已知数列{an}的前n项和为Sn,首项a1=1,且对于任意n∈N+都有nan+1=2Sn.(Ⅰ)求{an}的通项公式;(Ⅱ)设bn=4an+1an2an+22,且数列{bn}的前n项之和为Tn,求证:

题目简介

已知数列{an}的前n项和为Sn,首项a1=1,且对于任意n∈N+都有nan+1=2Sn.(Ⅰ)求{an}的通项公式;(Ⅱ)设bn=4an+1an2an+22,且数列{bn}的前n项之和为Tn,求证:

题目详情

已知数列{an}的前n项和为Sn,首项a1=1,且对于任意n∈N+都有nan+1=2Sn
(Ⅰ)求{an}的通项公式;
(Ⅱ)设bn=
4an+1
an2an+22
,且数列{bn}的前n项之和为Tn,求证:Tn
5
4
题型:解答题难度:中档来源:不详

答案

(Ⅰ)解法一:由nan+1=2Sn①
得当n≥2时,(n-1)an=2Sn-1②,
由①-②可得,nan+1-(n-1)an=2(Sn-Sn-1)=2an,
所以nan+1=(n+1)an,
即当n≥2时,
an+1
an
=class="stub"n+1
n

所以
a3
a2
=class="stub"3
2
a4
a3
=class="stub"4
3
a5
a4
=class="stub"5
4
,…,
an
an-1
=class="stub"n
n-1

将上面各式两边分别相乘得,
an
a2
=class="stub"n
2

an=class="stub"n
2
a2
(n≥3),
又a2=2S1=2a1=2,所以an=n(n≥3),
此结果也满足a1,a2,
故an=n对任意n∈N+都成立.…(7分)
解法二:由nan+1=2Sn及an+1=Sn+1-Sn,
得nSn+1=(n+2)Sn,
Sn+1
Sn
=class="stub"n+2
n

∴当n≥2时,Sn=S1
S2
S1
S3
S2
•…•
Sn
Sn-1
=1×class="stub"3
1
×class="stub"4
2
×class="stub"5
3
×…×class="stub"n+1
n-1
=
n(n+1)
2
(此式也适合S1),
∴对任意正整数n均有Sn=
n(n+1)
2

∴当n≥2时,an=Sn-Sn-1=n(此式也适合a1),
故an=n.…(7分)
(Ⅱ)依题意可得bn=
4an+1
an2an+22
=class="stub"4n+4
n2(n+2)2
=class="stub"1
n2
-class="stub"1
(n+2)2

Tn=class="stub"1
12
-class="stub"1
32
+class="stub"1
22
-class="stub"1
42
+class="stub"1
32
-class="stub"1
52
+…+class="stub"1
n2
-class="stub"1
(n+2)2
=1+class="stub"1
4
-class="stub"1
(n+1)2
-class="stub"1
(n+2)2
=class="stub"5
4
-
(n+1)2+(n+2)2
(n+1)2(n+2)2
<class="stub"5
4

Tn<class="stub"5
4
.…(13分)

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