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

Question

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 ${(1+{x}^{2})}^{2}(1+x{)}^{n}$, are in arithmetic progression is :

Accepted Answer

$\begin{gathered} {(1+{x}^{2})}^{2}(1+{x}^{n}) \ (1+{x}^{4}+2{x}^{2})(1+x{)}^{n} \ x={}^{n}{C}_{1} \ {x}^{2}={}^{n}{C}_{2}+2\cdot {}^{n}{C}_{0} \ {x}^{3}={}^{n}{C}_{3}+2\cdot {}^{n}{C}_{1} \ \because x,{x}^{2},{x}^{3}	o AP \ \Rightarrow 2({}^{n}{C}_{2}+2)={}^{n}{C}_{1}+{}^{n}{C}_{3}+2n \ {2}^{n}{C}_{2}+4={}^{n}{C}_{3}+3n \ \Rightarrow {n}^{2}-9{n}^{2}+26n-24=0 \ \Rightarrow n=2,3,4 \ \Rightarrow n9 \end{gathered}$

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