已知{an}为等差数列,且a5=14,a7=20,数列{bn}的前n项和为Sn,且bn=2-2Sn(1)求数列{an},{bn}的通项公

2025-06-21 20:26:08
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回答1:

(1)数列{an}为等差数列,公差 d=

1
2
(a7?a5)=3,可得an=3n-1
由bn=2-2Sn,令n=1,则b1=2-2S1,又S1=b1
所以 b1
2
3
.b2=2-2(b1+b2),则 b2
2
9

当n≥2时,由bn=2-2Sn,可得bn-bn-1=-2(Sn-Sn-1)=-2bn.即
bn
bn?1
1
3
,所以{bn}是以 b1
2
3
为首项,
1
3
为公比的等比数列,于是 bn=2?
1
3n

(2)cnan?bn=2(3n?1)?
1
3n
Tn=2[2?
1
3
+5?
1
32
+8?
1
33
++(3n?1)?
1
3n
]
=
7
2
?
7
2
?
1
3n
?
n
3n?1
7
2