The-coefficient-of-x-m-in-1-x-p-1-x-p-1-1-x-n-p-m-n-is-

Question Number 62086 by Cypher1207 last updated on 15/Jun/19

The coefficient of x^{m} in

{(1 + x)}^{p} + {(1 + x)}^{p + 1} + \dots + {(1 + x)}^{n}, p ⩽ m ⩽ n

is

Answered by mr W last updated on 15/Jun/19

{(1 + x)}^{p} + {(1 + x)}^{p + 1} + \dots + {(1 + x)}^{n} \leftarrow G P

= \frac{{(1 + x)}^{p} [{(1 + x)}^{n - p + 1} - 1]}{(1 + x) - 1}

= \frac{{(1 + x)}^{n + 1} - {(1 + x)}^{p}}{x}

= \frac{1}{x} [\underset{k = 0}{\sum^{n + 1}} C_{k}^{n + 1} x^{k} - \underset{k = 0}{\sum^{p}} C_{k}^{p} x^{k}]

= \frac{1}{x} [\underset{k = 0}{\sum^{p}} C_{k}^{n + 1} x^{k} - \underset{k = 0}{\sum^{p}} C_{k}^{p} x^{k} + \underset{k = p + 1}{\sum^{n + 1}} C_{k}^{n + 1} x^{k}]

= \frac{1}{x} [\underset{k = 0}{\sum^{p}} (C_{k}^{n + 1} - C_{k}^{p}) x^{k} + \underset{k = p + 1}{\sum^{n + 1}} C_{k}^{n + 1} x^{k}]

= \underset{k = 0}{\sum^{p}} (C_{k}^{n + 1} - C_{k}^{p}) x^{k - 1} + \underset{k = p + 1}{\sum^{n + 1}} C_{k}^{n + 1} x^{k - 1}

= \underset{k = 0}{\sum^{p - 1}} (C_{k + 1}^{n + 1} - C_{k + 1}^{p}) x^{k} + \underset{k = p}{\sum^{n}} C_{k + 1}^{n + 1} x^{k}

= \underset{k = 0}{\sum^{n}} a_{k} x^{k}

w i t h a_{k} = {\begin{cases} C_{k + 1}^{n + 1} - C_{k + 1}^{p} f o r 0 ⩽ k ⩽ p - 1 \\ C_{k + 1}^{n + 1} f o r p ⩽ k ⩽ n \end{cases}

f o r p ⩽ m ⩽ n,

c o e f . o f x^{m} i s C_{m + 1}^{n + 1} .

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