Question:
If $13^{\text {th }}$ term in the expansion of $\left(x^{2}+\frac{2}{x}\right)^{n}$ is independent of $x$, then the value of $n$ is _______________
Solution:
In $\left(x^{2}+\frac{2}{x}\right)^{n}$
$T_{r+1}={ }^{n} C_{r}\left(x^{2}\right)^{n-r}\left(\frac{2}{x}\right)^{r}$
$={ }^{n} C_{r} x^{2 n-2 r} \frac{(2)^{r}}{x^{r}}$
$={ }^{n} C_{r} x^{2 n-2 r} x^{-r}(2)^{r}$
$T_{r+1}={ }^{n} C_{r} 2^{r} x^{2 n-3 r}$
If for r = 12, i.e 13th term is independent of x
2n – 3r = 0
⇒ 2n = 3 × 12
i.e. n = 18