Question

Let the coefficients of powers of $x$ in the $2^{nd},3^{rd}$ and $4^{th}$ terms in the expansion of ${\left(1+x\right)}^n,$ 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)

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