n平方的前n项和，非零自然数的平方求和

2022-11-04 09:09 作者:三十名八千里 0人读过 | 我要投稿

$S_%7Bn%7D%3D1%2B2%2B3%2B%20...%2Bn%3D%5Cfrac%7Bn%2B1%7D%7B2%7D%20%5Ccdot%20n$ ①

$M_%7Bn%7D%3D1%5E2%2B2%5E2%2B3%5E2%2B...%2Bn%5E2$ ②

$%E7%94%B1(n%2B1)%5E3-n%5E3%3D3n%5E2%2B3n%2B1%E6%9C%89%E5%A6%82%E4%B8%8B%EF%BC%9A$

$(n%2B1)%5E3-n%5E3%3D3n%5E2%2B3n%2B1$

$n%5E3-(n-1)%5E3%3D3(n-1)%5E3%2B3(n-1)%2B1$

......

$2%5E3-1%5E3%3D3%5Ctimes%201%5E2%2B3%5Ctimes%201%2B1$

将上式累加后有： $(n%2B1)%5E3-1%5E3%3D%5Csum_%7Bk%3D1%7D%5En%20(3k%5E2%2B3k%2B1)%3D3M_%7Bn%7D%20%2B3S_%7Bn%7D%20%2Bn$

$%E5%8C%96%E7%AE%80%E5%90%8E%E5%BE%97%EF%BC%9AM_%7Bn%7D%20%3D%5Cfrac%7B1%7D%7B6%7D%20n%5Ccdot%20(n%2B1)%5Ccdot%20(2n%2B1)$

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n平方的前n项和，非零自然数的平方求和的评论 (共条)