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Term Independent of x

Which term is...

Question

Which term is independent of x, in the expansion of ${(x - \frac{1}{3 x^{2}})}^{9} ?$

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Solution

${(x - \frac{1}{3 x^{2}})}^{9}$

Let (r+1)th term be the independent of x in the given expression $T_{r + 1} = 9_{C_{r}} (x)^{9 - r} {(- \frac{1}{3 x^{2}})}^{r}$

$= 9_{C_{r}} {(- \frac{1}{3})}^{r} (x)^{9 - r - 2 r} = 9_{C_{r}} {(- \frac{1}{3})}^{r} (x)^{9 - 3 r}$

This is term is independent of x if

$9 - 3 r = 0 \Rightarrow r = 3$

So, (3+1)th i.e. 4th term is independent of x.

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