Question

Proving $\lim\limits_{x \to a} \frac{x^n-a^n}{x-a}=na^{n-1}$ by induction

score 241 · Answer 1 · 2018-06-26 15:08:49

241

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Proving $\lim\limits_{x \to a} \frac{x^n-a^n}{x-a}=na^{n-1}$ by induction

Published on 26 Jun 2018 - 15:08

score 104 · Answer 2 · 2018-06-27 03:29:47

104

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Strong Principle of Induction

Published on 27 Jun 2018 - 3:29

score 46 · Answer 3 · 2018-06-28 12:31:05

46

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What am I doing wrong in this induction question?

Published on 28 Jun 2018 - 12:31

score 993 · Answer 4 · 2018-06-28 22:16:46

993

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Prove by induction that $2^n\le (n+1)!$ for all $n\in\Bbb N$.

Published on 28 Jun 2018 - 22:16

score 314 · Answer 5 · 2026-03-27 00:09:51

314

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Asymptotic bound in induction

Published on 27 Mar 2026 - 0:09

score 113 · Answer 6 · 2018-06-30 11:27:27

113

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Principle of Strong Induction

Published on 30 Jun 2018 - 11:27

score 1.8k · Answer 7 · 2018-06-30 12:25:50

1.8k

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How to prove that $x^2+y^2+z^2=14^n$ holds for distinct integers $x,y,z$ for every natural $n$?

Published on 30 Jun 2018 - 12:25

score 118 · Answer 8 · 2018-06-30 12:41:33

118

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On Assumptions In Induction

Published on 30 Jun 2018 - 12:41

score 183 · Answer 9 · 2026-03-26 02:35:45

183

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Prove that $\left(\sum_{k=1}^n\,\left(\frac{k}{n}\right)^n\right)^{\frac{1}{n}}$ decreases as $n$ increases.

Published on 26 Mar 2026 - 2:35

score 354 · Answer 10 · 2026-03-25 05:59:48

354

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Solve a recursive equation about chess

Published on 25 Mar 2026 - 5:59

score 250 · Answer 11 · 2026-03-25 23:51:48

250

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Prove that $5^{2^n} - 1$ is divisible by $2^{n+1}$ for all $n ≥ 1$

Published on 25 Mar 2026 - 23:51

score 263 · Answer 12 · 2018-07-03 08:10:22

263

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Use mathematical induction to prove that for all positive integers n, $ \sum_{r=1}^n (2r-1)\cdot 2^{-r} = 3- \frac{2n+3}{2^n} $.

Published on 03 Jul 2018 - 8:10

score 171 · Answer 13 · 2018-07-03 20:33:27

171

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Von Neumann integers and induction

Published on 03 Jul 2018 - 20:33

score 97 · Answer 14 · 2018-07-04 03:44:31

97

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Summation of series: How can we prove that $\left(\sum_{m=1}^{n}{m}\right)^2=\sum_{m=1}^{n}{m^3}$ holds true?

Published on 04 Jul 2018 - 3:44

score 511 · Answer 15 · 2026-03-25 14:33:19

511

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Prove by induction $|u - y| < \delta \Rightarrow |u^{n} - y^{n}| < \epsilon$

Published on 25 Mar 2026 - 14:33

List Question

Proving $\lim\limits_{x \to a} \frac{x^n-a^n}{x-a}=na^{n-1}$ by induction

Strong Principle of Induction

What am I doing wrong in this induction question?

Prove by induction that $2^n\le (n+1)!$ for all $n\in\Bbb N$.

Asymptotic bound in induction

Principle of Strong Induction

How to prove that $x^2+y^2+z^2=14^n$ holds for distinct integers $x,y,z$ for every natural $n$?

On Assumptions In Induction

Prove that $\left(\sum_{k=1}^n\,\left(\frac{k}{n}\right)^n\right)^{\frac{1}{n}}$ decreases as $n$ increases.

Solve a recursive equation about chess

Prove that $5^{2^n} - 1$ is divisible by $2^{n+1}$ for all $n ≥ 1$

Use mathematical induction to prove that for all positive integers n, $ \sum_{r=1}^n (2r-1)\cdot 2^{-r} = 3- \frac{2n+3}{2^n} $.

Von Neumann integers and induction

Summation of series: How can we prove that $\left(\sum_{m=1}^{n}{m}\right)^2=\sum_{m=1}^{n}{m^3}$ holds true?

Prove by induction $|u - y| < \delta \Rightarrow |u^{n} - y^{n}| < \epsilon$

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