Question

prove or disprove $n > 0 ~\rightarrow~\prod \limits_{i=1}^{n} \left ( \frac{1}{2i~+~1} \cdot \frac{1}{2i~+~2} \right ) ~=~ \frac{1}{(2n~+~2)!}$

score 51 · Answer 1 · 2026-04-17 00:30:18

prove or disprove $n > 0 ~\rightarrow~\prod \limits_{i=1}^{n} \left ( \frac{1}{2i~+~1} \cdot \frac{1}{2i~+~2} \right ) ~=~ \frac{1}{(2n~+~2)!}$

score 116 · Answer 2 · 2026-04-16 19:57:26

Prove or disprove $n \geq 2 ~\rightarrow~ \prod \limits_{i=1}^{n} \left ( 1 - \frac{1}{i^2} \right ) ~=~ \frac{n+1}{2n}$

score 111 · Answer 3 · 2026-04-15 20:48:12

111

Views

Show $P(1), P(2),...,P(99)$ true statements but $P(100)$ is false.

Published on 15 Apr 2026 - 20:48

score 54 · Answer 4 · 2026-04-15 19:23:06

54

Views

Show by induction: $ \frac{a_{2n+4}}{a_{n+2}}=\frac{a_{2n+2}}{a_{n+1}}+\frac{a_{2n}}{a_{n}}$

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score 272 · Answer 5 · 2026-04-14 17:40:41

272

Views

Proving $n! < n^n$ by induction for all $n\geq 2$.

Published on 14 Apr 2026 - 17:40

score 20.3k · Answer 6 · 2026-04-15 08:30:26

20.3k

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score 213 · Answer 7 · 2026-04-12 08:09:59

213

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180

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score 618 · Answer 9 · 2026-04-12 10:33:08

618

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score 632 · Answer 10 · 2026-04-16 08:44:59

632

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score 5.1k · Answer 11 · 2026-04-12 08:03:39

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score 368 · Answer 12 · 2026-04-13 01:20:20

368

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score 80 · Answer 13 · 2026-04-11 17:36:44

80

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score 91 · Answer 14 · 2026-04-12 20:05:42

91

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score 114 · Answer 15 · 2026-04-14 14:07:42

114

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List Question

prove or disprove $n > 0 ~\rightarrow~\prod \limits_{i=1}^{n} \left ( \frac{1}{2i~+~1} \cdot \frac{1}{2i~+~2} \right ) ~=~ \frac{1}{(2n~+~2)!}$

Prove or disprove $n \geq 2 ~\rightarrow~ \prod \limits_{i=1}^{n} \left ( 1 - \frac{1}{i^2} \right ) ~=~ \frac{n+1}{2n}$

Show $P(1), P(2),...,P(99)$ true statements but $P(100)$ is false.

Show by induction: $ \frac{a_{2n+4}}{a_{n+2}}=\frac{a_{2n+2}}{a_{n+1}}+\frac{a_{2n}}{a_{n}}$

Proving $n! < n^n$ by induction for all $n\geq 2$.

Combinatorial proof of $\sum_{k=1}^n k k!=(n+1)!-1$

How can I fix this proof using transfinite induction of the existence of bases of normed vector spaces?

Proving some property of a Formal Logic Language

Induction step of a power of 2 with ceiling function

For all integers $n \ge 1$, prove 6 divides $n(n+1)(n+2)$ by PMI.

Avoiding proof by induction

finite intersection of a family of sets same as countable intersection?

Induction proof - not sure how to proceed with next step

Fifth root of an even number

Monotonicity of n-th root by induction

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