已知数列{an}的前n项和为Sn(n∈N*),a1=23,且当n≥2时,SnSn-1-3Sn+2=0.(Ⅰ)求a2,a3的值;(Ⅱ)若

2025-04-15 09:13:25
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回答1:

(Ⅰ)∵当n≥2时,snsn-1-3sn+2=0,a1

2
3

∴当n=2时,
2
3
s2?3s2+2=0
,解得s2
6
7

a2s2?a1
6
7
?
2
3
4
21

当n=3时,
6
7
s3?3s3+2=0
,解得s3
14
15
,可得a3
8
105

(Ⅱ)当n≥2时,snsn-1-3sn+2=0,由bn
1
Sn?1
sn=1+
1
bn

于是(1+
1
bn
)(1+
1
bn?1
)?3(1+
1
bn
)+2=0

化简,得bn=2bn-1-1,从而bn-1=2(bn-1-1),
∴{bn-1}是以2为公比的等比数列.∴bn-1=(b1-1)?2n-1=-2n+1,bn=-2n+1+1.
(Ⅲ)由(2),得
Sn+1?1
Sn?1
=
bn
bn+1
=
1?2n+1
1?2n+2
=
1
2
?
1
8?2n?2
=
1
2
?
1
7?2n