已知Sn为数列{an}的前n项和,且有a1=1,Sn+1=an+1(n∈N*).(1)求数列{an}的通项公式;(2)若bn=n4an

2025-05-19 01:08:06
推荐回答(1个)
回答1:

(1)当n=1时,a2=S1+1=a1+1=2;                      
当n≥2时,Sn+1=an+1,Sn-1+1=an,兩式相减得,an+1=2an
又a2=2a1
∴{an}是首项为1,公比为2的等比数列,
∴an=2n-1
(2)由(1)知an=2n-1
∴bn=

n
4an
=
n
4?2n?1
n
2n+1

Tn
1
22
+
2
23
+
3
24
+…+
n
2n+1
1
2
Tn
1
23
+
2
24
+
3
25
+…+
n?1
2n+1
+
n
2n+2

两式相减得,
1
2
Tn
1
22
+
1
23
+
1
24
+…+
1
2n+1
?
n
2n+2

=
1
22
(1?
1
2n
)
1?
1
2
-
n
2n+2
=
1
2