已知数列{an}中,a1=1,an+1=n/n+1an. (1)写出数列的前5项; (2)猜想数列的通项公式.

问题描述:

已知数列{an}中,a1=1,an+1=

n
n+1
an
(1)写出数列的前5项;
(2)猜想数列的通项公式.

(1)∵数列{an}中,a1=1,an+1=

n
n+1
an
∴a2=
1
2
×1
=
1
2

a3=
2
3
×
1
2
=
1
3

a4=
3
4
×
1
3
=
1
4

a5=
4
5
×
1
4
=
1
5

(2)由数列的前5项,猜想an
1
n

用数学归纳法证明:
①当n=1时,a1
1
1
=1,成立;
②假设n=k时,等式成立,即ak
1
k

当n=k+1时,ak+1=
k
k+1
×
1
k
=
1
k+1
,也成立.
an
1
n