Find, in terms of n and k,
圖片參考：http://imgcld.yimg.com/8/n/HA00266150/o/700911020012613873405840.jpg
where k is a positive integer
數學挑戰題
2009-11-02 9:36 am
回答 (3)
2009-11-13 2:24 am
✔ 最佳答案
Denote Sk (n)=1^k + 2^k + ... + n^k. There exist a recursive formula such that you can use the previous formula of Sk-1 (n), Sk-2 (n),... S1 (n) to find out Sk (n)k+1C1Sk (n)+ k+1C2Sk-1 (n)+...+k+1Ck+1S0 (n)=(n+1)k+1-1
where S0 (n)=n.
For example, 2C1S1 (n)+2C2S0 (n)=(n+1)2-1
S1 (n)=(n2+2n-n)/2=n(n+1)/2
There is a similar formula which is related to Bernoulli numbers. However, since to find out those Bernoulli numbers also requires using other recursive formulae. So, I omit them here.
2009-11-03 2:37 am
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