Question

If the number of integral terms in the expansion of ${(3^{1/2}+5^{1/8})}^n$ is exactly $33,$ then the least value of $n$ is:

• A. $128$
• B. $248$
• C. $256$
• D. $264$

Answer 1

The correct option is C $256$

The general term of the expansion ${(3^{1/2}+5^{1/8})}^n$ is
$T_{r+1}={}^n{C}_r\cdot {\left(3^{1/2}\right)}^{n-r}\cdot {\left(5^{1/8}\right)}^r$ $(n\ge r)$
$\Rightarrow T_{r+1}={}^n{C}_r\cdot (3{)}^{\frac{n-r}{2}}\cdot (5{)}^{\frac{r}{8}}$ $(n\ge r)$

For integral term in the expansion $r$ should be a multiple of $8$
$\Rightarrow r=0,8,16,24,32,\cdots \cdots$
$\Rightarrow n-r=0,2,4,6,8,\cdots \cdots$
Common $r=0,8,16,24,32,\cdots \cdots$
$\because$ There are exactly $33$ integral terms.
$\therefore$ Least value of $n=0+(33-1)\times 8=256$