Question

The number of integral terms in the expansion of $\left(\sqrt{3}+\sqrt[8]{5}\right)^{256}$ is

• A. $32$
• B. $33$
• C. $34$
• D. $35$

Answer 1

Answer: (B)

$T_{r+1}={ }^{256} C_{r}(\sqrt{3})^{256-r} 5^{\frac{r}{8}}$
For integral terms $\frac{256-r}{2}, \frac{r}{8}$ are both positive
integer
$\therefore r=0,8,16, \ldots 256$
$\therefore 256=0+(n-1) 8$ using $t_{n}=a+(n-1) d$
$\therefore \frac{256}{8}=n-1$
$\therefore n=\frac{256}{8}+1$
$n =32+1=> n =33$