已知公差不为零的等差数列{an}满足a3=5,且a1,a2,a5成等比数列.(1)求数列{an}的通项公式an;(2)设Sn为数列{an}的前n项和,数列{bn}满足bn=2n•Sn,求数列{bn}的

题目简介

已知公差不为零的等差数列{an}满足a3=5,且a1,a2,a5成等比数列.(1)求数列{an}的通项公式an;(2)设Sn为数列{an}的前n项和,数列{bn}满足bn=2n•Sn,求数列{bn}的

题目详情

已知公差不为零的等差数列{an}满足a3=5,且a1,a2,a5成等比数列.
(1)求数列{an}的通项公式an
(2)设Sn为数列{an}的前n项和,数列{bn}满足bn=2n
Sn
,求数列{bn}的前n项和Tn
题型:解答题难度:中档来源:不详

答案

(1)∵a1,a2,a5成等比数列,a3=5
a22=a1a3
∴(5-d)2=(5-2d)(5+2d)
∵d≠0
∴d=2
∴an=a3+(n-3)d=5+2(n-3)=2n-1
(2)由(1)可得,Sn=class="stub"1+2n-1
2
×n
=n2
bn=2n
Sn
=n•2n
Tn=1•2+2•22+3•23+…+n•2n
∴2Tn=1•22+2•23+…+(n-1)•2n+n•2n+1
两式相减可得,-Tn=2+22+23+…+2n-n•2n+1
=
2(1-2n)
1-2
-n•2n+1
=2n+1-2-n•2n+1
Tn=(n-2)•2n+1+2

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