已知等差数列{an}的首项a1=1,公差d>0,且a2,a5,a14成等比数列.(1)求数列{an}的通项公式;(2)设bn=1n(an+3)(n∈N*),Sn=b1+b2+…+bn,求Sn>136.

题目简介

已知等差数列{an}的首项a1=1,公差d>0,且a2,a5,a14成等比数列.(1)求数列{an}的通项公式;(2)设bn=1n(an+3)(n∈N*),Sn=b1+b2+…+bn,求Sn>136.

题目详情

已知等差数列{an}的首项a1=1,公差d>0,且a2,a5,a14成等比数列.
(1)求数列{an}的通项公式;
(2)设bn=
1
n(an+3)
(n∈N*),Sn=b1+b2+…+bn
,求Sn
1
36
题型:解答题难度:中档来源:不详

答案

(1)∵等差数列{an}的首项a1=1,公差d>0,且a2,a5,a14成等比数列,
(a1+d)(a1+13d)=(a1+4d)2
整理得:2a1d=d2
∵a1=1,解得d=2(d=0舍去)
an=2n-1(n∈N*)
(2)bn=class="stub"1
n(an+3)
=class="stub"1
2n(n+1)
=class="stub"1
2
(class="stub"1
n
-class="stub"1
n+1
)

∴Sn=b1+b2+…+bn
=class="stub"1
2
[(1-class="stub"1
2
)+(class="stub"1
2
-class="stub"1
3
)+…+(class="stub"1
n
-class="stub"1
n+1
)]

=class="stub"1
2
(1-class="stub"1
n+1
)

∴当n=1时,Sn取最小值S1=class="stub"1
2
(1-class="stub"1
2
)
=class="stub"1
4
>class="stub"1
36

Sn>class="stub"1
36

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