Question

The sum of coefficients of integral powers of $x$ in the binomial expansion $(1-2\sqrt{x}{)}^{50}$ is

• A. $\frac{1}{2}(3^{50})$
• B. $\frac{1}{2}(3^{50}+1)$
• C. $\frac{1}{2}(3^{50}-1)$
• D. $\frac{1}{2}(2^{50}+1)$

Answer 1

Answer: (B)

Let $T_{t+1}$ be the general term in the expension of
$(1-2\sqrt{x}{)}^{50}$
$T_{r+1}={}^{50}{C}_r(1{)}^{50-r}{\left(-2x^{1/2}\right)}^r$
$={}^{50}{C}_r{2}^r{x}^{r/2}(-1{)}^r$
For the integral power of $x,r$ should be even integer.
$\therefore$ Sum of coefficients $=\sum _{r=0}^{25}{}^{50}{C}_{2r}(2{)}^{2r}$
$=\frac{1}{2}\left[(1+2{)}^{50}+(1-2{)}^{50}\right]=\frac{1}{2}\left(3^{50}+1\right)$