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.
$\Rightarrow \frac{2 n(n-1)}{2 !}=n+\frac{n(n-1)(n-2)}{3 !}$
$\Rightarrow n^{2}-9 n+14=0$
$\Rightarrow (n-7)(n-2)=0$
$\Rightarrow n=7$ since $n \neq 2$
$\therefore $ The sum of the coefficients of odd powers of $x$ in the expansion of $(1+x)^{n}$ is
$\frac{2^{n}}{2}=\frac{2^{7}}{2}=2^{6}=64$