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
$(1-2 \sqrt{x})^{50}$
$T_{r+1}={ }^{50} C_{r}(1)^{50-r}\left(-2 x^{1 / 2}\right)^{r}$
$={ }^{50} C_{r} 2^{r} x^{r / 2}(-1)^{r}$
For the integral power of $x, r$ should be even integer.
$\therefore $ Sum of coefficients $=\displaystyle\sum_{r=0}^{25}{ }^{50} C_{2 r}(2)^{2 r} $
$=\frac{1}{2}\left[(1+2)^{50}+(1-2)^{50}\right]=\frac{1}{2}\left(3^{50}+1\right)$