已经等差数列｛an｝的前n项和为Sn,a5=5'S5=15,则数列｛1/an×an+1｝的前100项和为?
问题描述：
已经等差数列｛an｝的前n项和为Sn,a5=5'S5=15,则数列｛1/an×an+1｝的前100项和为?
答
设公差为d
S5=5a3=15
a3=3
a5-a3=2d=5-3=2
d=1
an=a1+(n-1)d=a3+(n-3)d=3+1×(n-3)=n
1/[an·a(n+1)]=1/[n(n+1)]=1/n -1/(n+1)
S100=1/(a1·a2)+1/(a2·a3)+...+1/[an·a(n+1)]
=1/1-1/2+1/2-1/3+...+1/100-1/101
=1- 1/101
=100/101