(1)由an+1=4an-3n+1
得[a(n+1)-(n+1)]/(an-n)=4
所以数列{an-n}是公比为4的等比数列
(2)设数列{an-n}的通项为bn,前n项的和为Tn
b1=a1-1=1
Tn=(4^n-1)/3
同时Tn=b1+b2+b3+...+bn=a1-1+a2-2+a3-3+...+an-n=Sn-n(n+1)/2
Sn-n(n+1)/2=(4^n-1)/3
Sn=(4^n-1)/3+n(n+1)/2
Sn+1=[4^(n+1)-1)]/3+(n+1)(n+2)/2
Sn+1- 4Sn=[4^(n+1)-1]/3+(n+1)(n+2)/2-4[(4^n-1)/3+n(n+1)/2]
化简
[4^(n+1)-1]/3+(n+1)(n+2)/2-4[(4^n-1)/3+n(n+1)/2]
=[4^(n+1)-1]/3+(n+1)(n+2)/2-[4^(n+1)-4]/3-4n(n+1)/2
=[4^(n+1)-1]/3-[4^(n+1)-4]/3+(n+1)(n+2)/2-4n(n+1)/2
={[4^(n+1)-1]-[4^(n+1)-4]}/3+[(n+1)(n+2)-4n(n+1)]/2
=[4^(n+1)-1-4^(n+1)+4]/3+(n^2+3n+2-4n^2-4n)/2
=[4^(n+1)-4^(n+1)+4-1]/3+(-3n^2-n+2)/2
=1+(-3n^2-n+2)/2
=(2-3n^2-n+2)/2
=(-3n^2-n+4)/2
=-3n^2/2-n/2+2
=-3/2(n^2+n/3-4/3)
=-3/2(n^2+n/3+1/36-1/36-4/3)
=-3/2[(n+1/6)^2-49/36]
=-3/2(n+1/6)^2+49/24 （因为太乱,所以特别仔细小心地算）
由于只能取正整数,所以还得分析抛物线
抛物线的对称轴为n=-1/6
由于抛物线开口向下,在对称轴的右边是减函数,所以当n=1时,函数值得了大
所以Sn+1- 4Sn的最大值是=-3/2-1/2+2=0