數 - ( 20 ) 3次方

Let n be the series

1 ^ 3 + 2 ^ 3 + 3 ^ 3 + ... + n ^ 3

when n=1
1 ^ 3 = 1

when n=2
1 ^ 3 + 2 ^ 3 =9 =(1+2)^2

when n=3
1 ^ 3 + 2 ^ 3 + 3 ^ 3 = 36 = (1+2+3)^2

So, we need to prove the sequence by mathmatical induction.
1 ^ 3 + 2 ^ 3 + 3 ^ 3 + ... + n^ 3 = (1+2+3+...+n)^2

Once we prove the above sequence, when n=200
1 ^ 3 + 2 ^ 3 + 3 ^ 3 + ... + 200 ^ 3
= (1+2+3+200)^2
= ((1+200)/2*200)^2
= 20100^2
= 404010000

As follows:

圖片參考：http://i707.photobucket.com/albums/ww74/stevieg90/01-111.jpg


2010-06-25 17:06:11 補充：
http://i707.photobucket.com/albums/ww74/stevieg90/01-111.jpg

2010-06-26 00:00:19 補充：
Alternative solution: (using formula to calculate)
http://i707.photobucket.com/albums/ww74/stevieg90/01-112.jpg

2010-06-26 00:01:31 補充：
For the derivation of Bernoulli's formula, you can refer to http://en.wikipedia.org/wiki/Bernoulli_number#Values_of_the_Bernoulli_numbers