1-x-2x-2-8-

Question Number 49220 by malwaan last updated on 04/Dec/18

{(1 + x - {2 x}^{2})}^{8} = ?

Commented by Abdo msup. last updated on 04/Dec/18

S = \sum_{k = 0}^{8} C_{8}^{k} {(x - 2 x^{2})}^{k} = \sum_{k = 0}^{8} C_{8}^{k} x^{k} {(1 - 2 x)}^{k}

= \sum_{k = 0}^{8} C_{8}^{k} x^{k} {\sum_{p = 0}^{k} C_{k}^{p} {(- 2 x)}^{p}}

= \sum_{k = 0}^{8} C_{8}^{k} (\sum_{p = 0}^{k} {(- 2)}^{p} C_{k}^{p} x^{p}) x^{k}

= 1 + C_{8}^{1} (1 - 2 C_{1}^{1} x) x + C_{8}^{2} (1 - 2 C_{2}^{1} x + 4 C_{2}^{2} x^{2}) x^{2} + \dots

+ C_{8}^{8} (\sum_{p = 0}^{8} {(- 2)}^{p} C_{8}^{p} x^{p}) x^{8} .

Commented by malwaan last updated on 05/Dec/18

thank you sir

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