设数列{an}的前n项和为Sn,a1=1,Sn=nan-2n(n-1).(I)求证:数列{an}是等差数列;(II)设数列{1anan+1}的前n项和为Tn,求Tn.-数学

题目简介

设数列{an}的前n项和为Sn,a1=1,Sn=nan-2n(n-1).(I)求证:数列{an}是等差数列;(II)设数列{1anan+1}的前n项和为Tn,求Tn.-数学

题目详情

设数列{an}的前n项和为Sn,a1=1,Sn=nan-2n(n-1).
(I)求证:数列{an}是等差数列;
(II)设数列{
1
anan+1
}
的前n项和为Tn,求Tn
题型:解答题难度:中档来源:不详

答案

(I)由Sn=nan-2n(n-1),
则Sn+1=nan+1-2(n+1)n,
又由an+1=Sn+1-Sn可得an+1=Sn+1-Sn=(n+1)an+1-nan-4n,
即an+1-an=4,
则数列{an}是以1为首项,4为公差的等差数列;
(II)由(1)可得an=4n-3.
Tn=class="stub"1
a1a2
+…+class="stub"1
anan+1

=class="stub"1
1×5
+class="stub"1
5×9
+class="stub"1
9×13
+…+class="stub"1
(4n-3)×(4n+1)

=class="stub"1
4
(1-class="stub"1
5
+class="stub"1
5
-class="stub"1
9
+class="stub"1
9
-class="stub"1
13
+…+class="stub"1
4n-3
-class="stub"1
4n+1
)

=class="stub"1
4
(1-class="stub"1
4n+1
)

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