Question

Prove using induction that $1^2 + 3^2 + 5^2 + . . . + (2n + 1)^2 = (n + 1) ∗ (2n + 1) ∗ (2n + 3)/3$

score 73 · Answer 1 · 2026-04-17 05:27:56

73

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Prove using induction that $1^2 + 3^2 + 5^2 + . . . + (2n + 1)^2 = (n + 1) ∗ (2n + 1) ∗ (2n + 3)/3$

Published on 17 Apr 2026 - 5:27

score 276 · Answer 2 · 2026-04-11 13:41:05

276

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Changing the order of finite double summations

Published on 11 Apr 2026 - 13:41

score 1.5k · Answer 3 · 2026-04-14 23:27:37

1.5k

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Prove that any integer amount of postage from 18 cents and up can be made using only 4-cent and 7-cent stamp. Assume an infinite supply of stamps

Published on 14 Apr 2026 - 23:27

score 108 · Answer 4 · 2026-04-11 03:57:52

108

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Randomize Fibonacci sequence

Published on 11 Apr 2026 - 3:57

score 32 · Answer 5 · 2026-03-27 01:13:43

32

Views

Summing cotangent pairs

Published on 27 Mar 2026 - 1:13

score 268 · Answer 6 · 2026-04-14 16:11:02

268

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Closed form for $\sum_{k=0}^{l}\binom{k}{n}\binom{k}{m}$

Published on 14 Apr 2026 - 16:11

score 210 · Answer 7 · 2026-03-26 02:53:01

210

Views

How to show convolution is associative?

Published on 26 Mar 2026 - 2:53

score 606 · Answer 8 · 2026-04-14 02:23:15

606

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Asymptotic expansion of $\sum _{k=1}^n \left(\frac{k}{n}\right)^k$

Published on 14 Apr 2026 - 2:23

score 93 · Answer 9 · 2026-04-16 14:11:55

93

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Simplifying $\arctan\frac{2}{1^2} + \arctan\frac{2}{2^2} + \arctan\frac{2}{3^2}+ \cdots+\arctan\frac{2}{n^2}$

Published on 16 Apr 2026 - 14:11

score 61 · Answer 10 · 2026-04-13 07:14:34

61

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Can every positive integer be a divisor of this sum?

Published on 13 Apr 2026 - 7:14

score 62 · Answer 11 · 2026-04-12 02:40:03

62

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Does $\ n\mid f(n)\ $ has infinite many solutions?

Published on 12 Apr 2026 - 2:40

score 125 · Answer 12 · 2026-04-17 14:39:39

125

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Proving identity related to $\sum_{k=0}^{\large\lfloor{\frac{n}{2}}\rfloor}\binom{n}{2k}\frac{x^{2k}}{2k+1}$

Published on 17 Apr 2026 - 14:39

score 270 · Answer 13 · 2026-04-13 12:00:36

270

Views

Quotient of two sums

Published on 13 Apr 2026 - 12:00

score 199 · Answer 14 · 2026-04-16 21:32:56

199

Views

How to prove: $\sum_{k=0}^{n}~(-1)^k{n \choose k} {2n-2k \choose n+m-1}=2^{n-m+1} {n \choose m-1}$

Published on 16 Apr 2026 - 21:32

score 87 · Answer 15 · 2026-04-15 15:34:26

87

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Evaluating $\sum_{i=0}^{n}\sum_{j=0}^{i-1}\binom{n}{i}\binom{n-1}{j}$

Published on 15 Apr 2026 - 15:34

List Question

Prove using induction that $1^2 + 3^2 + 5^2 + . . . + (2n + 1)^2 = (n + 1) ∗ (2n + 1) ∗ (2n + 3)/3$

Changing the order of finite double summations

Prove that any integer amount of postage from 18 cents and up can be made using only 4-cent and 7-cent stamp. Assume an infinite supply of stamps

Randomize Fibonacci sequence

Summing cotangent pairs

Closed form for $\sum_{k=0}^{l}\binom{k}{n}\binom{k}{m}$

How to show convolution is associative?

Asymptotic expansion of $\sum _{k=1}^n \left(\frac{k}{n}\right)^k$

Simplifying $\arctan\frac{2}{1^2} + \arctan\frac{2}{2^2} + \arctan\frac{2}{3^2}+ \cdots+\arctan\frac{2}{n^2}$

Can every positive integer be a divisor of this sum?

Does $\ n\mid f(n)\ $ has infinite many solutions?

Proving identity related to $\sum_{k=0}^{\large\lfloor{\frac{n}{2}}\rfloor}\binom{n}{2k}\frac{x^{2k}}{2k+1}$

Quotient of two sums

How to prove: $\sum_{k=0}^{n}~(-1)^k{n \choose k} {2n-2k \choose n+m-1}=2^{n-m+1} {n \choose m-1}$

Evaluating $\sum_{i=0}^{n}\sum_{j=0}^{i-1}\binom{n}{i}\binom{n-1}{j}$

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