The natural number m, for which the coefficient of x in the binomial expansion of (x^m+1x^…

Question

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left(x^m+\dfrac{1}{x^2}\right)^{22}$ is 1540, is \_\_\_\_\_\_\_.

Answer: 13

Accepted Answer

$$T_{r+1}=\binom{22}{r}x^{m(22-r)-2r}$$

For coefficient of $x$: $22m-mr-2r=1\Rightarrow r=\dfrac{22m-1}{m+2}$

Checking $m=13$: $r=\dfrac{285}{15}=19$; $\binom{22}{19}=\binom{22}{3}=1540$

$m=13$

Solutions · 1