f-x-x-a-1-x-a-2-x-a-3-x-a-n-find-the-coefficient-of-term-x-k-0-k-n-

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

f (x) = (x + a_{1}) (x + a_{2}) (x + a_{3}) \dots (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} \dots 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}} \dots \underset{p_{n - 1} = p_{n - 2} + 1}{\sum^{n}} a_{p_{1}} a_{p_{2}} \dots 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}} \dots \underset{p_{n - 2} = p_{n - 3} + 1}{\sum^{n}} a_{p_{1}} a_{p_{2}} \dots a_{p_{n - 2}}

\dots \dots

C_{k} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} \dots \underset{p_{k} = p_{k - 1} + 1}{\sum^{n}} \frac{P}{a_{p_{1}} a_{p_{2}} \dots a_{p_{k}}} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} \dots \underset{p_{n - k} = p_{n - k - 1} + 1}{\sum^{n}} a_{p_{1}} a_{p_{2}} \dots a_{p_{n - k}}

(1 ⩽ k ⩽ n)

\dots \dots

C_{n - 1} = \underset{p_{1} = 1}{\sum^{n}} \underset{p_{2} = p_{1} + 1}{\sum^{n}} \dots \underset{p_{n - 1} = p_{n - 2} + 1}{\sum^{n}} \frac{P}{a_{p_{1}} a_{p_{2}} \dots a_{p_{n - 1}}} = \underset{i = 1}{\sum^{n}} a_{i}

C_{n} = 1

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