Question

Prove by mathematical induction that $(7n + 1)6^n + (−1)^{n+1}$ is divisible by $49$ for all $n \in \mathbb{Z}^+$.

score 183 · Answer 1 · 2026-04-14 12:06:42

Prove by mathematical induction that $(7n + 1)6^n + (−1)^{n+1}$ is divisible by $49$ for all $n \in \mathbb{Z}^+$.

score 121 · Answer 2 · 2026-04-24 18:00:56

121

Views

Prove that $a_n = \sum_{k=0}^{\lfloor n/2\rfloor} 1/(k+1) {2k\choose k} {n\choose 2k}$

Published on 24 Apr 2026 - 18:00

score 36 · Answer 3 · 2026-04-16 20:56:09

36

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Express matrix whose components are coefficients of convex combination that generates $m^n$

Published on 16 Apr 2026 - 20:56

score 94 · Answer 4 · 2026-04-10 18:59:55

94

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Proof by induction for positive real numbers

Published on 10 Apr 2026 - 18:59

score 93 · Answer 5 · 2026-04-14 12:04:26

93

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Prove that the solution to Moser's circle problem is not $2^{n-1}$

Published on 14 Apr 2026 - 12:04

score 139 · Answer 6 · 2026-04-16 14:46:10

139

Views

Proving the Number Theory Property: $\delta(n^k) = \delta(\delta(n)^k)$ for natural $n$ and $k$

Published on 16 Apr 2026 - 14:46

score 347 · Answer 7 · 2026-04-15 10:07:24

347

Views

Induction proof that for all $n$ girls and $n$ boys there is never less girls than boys

Published on 15 Apr 2026 - 10:07

score 291 · Answer 8 · 2026-04-15 17:55:31

291

Views

A finite group $G$ with $|G|$ divisible by $2^n$ has a subgroup of order $2^n$

Published on 15 Apr 2026 - 17:55

score 100 · Answer 9 · 2026-04-14 02:19:41

100

Views

Proof by induction by length of the formula:

Published on 14 Apr 2026 - 2:19

score 52 · Answer 10 · 2026-04-14 15:30:15

52

Views

Proving convergence of a sequence using an initial assumption that it does indeed converge. How to jump from step 5 to step 6?

Published on 14 Apr 2026 - 15:30

score 43 · Answer 11 · 2026-04-15 10:34:38

43

Views

Using a theorem proved inductively to go "downards"

Published on 15 Apr 2026 - 10:34

score 27 · Answer 12 · 2026-04-16 01:34:10

27

Views

Proving that if $n$ hemispheres cover a sphere, it is possible to choose 4 hemisphere that also cover the sphere.

Published on 16 Apr 2026 - 1:34

score 74 · Answer 13 · 2026-04-14 16:37:38

Use mathematical induction to prove that $5^{2n+1}$ + $3^{2n+1}$ is divisible by $4$ for all natural numbers $n$.

score 115 · Answer 14 · 2026-04-24 14:27:26

115

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Proof identities (i)$\sum_{k=0}^{n}\frac{1}{k+1}\binom{n}{k}=\frac{2^{n+1}-1}{n+1}$ and (ii) $\sum_{k=0}^{m}\binom{n+k}{k}=\binom{n+m+1}{n+1}$

Published on 24 Apr 2026 - 14:27

score 174 · Answer 15 · 2026-04-16 03:10:02

174

Views

Show that if $f\{1,\dots,n\}\to\{1,\dots,n\}$ is surjective then it is also bijective.

Published on 16 Apr 2026 - 3:10

List Question

Prove by mathematical induction that $(7n + 1)6^n + (−1)^{n+1}$ is divisible by $49$ for all $n \in \mathbb{Z}^+$.

Prove that $a_n = \sum_{k=0}^{\lfloor n/2\rfloor} 1/(k+1) {2k\choose k} {n\choose 2k}$

Express matrix whose components are coefficients of convex combination that generates $m^n$

Proof by induction for positive real numbers

Prove that the solution to Moser's circle problem is not $2^{n-1}$

Proving the Number Theory Property: $\delta(n^k) = \delta(\delta(n)^k)$ for natural $n$ and $k$

Induction proof that for all $n$ girls and $n$ boys there is never less girls than boys

A finite group $G$ with $|G|$ divisible by $2^n$ has a subgroup of order $2^n$

Proof by induction by length of the formula:

Proving convergence of a sequence using an initial assumption that it does indeed converge. How to jump from step 5 to step 6?

Using a theorem proved inductively to go "downards"

Proving that if $n$ hemispheres cover a sphere, it is possible to choose 4 hemisphere that also cover the sphere.

Use mathematical induction to prove that $5^{2n+1}$ + $3^{2n+1}$ is divisible by $4$ for all natural numbers $n$.

Proof identities (i)$\sum_{k=0}^{n}\frac{1}{k+1}\binom{n}{k}=\frac{2^{n+1}-1}{n+1}$ and (ii) $\sum_{k=0}^{m}\binom{n+k}{k}=\binom{n+m+1}{n+1}$

Show that if $f\{1,\dots,n\}\to\{1,\dots,n\}$ is surjective then it is also bijective.

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