已知数列{an}中,a3=2,a7=1,若{1/(an+1)}为等差数列,则a11=?

已知数列{an}中,a3=2,a7=1,若{1/(an+1)}为等差数列,则a11=?
已知数列{an}中,a3=2,a7=1,若{1/(an+1)}为等差数列,则a11=?

已知数列{an}中,a3=2,a7=1,若{1/(an+1)}为等差数列,则a11=?
a3=2,a7=1,若{1/(an+1)}为等差数列
公差d=（1/a7+1 -1/a3+1)/(7-3) =1/24
1/(a11 +1) =1/3 +1/24 *(11-3) =2/3
a11=1/2

可设bn=1/(1+an) n=1,2,3,..
易知, bn是等差数列
b3=1/3=b1+2d
b7=1/2=b1+6d
解得d=1/24 b1=1/4
∴bn=(n+5)/24
b11=16/24=2/3
∴2/3=1/(1+a11)
∴a11=1/2

设数列 {1an+1}的公差为d
∵数列{an}中， a3=2，a7=1，数列{1an+1}是等差数列
∴ 1a7+1=1a3+1+4d
将a3=2，a7=1代入得：d= 124
∵ 1a11+1=1a7+1+4d
∴a11= 12

1/2-1/3=1/6
1/2+1/6=1/(a11+1)=2/3
a11=1/2