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=== Powers and logarithm of arithmetic progressions === : <math>\sum_{i=1}^n c = nc\quad</math> for every {{mvar|c}} that does not depend on {{mvar|i}} : <math>\sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}\qquad</math> (Sum of the simplest [[arithmetic progression]], consisting of the first ''n'' natural numbers.){{r|CRC|p=52}} : <math>\sum_{i=1}^n (2i-1) = n^2\qquad</math> (Sum of first odd natural numbers) : <math>\sum_{i=0}^{n} 2i = n(n+1)\qquad</math> (Sum of first even natural numbers) : <math>\sum_{i=1}^{n} \log i = \log (n!)\qquad</math> (A sum of [[logarithm]]s is the logarithm of the product) : <math>\sum_{i=0}^n i^2 = \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}\qquad</math> (Sum of the first [[square number|squares]], see [[square pyramidal number]].) {{r|CRC|p=52}} : <math>\sum_{i=0}^n i^3 = \biggl(\sum_{i=0}^n i \biggr)^2 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4}\qquad</math> ([[Nicomachus's theorem]]) {{r|CRC|p=52}} More generally, one has [[Faulhaber's formula]] for <math>p>1</math> : <math> \sum_{k=1}^n k^{p} = \frac{n^{p+1}}{p+1} + \frac{1}{2}n^p + \sum_{k=2}^p \binom p k \frac{B_k}{p-k+1}\,n^{p-k+1},</math> where <math>B_k</math> denotes a [[Bernoulli number]], and <math>\binom p k</math> is a [[binomial coefficient]].
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