f(x)=(x+a_1 )(x+a_2 )(x+a_3 )...(x+a_n ) find the coefficient of term x^k (0≤k≤n)


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Question Number 29805 by mrW2 last updated on 12/Feb/18

$f (x) = (x + a_{1}) (x + a_{2}) (x + a_{3}) . . . (x + a_{n})$ $f i n d t h e c o e f f i c i e n t o f t e r m x^{k} (0 ⩽ k ⩽ n)$
Commented by mrW2 last updated on 13/Feb/18

$A f i r s t t r y :$ $l e t P = a_{1} a_{2} . . . a_{n}$ $l e t C_{k} = c o e f f i c i e n t o f t e r m x^{k} i n f (x)$ $C_{0} = P$ $C_{1} = \underset{i = 1}{\sum^{n}} \frac{P}{a_{i}} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} . . . \underset{p_{n - 1} = p_{n - 2} + 1}{\sum^{n}} a_{p_{1}} a_{p_{2}} . . . a_{p_{n - 1}}$ $C_{2} = \underset{i = 1}{\sum^{n}} \underset{j = i + 1}{\sum^{n}} \frac{P}{a_{i} a_{j}} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} . . . \underset{p_{n - 2} = p_{n - 3} + 1}{\sum^{n}} a_{p_{1}} a_{p_{2}} . . . a_{p_{n - 2}}$ $. . . . . .$ $C_{k} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} . . . \underset{p_{k} = p_{k - 1} + 1}{\sum^{n}} \frac{P}{a_{p_{1}} a_{p_{2}} . . . a_{p_{k}}} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} . . . \underset{p_{n - k} = p_{n - k - 1} + 1}{\sum^{n}} a_{p_{1}} a_{p_{2}} . . . a_{p_{n - k}}$ $(1 ⩽ k ⩽ n)$ $. . . . . .$ $C_{n - 1} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} . . . \underset{p_{n - 1} = p_{n - 2} + 1}{\sum^{n}} \frac{P}{a_{p_{1}} a_{p_{2}} . . . a_{p_{n - 1}}} = \underset{i = 1}{\sum^{n}} a_{i}$ $C_{n} = 1$
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