 2020, 2 Sep (Shift 1)]
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Five numbers are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers is \( -\frac{1}{2} \), then the greatest number amongst them is
[JEE Main 2020, 7 Jan (Shift 1)]
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If \( \alpha, \beta \) are natural numbers such that \( (100)^{\alpha}- \) \( 199 \beta=(100)(100)+(99)(101)+(98)(102) \) \( +\ldots \ldots . .+(1)(199) \), then the slope of the line passing through \( (\alpha, \beta) \) and origin is:
[JEE Main 2021, 18 Mar (Shift 1)]
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\(\text{ The }sum\sum _{n=1}^{\infty }\frac{2{n}^{2}+3n+4}{(2n)!}\text{ is equal to: }\)
[JEE Main 2023, 1 Feb (Shift 2)]
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The minimum value of \( f(x)=a^{a^{x}}+a^{1-a^{x}} \), where
\( a, x \in R \) and \( a>0 \), is equal to
[JEE Main 2021, 25 Feb (Shift 2)]
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If \(S_n=4+11+21+34+50+\ldots \ldots\) to \(n\) terms, then \(\frac{1}{60}\left(S_{29}-S_9\right)\) \
[JEE Main 2023, 10 Apr (Shift 2)]
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If \( a_{n}=\frac{-2}{4 n^{2}-16 n+15} \), then \( a_{1}+a_{2}+\ldots \ldots+a_{25} \) is equal to
[JEE Main 2023, 30 Jan (Shift 1)]
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The minimum value of \(2^{\sin x}+2^{\cos x}\) is:
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Let \(a_1, a_2, \ldots ., a_{10}\) be an A.P. with common difference -3 and \(b_1, b_2, \ldots \ldots, b_{10}\) be a GP with common ratio 2 . Let \(c_k=a_k+b_k, k=1,2, \ldots ., 10\). If \(c_2=12\) and \(c_3=13\), then \(\sum_{k=1}^{10} c_k\) is equal to _______ .
[JEE Main 2021, 26 Aug (Shift 2)]
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Let \(x_1, x_2 \ldots, x_{100}\) be in an arithmetic progression, with \(x_1=2\) and their mean equal to \(200.\) If \(y_{ i }=i\left(x_{ i }-i\right), 1 \leq i \leq 100 ,\) then the mean of \(y_1, y_2, \ldots, y_{100}\) is
[JEE Main 2023, 11 Apr (Shift 1)]
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Let \(a, b, c>1, a^3, b^3\) and \(c^3\) be in A.P., and \(\log _a b, \log _c a\) and \(\log _b c\) be in G.P. If the sum of first 20 terms of an A.P., whose first term is \(\frac{a+4 b+c}{3}\) and the common difference is \(\frac{a-8 b+c}{10}\) is \(-444\), then \(a b c\) is equal to
[JEE Main 2023, 30 Jan (Shift 2)]
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Let \( S_{K}=\frac{1+2+\ldots .+K}{K} \) and
\( \sum_{j=1}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right) \), where \( A, B, C, D \in N \)
and \( A \) has least value. Then
[JEE Main 2023, 08 Apr (Shift 1)]
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If \(H\) is the harmonic mean between \(p\) and \(q\), then the value of \(\frac{H}{p}+\frac{H}{q}\) is
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Let \(s _1, s _2, s _3, \ldots ., s _{10}\) respectively be the sum to \(12\) terms of \(10\) A.P.s whose first terms are \(1,2,3, \ldots . ., 10\) and the common differences are \(1,3,5, \ldots . ., 19\) respectively. Then \(\sum_{i=1}^{10} s _i\) is equal to
[JEE Main 2023, 13 Apr (Shift 1)]
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Let a, b, c and d be positive real numbers such that \(a+b +c+d=11.\) If the maximum value of \({a}^{5}{b}^{3}{c}^{2}d\) is \(3750\beta\), then the value of \(\beta\) is
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Let \(S_n(x)=\log _{a^{\frac{1}{2}}} x+\log _{a^{\frac{1} { 3}}} x+\log _{a^{\frac{1}{6}}} x+\) \(\log _{a^{\frac{1} { 11}}} x+\log _{a^{\frac{1} { 18}}} x+\log _{a^{\frac{1} { 27}}} x+\ldots\) up to \(n\)-terms. Where \(a>1\). If \(S _{24}(x)=1093\) and \(S _{12}(2x)=265\), then the value of \(a\) is equal to ___
[JEE Main 2021, 16 Mar (Shift 2)]
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The minimum of \(f(x)=a^{a^x}+a^{1-a^x}\), where \(a, x \in R\) and \( a>0\), is equal to:
[JEE Main 2021, 25 Feb (Shift 2)]
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If the sum of the series \( 20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots . \). upto \( n \) terms is 488 and the \( n^{\text {th }} \) term is negative, then the value of \( n^{\text {th }} \) term is
[JEE Main 2020, 3 Sep (Shift 2)]
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