Question

Prove $\lceil(3 + \sqrt{5})^n\rceil$ is divisible by $2^n$ for all natural numbers $n$

score 264 · Answer 1 · 2026-03-28 16:58:14

264

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Prove $\lceil(3 + \sqrt{5})^n\rceil$ is divisible by $2^n$ for all natural numbers $n$

Published on 28 Mar 2026 - 16:58

score 79 · Answer 2 · 2019-09-17 03:21:04

79

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Let $x,y \in \mathbb{R}$..Show that $x^{n+1}-y^{n+1}=(x-y) \sum\limits_{k=0}^n x^k y^{n-k}$ for all $n \in \mathbb{N}_0$

Published on 17 Sep 2019 - 3:21

score 45 · Answer 3 · 2019-09-17 11:24:18

45

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Prove recursively function f(x) is O(n)

Published on 17 Sep 2019 - 11:24

score 31 · Answer 4 · 2019-09-17 11:39:26

Prove that $\forall n\geq2,\;\exists \text{unique} \; t\in N $ such that $1+2+\ldots t < n \leq 1+2+3\ldots (1+t)$

score 59 · Answer 5 · 2026-03-26 22:55:33

59

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Then $F_a$ divides $F_{an}$, $n \in \Bbb{N}$ where $F_i$ denotes the $i$-th Fibonacci number.

Published on 26 Mar 2026 - 22:55

score 435 · Answer 6 · 2026-03-26 12:35:48

435

Views

Mathematical Induction: Why don't we just substitute $n$ by $(n+1)$?

Published on 26 Mar 2026 - 12:35

score 63 · Answer 7 · 2026-03-26 12:35:51

63

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Is $n\in\mathbb{Z^{+}}$ or $n\in\mathbb{N}$ more appropriate for proof by induction?

Published on 26 Mar 2026 - 12:35

score 69 · Answer 8 · 2019-09-18 17:44:51

69

Views

Proving by Mathematical Induction

Published on 18 Sep 2019 - 17:44

score 151 · Answer 9 · 2026-03-26 08:00:28

151

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$a_{n+1}\ge a_n$ for $a_n=\frac1{n+1}+\frac1{n+2}+\dots+\frac1{2n}$

Published on 26 Mar 2026 - 8:00

score 70 · Answer 10 · 2019-09-20 01:54:21

70

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How do you prove that $n^{1/n}>1$ for all $n\geq2$?

Published on 20 Sep 2019 - 1:54

score 352 · Answer 11 · 2026-03-26 04:30:43

352

Views

How can I prove this by well ordering principle?

Published on 26 Mar 2026 - 4:30

score 126 · Answer 12 · 2019-09-21 08:43:05

126

Views

$\sum_{k=1} ^n (k^2 +1)k!=n(n+1)!$

Published on 21 Sep 2019 - 8:43

score 65 · Answer 13 · 2019-09-21 11:13:32

65

Views

Mathematics Induction

Published on 21 Sep 2019 - 11:13

score 100 · Answer 14 · 2026-03-26 04:34:19

100

Views

Prove that the sequence $C_{n+1} = 1 + \frac{C_n}{C_n + 1}$ is monotonically increasing by induction

Published on 26 Mar 2026 - 4:34

score 59 · Answer 15 · 2019-09-22 15:11:45

59

Views

Is this a valid operation on binomial coefficients?

Published on 22 Sep 2019 - 15:11

List Question

Prove $\lceil(3 + \sqrt{5})^n\rceil$ is divisible by $2^n$ for all natural numbers $n$

Let $x,y \in \mathbb{R}$..Show that $x^{n+1}-y^{n+1}=(x-y) \sum\limits_{k=0}^n x^k y^{n-k}$ for all $n \in \mathbb{N}_0$

Prove recursively function f(x) is O(n)

Prove that $\forall n\geq2,\;\exists \text{unique} \; t\in N $ such that $1+2+\ldots t < n \leq 1+2+3\ldots (1+t)$

Then $F_a$ divides $F_{an}$, $n \in \Bbb{N}$ where $F_i$ denotes the $i$-th Fibonacci number.

Mathematical Induction: Why don't we just substitute $n$ by $(n+1)$?

Is $n\in\mathbb{Z^{+}}$ or $n\in\mathbb{N}$ more appropriate for proof by induction?

Proving by Mathematical Induction

$a_{n+1}\ge a_n$ for $a_n=\frac1{n+1}+\frac1{n+2}+\dots+\frac1{2n}$

How do you prove that $n^{1/n}>1$ for all $n\geq2$?

How can I prove this by well ordering principle?

$\sum_{k=1} ^n (k^2 +1)k!=n(n+1)!$

Mathematics Induction

Prove that the sequence $C_{n+1} = 1 + \frac{C_n}{C_n + 1}$ is monotonically increasing by induction

Is this a valid operation on binomial coefficients?

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