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1/1x2+1/2x3+1/3x4+1/4x5+……1/8x9+1/9x10=?(简便算...

1/1x2+1/2x3+1/3x4+1/4x5+……1/8x9+1/9x10 =1-1/2+1/2-1/3+1/3-1/4+....+1/8-1/9+1/9-1/10 =1-1/10 =9/10

1/1x2+1/2x3+1/3x4+1/4x5+1/5x6+1/6x7+1/7x8+1/8x9+1/9x10 =1-1/2+1/2-1/3+1/3-....-1/9+1/9-1/10 =1-1/10 =9/10

做了将近半个小时没做出来,突然想出来了,该题好像在课本上有,觉得太简单了。 将题目化为:1/nx(n+1)=1/n---1/(n+1)带入式子得:中间的全消掉了, 最后剩下的是1-1/2011=2010/2011!

=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7 =1-1/7 =6/7

=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9 =1-1/9 =8/9

=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5******+1/98-1/99+1/99-1/100 =1-1/100 =99/100

一般的,有: (n-1)n(n+1) =n^3-n {n^3}求和公式:Sn=[n(n+1)/2]^2 {n}求和公式:Sn=n(n+1)/2 1x2x3+2x3x4+3x4x5+....+7x8x9 =2^3-2+3^3-3+...+8^3-8 =(2^3+3^3+...+8^3)-(2+3+...+8) =[(8*9/2)^2-1]-8*9/2+1 =1260

1x2+2x3+3x4+4x5+...+n(n+1) =(1^2+2^2+……n^2)+(1+2+3+……n) =n(n+1)(2n+1)/6+(1+n)xn/2 =n(n+1)(n+2)/3

=2^2-1^2+3^2-2^2+.....33^2-32^2 =33^2-1

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