Question

In the expansion of $(5^{\frac{1}{2}}+2^{\frac{1}{8}}{)}^{1024}$, the number of integral terms is

• A. $128$
• B. $129$
• C. $130$
• D. $131$

Answer 1

The correct option is B $129$

$T_{r+1}{=}^{1024}{C}_r\times 5^{\frac{r}{2}}\times 2^{\frac{(1024-r)}{8}}$
For $T_{r+1}$ to be integral, the powers of $5$ and $2$ must be integers.
Hence,
$\frac{r}{2}=a$ and $\frac{1024-r}{8}=b$, where $a$ and $b$ are integers.
Hence, $r$ must be a factor of $8$.
$r$ can assume values from $0$ to $1024$ which are factors of $8$.
Hence, the total number of values $r$ can acquire is $1+\frac{1024}{8}=129$.