Question

3/1x2x3+5/2x3x4+7/3x4x5+....+197/98x99x100

Answer 1

3/铅含1x2x3+5/2x3x4+7/3x4x5+....+197/98x99x100
是：裤橡
3/(1x2x3)+5/(2x3x4)+7/(3x4x5)+....+197/(98x99x100)吧。
3/(1x2x3)+5/(2x3x4)+7/(3x4x5)+....+197/(98x99x100)
即：
3/(1x2x3)+5/(2x3x4)+7/(3x4x5)+..(2n+1)/((n)(n+1)(n+2))..+197/(98x99x100) n=1,2...98
对式子中的任意一项：
(2n+1)/((n)(n+1)(n+2))=(n+(n+1))/((n)(n+1)(n+2))
=1/((n+1)(n+2))+1/(n(n+2))
对上式中的：
1/胡激旁((n+1)(n+2))=((n+2)-(n+1))/((n+1)(n+2))=1/(n+1)-1/(n+2)
同理，对上式中的：
1/(n(n+2))=(1/2)(2/(n(n+2)))=(1/2)((n+2-n)/(n(n+2)))=(1/2)(1/n-1/(n+2))
所以：
3/(1x2x3)+5/(2x3x4)+7/(3x4x5)+..(2n+1)/((n)(n+1)(n+2))..+197/(98x99x100) n=1,2...98
变形为：
(1/(2*3)+1/(1*3))+(1/(3*4)+1/(2*4))+...+(1/((n+1)(n+2))+1/(n(n+2)))+...+(1/(99*100)+1/(98+100))
=(1/(2*3)+(1/(3*4)+...+1/((n+1)(n+2))+...+1/(99*100))
+(1/(1*3)+1/(2*4)+...+1/(n(n+2))+...+1/(98*100))
=((1/2-1/3)+(1/3-1/4)+...+(1/(n+1)-1/(n+2))+...+(1/99-1/100))
+(1/2)((1-1/3)+(1/2-1/4)+(1/3-1/5)+(1/4-1/6)+...+(1/n-1/(n+2))+...+(1/98-1/100))
=(1/2-1/100)+(1/2)(1-1/99+1/2-1/100)
=1/2-1/100+3/4-1/198-1/200
=247/200-1/198
=24353/19800