Question

If the constant term in the binomial expansion of ${(x^2-\frac{1}{x})}^n,nϵN$ is 15 then the value of n is equal to

Answer 1

Expansion is ${(x^2-\frac{1}{x})}^n$

constant term is 15th

$\Rightarrow T_{15}={}^n{C}_{14}(x^2{)}^{n-14}{(-\frac{1}{x})}^{14}$

is independent of x.

$\Rightarrow T_{15}={}^n{C}_{14}{x}^{2n-28}.\frac{1}{x^{14}}$

$={}^n{C}_{14}\left(x^{2n-28}\right)\left(x^{-14}\right)$

$={}^n{C}_{14}{x}^{2n-42}$

For term to be independent of x exponent of x should be zero.

i.e. $2n-42=0$

$\Rightarrow 2n=42$

$\Rightarrow n=\frac{42}{2}=21$