Question

In the expansion of ${(5^{\frac{1}{2}}+7^{\frac{1}{8}})}^{1024}$, the number of integral terms is

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

Answer 1

The correct option is B $129$

The general term in the expansion of $(5^{\frac{1}{2}}+7^{\frac{1}{8}}{)}^{1024}$ is
$T_{r+1}{=}^{1024}{C}_r\left(5^{\frac{1}{2}}{)}^{1024-r}\right(7^{\frac{1}{8}}{)}^r$
${=}^{1024}{C}_r{5}^{512-\frac{r}{2}}{7}^{\frac{r}{8}}$
$={(}^{1024}{C}_r{5}^{512-r}\left)\right(5^{\frac{r}{2}}{7}^{\frac{r}{8}})$
$={(}^{1024}{C}_r{5}^{512r}\left)\right(5^4\times 7{)}^{\frac{r}{8}}$
Clearly $T_{r+1}$ will be an integer, if $\frac{r}{8}$ is an integer such that $0\le r\le 1024$
$\Rightarrow r$ is a multiple of $8$ lying satisfying $0\le r\le 1024$
$r=0,8,16,...1024$
$\Rightarrow r$ can assume $129$ values.
Hence, there are $129$ integral terms in the expansion of $(5^{\frac{1}{2}}+7^{\frac{1}{8}}{)}^{1024}$