数列{an}的前n项和Sn,若an=1/n(n+1),则S100

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数列{an}的前n项和Sn,若an=1/n(n+1),则S100
数列{an}的前n项和Sn,若an=1/n(n+1),则S100

数列{an}的前n项和Sn,若an=1/n(n+1),则S100
S100
=1/(1*2)+1/(2*3)+...+1/(100*101)
=(1-1/2)+(1/2-1/3)+...+(1/100-1/101)
=1-1/101
=100/101

a(n)=1/n-1/(n+1) (1)
a1=1-1/2 (2)
a2=1/2-1/3 (3)
......
. ......
a100=1/99-1/100. (100)
从(1)加到(100),左边是是S100, 右边是1-1/100
左边等于右边即是 S100=1-1/100=99/100

由题可知 a1=1/2;
an=1/n(n+1)={(n+1)-1}/n(n+1)=1/n-1/(n+1);
S100=(1-1/2)+(1/2-1/3)+...+(1/100-1/101)
=1-1/101
=100/101