Question

Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, ... ,n$, $f^{(r)}(x)$ is a polynomial with value $0$ at no fewer than $r$ distinct points on $(-1,1)$.

score 935 · Answer 1 · 2026-04-14 20:32:06

935

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Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, ... ,n$, $f^{(r)}(x)$ is a polynomial with value $0$ at no fewer than $r$ distinct points on $(-1,1)$.

Published on 14 Apr 2026 - 20:32

score 142 · Answer 2 · 2026-04-16 05:45:23

142

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Proving $2^n > (n+1)^2$ for $n\geq 6$ by induction.

Published on 16 Apr 2026 - 5:45

score 229 · Answer 3 · 2026-04-17 08:41:45

229

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Induction Proof: $\frac{1\cdot 3\cdots (2n-1)}{2\cdot 4\cdots (2n)} \geq \frac{1}{2n}$

Published on 17 Apr 2026 - 8:41

score 965 · Answer 4 · 2026-04-14 12:00:06

965

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Prove that a sequence is increasing

Published on 14 Apr 2026 - 12:00

score 489 · Answer 5 · 2026-04-15 18:59:26

489

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Induction Question. Suppose there are n teams in a football league

Published on 15 Apr 2026 - 18:59

score 483 · Answer 6 · 2026-04-14 20:14:04

Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $

score 1.8k · Answer 7 · 2026-04-14 22:47:29

1.8k

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Prove $k^2>k+1$ by induction

Published on 14 Apr 2026 - 22:47

score 173 · Answer 8 · 2026-04-11 23:07:52

173

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Proving a function by induction

Published on 11 Apr 2026 - 23:07

score 343 · Answer 9 · 2026-04-14 20:35:54

343

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Using induction to study the sequence $\sqrt{6} , \sqrt{6 +\sqrt{6}}, \dots$

Published on 14 Apr 2026 - 20:35

score 1.6k · Answer 10 · 2026-04-15 10:32:05

1.6k

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Proving that $7^n(3n+1)-1$ is divisible by 9

Published on 15 Apr 2026 - 10:32

score 164 · Answer 11 · 2026-04-11 17:00:09

164

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Proof by exhaustion:

Published on 11 Apr 2026 - 17:00

score 78 · Answer 12 · 2026-04-16 07:15:28

78

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How to prove $2^{n+1} * 2^{n+1} = (2^n2^n)+(2^n2^n)+(2^n2^n)+(2^n2^n)$

Published on 16 Apr 2026 - 7:15

score 412 · Answer 13 · 2026-04-16 00:48:51

412

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Proving something about the sequence of powers of 3 mod 10. Oh boy

Published on 16 Apr 2026 - 0:48

score 104 · Answer 14 · 2026-04-10 17:11:11

104

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Mathematical induction problem with inequality

Published on 10 Apr 2026 - 17:11

score 69 · Answer 15 · 2026-04-16 21:03:23

69

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Showing that if $xf(x)=\log x$ for $x>0$, then $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac12+\cdots+\frac{1}{n}\right)$

Published on 16 Apr 2026 - 21:03

List Question

Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, ... ,n$, $f^{(r)}(x)$ is a polynomial with value $0$ at no fewer than $r$ distinct points on $(-1,1)$.

Proving $2^n > (n+1)^2$ for $n\geq 6$ by induction.

Induction Proof: $\frac{1\cdot 3\cdots (2n-1)}{2\cdot 4\cdots (2n)} \geq \frac{1}{2n}$

Prove that a sequence is increasing

Induction Question. Suppose there are n teams in a football league

Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $

Prove $k^2>k+1$ by induction

Proving a function by induction

Using induction to study the sequence $\sqrt{6} , \sqrt{6 +\sqrt{6}}, \dots$

Proving that $7^n(3n+1)-1$ is divisible by 9

Proof by exhaustion:

How to prove $2^{n+1} * 2^{n+1} = (2^n2^n)+(2^n2^n)+(2^n2^n)+(2^n2^n)$

Proving something about the sequence of powers of 3 mod 10. Oh boy

Mathematical induction problem with inequality

Showing that if $xf(x)=\log x$ for $x>0$, then $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac12+\cdots+\frac{1}{n}\right)$

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