Question

Let the coefficients of powers of $x$ in the second, third and fourth terms in the binomial expansion of $(1+x{)}^n,$ where $n$ is a positive integer, be in arithmetic progression. The sum of the coefficients of odd powers of $x$ in the expansion is

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

Answer 1

The correct option is B $64$

Given that ${}^n{C}_1,{}^n{C}_2,{}^n{C}_3$ are in A.P.
$\Rightarrow 2{}^n{C}_2={}^n{C}_1+{}^n{C}_3$
$\Rightarrow \frac{2n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}$
$\Rightarrow n^2-9n+14=0$
$\Rightarrow n=7[n=2\text{rejected}]$
Sum of the coefficients of odd powers of $x$ in the expansion is $2^{n-1}=2^6=64$