$\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}$is equal to: [JEE Main 2026, 28 Jan (Shift 2)]

Question

YESPYQ · Accepted Answer

Correct answer: d) ${2}^{26}$. $S=\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}\ =\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[1+6+{6}^{2}+...+{6}^{24}\right]$ $=\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[\frac{{\left(6\right)}^{25}−1}{6−1}\right]$ $=\frac{6}{{3}^{26}}+\frac{10}{{3}^{25}}\left[\frac{{6}^{25}−1}{5}\right]$ $=\frac{2}{{3}^{25}}+2\left[{2}^{25}−\frac{1}{{3}^{25}}\right]$ $={2}^{26}$

\(\frac{6}{{3}^{26}}+\frac{10⋅1}{{3}^{25}}+\frac{10⋅2}{{3}^{24}}+\frac{10⋅{2}^{2}}{{3}^{23}}+...+\frac{10⋅{2}^{24}}{3}\)…

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