Question

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

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

Answer 1

The correct option is A

$\frac{1}{2}(3^{50}+1)$

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