If the expansion ${(x^{2} + \frac{1}{x^{3}})}^{n}$ is to contain an independent term, then what should be the value of n?

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Solution

General term, $T_{r + 1} =^{n} C_{r} \cdot x^{n - r} \cdot y^{r}$ , for $(x + y)^{n}$
$⟹$ general term of ${(x^{2} + \frac{1}{x^{3}})}^{n}$ is
$^{n} C_{r} \cdot x^{2 n - 2 r} \cdot \frac{1}{x^{3 r}} =^{n} C_{r} \cdot x^{2 n - 5 r}$
For a term to be independent of x, $2 n - 5 r$ should be equal to zero.
$r = \frac{2}{5} n$ , since r can take only integral values, n has to be a multiple.of 5.

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