What is the proof of the Binomial Theorem, other than the induction method? How can we find the expansion of binomails with indices like 2n, 3n, 4n..?

score 153 · Answer 7 · 2026-03-31 07:48:53

153

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Combinatorial identity from squaring the binomial expansion

Published on 31 Mar 2026 - 7:48

score 71 · Answer 8 · 2021-12-14 18:56:21

71

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Show that $\sum_{k=1}^n \binom{n}{k}k^2=n^2\cdot \:2^{n-2}+n\cdot \:2^{n-2}$.

Published on 14 Dec 2021 - 18:56

score 527 · Answer 9 · 2021-12-17 06:55:37

527

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Ways to find $\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\cdots$

Published on 17 Dec 2021 - 6:55

score 400 · Answer 10 · 2021-12-18 21:26:31

400

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Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$. What is $\log_{2}|S|$?

Published on 18 Dec 2021 - 21:26

score 105 · Answer 11 · 2026-03-31 10:06:08

105

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Alternative way to deduce combinatorial identity?

Published on 31 Mar 2026 - 10:06

score 27 · Answer 12 · 2021-12-19 09:24:43

27

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How to show that $ \left(1+\frac{1}{n} \right)^n = \sum_{i=0}^{n}\frac{1}{i!}\left(\prod_{j=0}^{i-1}\left(1 - \frac{j}{n}\right)\right)$

Published on 19 Dec 2021 - 9:24

score 114 · Answer 13 · 2021-12-24 18:01:11

114

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Binomial theorem for $(1-x^2)^{n-\frac12}$

Published on 24 Dec 2021 - 18:01

score 80 · Answer 14 · 2021-12-25 06:18:43

80

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Where is an expository article on proving Newton's binomial theorem?

Published on 25 Dec 2021 - 6:18

score 138 · Answer 15 · 2021-12-29 04:02:35

138

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Having trouble with proving this binomial identity. I have taken it as far as I can go.

Published on 29 Dec 2021 - 4:02

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What is the proof of the Binomial Theorem, other than the induction method? How can we find the expansion of binomails with indices like 2n, 3n, 4n..?

Combinatorial identity from squaring the binomial expansion

Show that $\sum_{k=1}^n \binom{n}{k}k^2=n^2\cdot \:2^{n-2}+n\cdot \:2^{n-2}$.

Ways to find $\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\cdots$

Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$. What is $\log_{2}|S|$?

Alternative way to deduce combinatorial identity?

How to show that $ \left(1+\frac{1}{n} \right)^n = \sum_{i=0}^{n}\frac{1}{i!}\left(\prod_{j=0}^{i-1}\left(1 - \frac{j}{n}\right)\right)$

Binomial theorem for $(1-x^2)^{n-\frac12}$

Where is an expository article on proving Newton's binomial theorem?

Having trouble with proving this binomial identity. I have taken it as far as I can go.

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