学帮网1/1*2*3+1/2*3*4+1/3*4*5+...+1/98*99*100等于多少?
来源：学生作业学帮网 编辑：学帮网 时间：2024/05/05 20:50:30
1/1*2*3+1/2*3*4+1/3*4*5+...+1/98*99*100等于多少?
an=1/2n+1/2(n+2)-1/(n+1)
1/1*2*3+1/2*3*4+1/3*4*5+...+1/98*99*100
=(1/2+1/6-1/2)+(1/4+1/8-1/3)+...+(1/196+1/200-1/99)
=0.5*(1+1/2+...+1/98)+0.5*(1/3+1/4+...+1/100)-(1/2+1/3+1/4+...+1/99)
=0.5*(1+1/2+1/99+1/100)-1/2-1/99
=4949/19800
1/1*2*3+1/2*3*4+1/3*4*5+...+1/98*99*100
=(1-1/2-1/3)+(1/2-1/3-1/4)+...+(1/98-1/99-1/100)
=1-1/99-1/99-1/100
=9900/9900-200/9900-99/9900
=9601/9900
大概就是这样了
an=(1/n*(n+1)-1/(n+1)(n+2))/2
原式=[(1/1*2-1/2*3)+(1/2*3-1/3*4)+...+(1/(98*99-1/99*100)]/2
=(1/2-1/99*100)/2
=(4950-1)/9900/2
=4949/19800