Question

Find the constant term in expansion of
${(x+\frac{2}{x^2})}^{15}$

Answer 1

As we know that the general term of the expansion is given as-

${(a+b)}^{15}$ is given as-

$T_{r+1}={}^n{C}_r{a}^r{b}^{n-r}$

Given expansion:- ${(x+\frac{2}{x^2})}^{15}$

Here, $a=x,b=\frac{2}{x^2},n=15$

Therefore,

$T_{r+1}={}^{15}{C}_r{x}^r{\left(\frac{2}{x^2}\right)}^{15-r}$

$\Rightarrow T_{r+1}={}^{15}{C}_r{x}^{3(r-10)}{2}^{15-r}$

For constant term,

$3(r-10)=0$

$\Rightarrow r=10$

Therefore,

Constant term $=T_{11}={}^{15}{C}_{10}{x}^0{2}^{15-10}={}^{15}{C}_{10}{2}^5$