If-x-m-occurs-in-the-expansion-of-x-1-x-2-2n-the-coefficient-of-x-m-is-

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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