Question

Number of integral terms in the expansion of $\left(\sqrt{5} + \sqrt[8]{7}\right)^{1024}$ is

Answer 1

We have, $T_{r = 1}=^{1024}C_{r}\left(5\right)^{\frac{\left(1024 - r\right)}{2}}\left(7\right)^{\frac{r}{8}}$
This is an integer, if $1024-r$ is an even number and $r$ is a multiple of $8$
i.e. when $r$ is a multiple of $8$ since $0\leq r\leq 1024$
The number of multiples of $8$ which lies between $0$ and $1024$ is
$1024=0+\left(n - 1\right)8$
$\Rightarrow n-1=\frac{1024}{8}=128$
$\Rightarrow n=129$