2Sn+3=3An
2Sn=3An-3 ①
2S(n-1)=3A(n-1)-3 ②
①-②得
2[Sn-S(n-1)]=3An-3A(n-1)
2An=3An-3A(n-1)
An=3A(n-1)
An/A(n-1)=3
所以数列An是等比数列
2S1+3=3A1
2A1+3=3A1
A1=3
所以
An=A1q^(n-1)=3*3^(n-1)=3^n
Bn=(4n+1)/An
Bn=(4n+1)/3^n
Bn=(4n+1)(1/3)^n
Tn=B1+B2+……+B(n-1)+Bn
Tn=5(1/3)^1+9(1/3)^2+……+(4n-3)(1/3)^(n-1)+(4n+1)(1/3)^n ①
1/3Tn=5(1/3)^2+9(1/3)^3+……+(4n-3)(1/3)^n+(4n+1)(1/3)^(n+1) ②
①-②得
2/3Tn=5(1/3)^1+4(1/3)^2+……+4(1/3)^(n-1)+4(1/3)^n-(4n+1)(1/3)^(n+1)
=1/3+4(1/3)^1+4(1/3)^2+……+4(1/3)^(n-1)+4(1/3)^n-(4n+1)(1/3)^(n+1)
=1/3+4[1/3×(1-1/3^n)/1-1/3]-(4n+1)(1/3)^(n+1)
=1/3+2(1-1/3^n)-(4n+1)(1/3)^(n+1)
Tn=7/2-1/2(4n+7)×(1/3)^n＜7/2