Question

已知数列 1/1*2 ,1/2*3,1/3*4,…,1/n(n+1),… Sn为其前n项和

Answer 1

S1=1/2

S2=2/3

S3=3/4

Sn=n/(n+1)

证明：

当n=k时,若假设成立,则

Sk=k/(k+1)

S(k+1)=Sk+a(k+1)

a(k+1)=1/(k+1)(k+2)

S(k+1)=Sk+a(k+1)

=k/(k+1)+1/(k+1)(k+2)

=1-1/(k+1)+1/(k+1)-1/(k+2)

=1-1/(k+2)

=(k+1)/(k+2)

假设成立