在数列{an}中,a1＝2,a n＋1＝4an-3n＋1,n∈正整数

(1)A(n+1)-(n+1)=4An-4n=4(An-n)
A1-1=1≠0
∴{An-n}是等比数列
(2)An-n=4^(n-1)*(A1-1)=4^(n-1)
∴An=n+4^(n-1)
Sn=A1+A2+A3+……+An
=1+4^0+2+4^1+3+4^2+……+n+4^(n-1)
=(1+2+3+……+n)+(1+4+4^2+……+4^(n-1))
=n(n+1)/2+(1-4^n)/(1-4)
=1/3*4^n+1/2*n^2+1/2*n-1/3
(3)4Sn=1/3*4^(n+1)+2*n^2+2*n-4/3
S(n+1)=1/3*4^(n+1)+1/2*(n+1)^2+1/2*(n+1)-1/3
4Sn-S(n+1)=(1/3*4^(n+1)+2*n^2+2*n-4/3)-(1/3*4^(n+1)+1/2*(n+1)^2+1/2*(n+1)-1/3)
=3/2*n^2+1/2n-2
=1/2(n-1)(3n+4)
n≥1
所以4Sn-S(n+1)=1/2(n-1)(3n+4)≥0
∴S（n+1）≤4Sn
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