若bn=1/anan+1,求数列｛bn｝的前100项和已知等差数列｛an｝的前n项和为Sn,a5=5,S5=15

Question

若bn=1/anan+1,求数列｛bn｝的前100项和
已知等差数列｛an｝的前n项和为Sn,a5=5,S5=15

Answer 1

an=a1+(n-1)d
a5=5
a1+4d=5 (1)
S5=15
a1+2d=3 (2)
(1)-(2)
d=1
from (1) =>a1=1
ie
an=n
bn = 1/[an.a(n+1)]
= 1/n -1/(n+1)
Tn =b1+b2+...+bn
= 1- 1/(n+1)
=n/(n+1)
T100=b1+b2+...+b100 = 100/101