设数列{an}满足Sn=n^2+1,Pn=1/a1.a2+1/a2.a3+.+1/an.an+1,求n=?

Question

设数列{an}满足Sn=n^2+1,Pn=1/a1.a2+1/a2.a3+.+1/an.an+1,求n=?

Answer 1

Sn=n^2+1 S(n-1)=(n-1)^2+1
an=Sn-S(n-1)=2n-1
1/an*a(n+1)=1/(2n-1)(2n+1)=(1/2)[1/(2n-1)-1/(2n+1)]
Pn=1/1*3+1/3*5+...+1/(2n-1)(2n+1)
=(1/2)(1-1/3)+(1/2)(1/3-1/5)+(1/2)[1/(2n-1)-1/(2n+1)]
=(1/2)[1-1/(2n+1)]
=n/(2n+1)