问题 解答题

在数列{an}中,a1=2,an+1=4an-3n+1,n∈N*

(Ⅰ)证明数列{an-n}是等比数列;

(Ⅱ)求数列{an}的前n项和Sn

(Ⅲ)证明不等式Sn+1≤4Sn,对任意n∈N*皆成立.

答案

(Ⅰ)证明:由题设an+1=4an-3n+1,得an+1-(n+1)=4(an-n),n∈N*

又a1-1=1,所以数列{an-n}是首项为1,且公比为4的等比数列.

(Ⅱ)由(Ⅰ)可知an-n=4n-1,于是数列{an}的通项公式为an=4n-1+n.

所以数列{an}的前n项和Sn=

4n-1
3
+
n(n+1)
2

(Ⅲ)证明:对任意的n∈N*Sn+1-4Sn=

4n+1-1
3
+
(n+1)(n+2)
2
-4(
4n-1
3
+
n(n+1)
2
)=-
1
2
(3n2+n-4)≤0

所以不等式Sn+1≤4Sn,对任意n∈N*皆成立.

完形填空
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