Question

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

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

Answer 1

The correct option is B $129$

$T_{r+1}={}^{1024}{C}_r{\left(\right(5{)}^{\frac{1}{2}})}^{1024-r}{\left(\right(7{)}^{\frac{1}{8}})}^r\Rightarrow T_{r+1}={}^{1024}{C}_r{\left(5\right)}^{(1024-r)/2}(7{)}^{r/8}$
Now, this term will be an integer if
$\frac{1024-r}{2},\frac{r}{8}$ will be an integer, for which $r$ has to be multiple of $8.$
$\therefore r=0,8,16,\dots ,1024$
For A.P. $0,8,16,\dots ,1024,$
number of terms will be $\frac{1024}{8}+1=129$