2Sn=2Sn-1+2a(n-1)+1
2Sn-2Sn-1=2an=2a(n-1)+1
an-a(n-1)=1/2,为定值.
a1=1/4
数列{an}是以1/4为首项,1/2为公差的等差数列
an=1/4+(n-1)(1/2)=n/2 -1/4
3bn-b(n-1)=n
3bn-3(1/2)n+3/4=b(n-1)-(n-1)/2+1/4
[bn-n/2 +1/4]/[b(n-1)-(n-1)/2+1/4]=1/3,为定值.
b1-1/2+1/4=3/4-1/2+1/4=1/2
数列{bn-n/2 +1/4}是以1/2为首项,1/3为公比的等比数列
bn- n/2+1/4=(1/2)(1/3)^(n-1)
bn=n/2 +(1/2)(1/3)^(n-1) -1/4
bn-an=n/2+(1/2)(1/3)^(n-1) -1/4 -n/2+1/4=(1/2)(1/3)^(n-1)
b1-a1=3/4-1/4=1/2,同样满足.
(bn-an)/[b(n-1)-a(n-1)]=1/3,为定值.
数列{bn-an}是以1/2为首项,1/3为公比的等比数列.
bn=n/2+(1/2)(1/3)^(n-1) -1/4
Tn=(1+2+...+n)/2+(1/2)[1-(1/3)^n]/(1-1/3)-n/4
=n^2/4-(1/4)/3^(n-1)+3/4