Question

己知数列{an}中，a3=2.a5=1.若{1/1+an}是等差数列，则a11=?请写一下过程，谢谢！

Answer 1

1/(1+an)是等差数列,则设公差是d

1/(1+a5)=1/(1+a3)+2d
1/(1+1)=1/(1+2)+2d
2d=1/2-1/3=1/6
d=1/12
那么有1/(1+a11)=1/(1+a3)+8d=1/(1+2)+8*1/12=1/3+2/3=1
故有a11=0

Answer 2

let
bn = 1/(1+an)
bn = b1+(n-1)d
b3 = b1+2d = 1/3 (1)
b5= b1+4d = 1/2 (2)
bn = n/6
b11 = 11/6 = 1/(1+a11)
11+11a11=6
a11=-5/11

Answer 3

1/1+a3=1/3 1/1+a5=1/2
2d=1/6 d=1/12
1/1+a11=1/(1+a5)+6d=1
a11=0

Answer 4

设bn=1/(1+an),由题目知道bn为等差数列，设其公差为d
有：b3=1/3
b5=1/2
2d=b5-b3=1/6
d=1/12
所以b11=b5+6d=1
所以1/(1+a11)=1
a11=0

Answer 5

设bn=1/(1+an)，已知bn是等差数列,则设公差是d
b5-b3=1/(1+a5)-1/(1+a3)=2d
b5-b3=1/(1+1)-1/(1+2)=2d
解得d=1/12，b11=b5+6d=1/2+1/2=1
所以1/（1+a11）=1，解得a11=0