The sum of all possible values of \(n \in N\), so that the coefficients of \(x, x^2\) and \(x^3\) in the expansion of \(…

Given expansion:

\(\left({x}^{4}+2{x}^{2}+1\right)\left({}^{n}C_{0}{x}^{0}+{}^{n}C_{1}{x}^{1}+{}^{n}C_{2}{x}^{2}+{}^{n}C_{3}{x}^{3}+\ldots \right)\)

Coefficient \(x\Rightarrow {}^{n}C_{1}\),
coeff. of \({x}^{2}\Rightarrow 2+{}^{n}C_{2}\)\(=2+\frac{n(n-1)}{2}\)

Coeff. of \({x}^{3}=2.{}^{n}C_{1}+{}^{n}C_{3}\)

\(=2n+\frac{n(n-1)(n-2)}{6}\)

Now according to question

\(n+2n+\frac{n(n-1)(n-2)}{6}=2\left[2+\frac{n(n-1)}{2}\right]\\ \Rightarrow 3n+\frac{n(n-1)(n-2)}{6}=4+n(n-1)\\ \Rightarrow {n}^{3}-9{n}^{2}+26n-24=0\\ \Rightarrow n=2,3,4\)

Now checking for \(n = 2\)

Coeff. of \(x=2\), Coeff. of \(x^2=3\), Coeff. of \(x^3=4\)

are in A.P.
\(\Rightarrow n=2\) is also the correct choice
Required sum of values of \(n=2+3+4=9\)