已知数列{an}是等差数列,且a3=5,a5=9,Sn是数列{an}的前n项和.(1)求数列{an}的通项公式an及前n项和Sn;(2)若数列{bn}满足bn=1Sn•Sn+1,且Tn是数列{bn}的

题目简介

已知数列{an}是等差数列,且a3=5,a5=9,Sn是数列{an}的前n项和.(1)求数列{an}的通项公式an及前n项和Sn;(2)若数列{bn}满足bn=1Sn•Sn+1,且Tn是数列{bn}的

题目详情

已知数列{an}是等差数列,且a3=5,a5=9,Sn是数列{an}的前n项和.
(1)求数列{an}的通项公式an及前n项和Sn
(2)若数列{bn}满足bn=
1
Sn
Sn+1
,且Tn是数列{bn}的前n项和,求bn与Tn
题型:解答题难度:中档来源:不详

答案

(1)设等差数列{an}的公差为d,
由题意可知:
a3=a1+2d=5
a5=a1+4d=9
,解得:a1=1,d=2
∴an=a1+(n-1)d=1+2(n-1)=2n-1,
Sn=
(a1+an)n
2
=
(1+2n-1)n
2
=n2

(2)由(1)得,Sn=n2,
bn=class="stub"1
Sn
Sn+1
=class="stub"1
n(n+1)
=class="stub"1
n
-class="stub"1
n+1

Tn=b1+b2+b3+…+bn
=(class="stub"1
1
-class="stub"1
2
)+(class="stub"1
2
-class="stub"1
3
)+(class="stub"1
3
-class="stub"1
4
)+…+(class="stub"1
n
-class="stub"1
n+1
)=1-class="stub"1
n+1
=class="stub"n
n+1
.

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