If x^m occurs in the expansion of (x + (1/x^2 ))^(2n) , the coefficient of x^m is


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Question Number 33147 by Ahmad Hajjaj last updated on 11/Apr/18

$If x^{m} occurs in the expansion of$ ${(x + \frac{1}{x^{2}})}^{2 n}, the coefficient of x^{m} is$
Commented by prof Abdo imad last updated on 12/Apr/18

$w e h a v e {(x + \frac{1}{x^{2}})}^{2 n} = \sum_{k = 0}^{2 n} C_{2 n}^{k} x^{k} {(x^{- 2})}^{2 n - k}$ $= \sum_{k = 0}^{2 n} C_{2 n}^{k} x^{k - 4 n + 2 k} = \sum_{k} C_{2 n}^{k} x^{3 k - 4 n} s o i f x^{m}$ $a p p e a r s i n t h e e x p a n s i o n w e g e t 3 k - 4 n = m \Rightarrow$ $3 k = m + 4 n \Rightarrow k = [\frac{m + 4 n}{3}] a n d t h e c o e f f i c i e n t$ $i s C_{2 n}^{[\frac{m + 4 n}{3}]} .$
	Answered by Rio Mike last updated on 11/Apr/18

	$\underset{r = 1}{\sum^{2 n}}^{2 n} C_{r} . x^{2 n - r} {(\frac{1}{x^{2}})}^{r}$ $^{2 n} C_{r .} x^{2 n - r} . 1^{r} . x^{- 2 r}$ $^{2 n} C_{r .} x^{2 n - 3 n}$ $h e n c e, x^{m} = x^{2 (m) - 3 (m)}$ $m = 2 m - 3 m$ $m = - m$
	Answered by MWSuSon last updated on 25/Apr/20

	$= C_{r}^{2 n} {(x^{- 2})}^{r} x^{2 n - r}$ $= C_{r}^{2 n} x^{- 2 r} x^{2 n - r}$ $- 2 r + 2 n - r = m$ $2 n - m = 3 r$ $\frac{2 n - m}{3} = r$ $r e p l a c i n g r w i t h \frac{2 n - m}{3}$ $w e h a v e \frac{(2 n)!}{(\frac{2 n - m}{3})! (\frac{6 n - 2 n + m}{3})!}$ $\frac{(2 n)!}{(\frac{2 n - m}{3})! (\frac{4 n + m}{3})!}$
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