Question

Prove by induction $(n+1) + (n + 2) +\dots + (2n - 1 ) + (2n) = \frac{n(3n + 1)}{2}$, $n \geq 1$

score 68 · Answer 1 · 2026-03-25 22:05:36

68

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Prove by induction $(n+1) + (n + 2) +\dots + (2n - 1 ) + (2n) = \frac{n(3n + 1)}{2}$, $n \geq 1$

Published on 25 Mar 2026 - 22:05

score 474 · Answer 2 · 2026-03-25 20:53:12

474

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How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

Published on 25 Mar 2026 - 20:53

score 31 · Answer 3 · 2026-03-29 02:17:32

31

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Analytical form of the binomial moments

Published on 29 Mar 2026 - 2:17

score 81 · Answer 4 · 2019-01-17 19:03:37

81

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What is the goal behind defining the notion of summability

Published on 17 Jan 2019 - 19:03

score 225 · Answer 5 · 2019-01-18 05:31:43

225

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Roots of unity (group theory)

Published on 18 Jan 2019 - 5:31

score 6.9k · Answer 6 · 2026-03-28 17:05:32

6.9k

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Let X be a Poisson random variable, show that $E[X^n] = \lambda E[(X+1)^{n-1}]$

Published on 28 Mar 2026 - 17:05

score 582 · Answer 7 · 2026-03-26 07:51:31

582

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How do we prove that $[n+1/2]+[n+2/4]+[n+4/8]+[n+8/16] ...=n$?

Published on 26 Mar 2026 - 7:51

score 1k · Answer 8 · 2019-01-19 03:52:19

1k

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Double Summation: Help needed with proof

Published on 19 Jan 2019 - 3:52

score 45 · Answer 9 · 2019-01-19 17:17:03

45

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computing the following sum with few calculations

Published on 19 Jan 2019 - 17:17

score 91 · Answer 10 · 2019-01-20 00:05:05

91

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How to define a function with summation?

Published on 20 Jan 2019 - 0:05

score 351 · Answer 11 · 2019-01-20 11:52:47

351

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Verifying $ \sum_{k=0}^{\infty}{3k \choose k}\frac{9k^2-3k-1}{(3k-1)(3k-2)}\left(\frac{2}{27}\right)^k=\frac{1}{4}$

Published on 20 Jan 2019 - 11:52

score 1.3k · Answer 12 · 2019-01-20 14:18:11

1.3k

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Notation and major confusion with the definition of the probability simplex

Published on 20 Jan 2019 - 14:18

score 468 · Answer 13 · 2019-01-20 22:19:35

468

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How many possible ways can $i+j+k=n$ without taking the order into account?

Published on 20 Jan 2019 - 22:19

score 52 · Answer 14 · 2026-03-27 10:30:15

52

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Summation using binomial theorem

Published on 27 Mar 2026 - 10:30

score 98 · Answer 15 · 2026-03-26 07:52:15

98

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Prove that $\sum\limits_{k=1}^n\lfloor n/k\rfloor+\lfloor \sqrt{n} \rfloor$ is even.

Published on 26 Mar 2026 - 7:52

List Question

Prove by induction $(n+1) + (n + 2) +\dots + (2n - 1 ) + (2n) = \frac{n(3n + 1)}{2}$, $n \geq 1$

How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

Analytical form of the binomial moments

What is the goal behind defining the notion of summability

Roots of unity (group theory)

Let X be a Poisson random variable, show that $E[X^n] = \lambda E[(X+1)^{n-1}]$

How do we prove that $[n+1/2]+[n+2/4]+[n+4/8]+[n+8/16] ...=n$?

Double Summation: Help needed with proof

computing the following sum with few calculations

How to define a function with summation?

Verifying $ \sum_{k=0}^{\infty}{3k \choose k}\frac{9k^2-3k-1}{(3k-1)(3k-2)}\left(\frac{2}{27}\right)^k=\frac{1}{4}$

Notation and major confusion with the definition of the probability simplex

How many possible ways can $i+j+k=n$ without taking the order into account?

Summation using binomial theorem

Prove that $\sum\limits_{k=1}^n\lfloor n/k\rfloor+\lfloor \sqrt{n} \rfloor$ is even.

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