Answer:
Option A
Explanation:
Let T_{r+1} be the general term in the expansion of (1-2\sqrt{x})^{50}
\therefore T_{r+1} = ^{50}C_{r}(1)^{50-r}(-2x^{1/2})^{r}
= ^{50}C_{r}2^{r}x^{1/2}(-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}[(1+2)^{50}+(1-2)^{50}]
= \frac{1}{2}[3^{50}+1]
Alter:
We have ,
(1-2\sqrt{x})^{50}=C_{0}-C_{1}2\sqrt{x}+C_{2}(2\sqrt{x})^{2}+...+C_{50}(2\sqrt{x})^{50}......(i)
(1+2\sqrt{x})^{50}=C_{0}+C_{1}2\sqrt{x}+C_{2}(2\sqrt{x})^{2}+...+C_{50}(2\sqrt{x})^{50}......(ii)
On adding Eqs.(i) and (ii) , we get
(1-2\sqrt{x})^{50}+(1+2\sqrt{x})^{50}=2[C_{0}+C_{2}(2\sqrt{x})^{2}+...+C_{50}(2\sqrt{x})^{50}]
\Rightarrow \frac{(1-2\sqrt{x)}^{50}+(1+2\sqrt{x})^{50}}{2}=C_{0}
+C_{2}(2\sqrt{x})^{2}+......+C_{50}(2\sqrt{x})^{50}
On putting x=1, we get
\frac{(1-2\sqrt{1})^{50}+(1+2\sqrt{1})^{50}}{2}=C_{0}+C_{2}(2)^{2}+...+C_{50}(2)^{50}
\Rightarrow \frac{(-1)^{50}+(3)^{50}}{2}=C_{0}+C_{2}(2)^{2}+......+C_{50}(2)^{50}
\Rightarrow \frac{1+3^{50}}{2}=C_{0}+C_{2}(2)^{2}+....+C_{50}(2)^{50}