Question

Can someone explain the logic behind this step in a induction problem

score 47 · Answer 1 · 2014-07-23 20:08:38

47

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Can someone explain the logic behind this step in a induction problem

Published on 23 Jul 2014 - 20:08

score 83 · Answer 2 · 2014-07-24 05:24:59

83

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Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

Published on 24 Jul 2014 - 5:24

score 1.1k · Answer 3 · 2014-07-24 06:19:30

1.1k

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Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

Published on 24 Jul 2014 - 6:19

score 1.1k · Answer 4 · 2014-07-24 17:04:52

1.1k

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Inductive proof that if there is an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$

Published on 24 Jul 2014 - 17:04

score 617 · Answer 5 · 2014-07-24 18:20:09

617

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Sum of nth powers and generalized polynomial sum

Published on 24 Jul 2014 - 18:20

score 83 · Answer 6 · 2014-07-25 02:38:56

83

Views

If $a_i>o$ then $(a_1a_2\cdots a_{2^n})^{1/2^n}\leq \frac{a_1+a_2+\cdots+a_{2^n}}{2^n}$

Published on 25 Jul 2014 - 2:38

score 158 · Answer 7 · 2026-04-01 07:35:10

158

Views

Uniqueness in transfinite recursion.

Published on 01 Apr 2026 - 7:35

score 177 · Answer 8 · 2014-07-25 17:28:11

177

Views

Proving properties of closures using intersection of indexed sets and topology

Published on 25 Jul 2014 - 17:28

score 135 · Answer 9 · 2026-03-26 04:34:14

135

Views

Elementary Proof of sum of alternating factorials

Published on 26 Mar 2026 - 4:34

score 133 · Answer 10 · 2014-07-28 06:12:25

133

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A sequence defined as $a(n)=n-a(a(n-1))$ $n\geq 1,\ a(0)=0$, how to prove that $a(n)=⌊(n+1)(-1+√5)/2⌋$

Published on 28 Jul 2014 - 6:12

score 2.1k · Answer 11 · 2026-03-27 06:52:00

2.1k

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Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

Published on 27 Mar 2026 - 6:52

score 45 · Answer 12 · 2014-07-30 05:44:55

45

Views

Expanding Base 2B representation of an integer

Published on 30 Jul 2014 - 5:44

score 481 · Answer 13 · 2026-03-26 11:04:19

481

Views

Induction inequality on sum of reciprocals

Published on 26 Mar 2026 - 11:04

score 535 · Answer 14 · 2026-03-31 08:10:52

535

Views

Prove that $(2n+1)+(2n+3)+\dots +(4n-1) = 3n^2$ by induction

Published on 31 Mar 2026 - 8:10

score 354 · Answer 15 · 2026-04-03 17:37:01

354

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Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

Published on 03 Apr 2026 - 17:37

List Question

Can someone explain the logic behind this step in a induction problem

Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

Inductive proof that if there is an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$

Sum of nth powers and generalized polynomial sum

If $a_i>o$ then $(a_1a_2\cdots a_{2^n})^{1/2^n}\leq \frac{a_1+a_2+\cdots+a_{2^n}}{2^n}$

Uniqueness in transfinite recursion.

Proving properties of closures using intersection of indexed sets and topology

Elementary Proof of sum of alternating factorials

A sequence defined as $a(n)=n-a(a(n-1))$ $n\geq 1,\ a(0)=0$, how to prove that $a(n)=⌊(n+1)(-1+√5)/2⌋$

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

Expanding Base 2B representation of an integer

Induction inequality on sum of reciprocals

Prove that $(2n+1)+(2n+3)+\dots +(4n-1) = 3n^2$ by induction

Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

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