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

Answer 1

(c) 10

$$\begin{gathered} T_5=T_{4+1} \\ ={}^{n}C_4(a^{2/3}{)}^{n-4}(a^{-1}{)}^4 \\ ={}^{n}C_4{a}^{\left(\frac{2n-8}{3}-4\right)} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{be}\mathrm{independent}\mathrm{of}\mathrm{a},\mathrm{we}\mathrm{must}\mathrm{have} \\ \frac{2n-8}{3}-4=0 \\ \Rightarrow 2n-20=0 \\ \Rightarrow n=10 \end{gathered}$$