# If the fifth term of the expansion

Question:

If the fifth term of the expansion $\left(a^{2 / 3}+a^{-1}\right)^{n}$ does not contain ' $a$ '. Then $n$ is equal to

(a) 2

(b) 5

(c) 10

(d) none of these

Solution:

(c) 10

$T_{5}=T_{4+1}$

$={ }^{n} C_{4}\left(a^{2 / 3}\right)^{n-4}\left(a^{-1}\right)^{4}$

$={ }^{n} C_{4} a^{\left(\frac{2 n-8}{3}-4\right)}$

For this term to be independent of a, we must have

$\frac{2 n-8}{3}-4=0$

$\Rightarrow 2 n-20=0$

$\Rightarrow n=10$