Question

If 13^th term in the expansion of ${\left(x^2+\frac{2}{x}\right)}^n$ is independent of x, then the value of n is ___________.

Answer 1

$$\begin{gathered} \mathrm{In}{\left(x^2+\frac{2}{x}\right)}^n \\ {T}_{r+1}={}^{n}C_r{\left(x^2\right)}^{n-r}{\left(\frac{2}{x}\right)}^r \\ ={}^{n}C_r{x}^{2n-2r}\frac{{\left(2\right)}^r}{x^r} \\ ={}^{n}C_r{x}^{2n-2r}{x}^{-r}{\left(2\right)}^r \\ {T}_{r+1}={}^{n}C_r{2}^r{x}^{2n-3r} \end{gathered}$$
If for r = 12, i.e 13^th term is independent of x
2n – 3r = 0
⇒ 2n = 3 × 12
i.e. n = 18