Question

Let the coefficients of powers of $x$ in the $2^{nd},3^{rd}$ and $4^{th}$ terms in the expansion of $(1+x{)}^n$, where $n$ is a positive integer, be in arithmetic progression. Then, the sum of the coefficients of odd powers of $x$ in the expansion is

• A. $32$
• B. $64$
• C. $128$
• D. $256$

Answer 1

Answer: (B)

$64$

According to question, ${}^n{C}_1,{}^n{C}_2$ and ${}^n{C}_3$ are in AP.

$\Rightarrow \frac{2n(n-1)}{2!}=n+\frac{n(n-1)(n-2)}{3!}$

$\Rightarrow n^2-9n+14=0$

$\Rightarrow (n-7)(n-2)=0$

$\Rightarrow n=7$ since $n\ne 2$

$\therefore$ The sum of the coefficients of odd powers of $x$ in the expansion of $(1+x{)}^n$ is

$\frac{2^n}{2}=\frac{2^7}{2}=2^6=64$