Question

急求大神解答数学 第17题 步骤越详细越好

Answer 1

a(n+1)=2Sn+1 a1=t

S1=a1
a2=2a1+1
a(n+1)-an=2[Sn+1-Sn]=2an

a(n+1)=3an
a(n+1)/an=3
a2/a1=3
(2a1+1)/a1=3
2a1+1=3a1
a1=1
即t=1
2)an=3^(n-1)
a(n+1)=3^n
bn =log3[a(n+1)]=log3[3^n]=n
bn=n
bn/an=n(1/3)^(n-1)
Tn=1+2*1/3+3*(1/3)^2+4*(1/3)^3+....+(n-1)*(1/3)^(n-2)+n(1/3)^(n-1)
1/3Tn=1/3 +2*(1/3)^2+3*(1/3)^3+....+(n-2)*(1/3)^(n-2)+n(1/3)^n
两式相减得
2/3Tn=1+1/3+(1/3)^2+(1/3)^3+...+(1/3)^(n-1)-n(1/3)^n=3/2[1-(1/3)^n]-n(1/3)^n
=3/2-[(2n+3)/2*(1/3)^n]
2/3Tn=3/2-[(2n+3)/2*(1/3)^n]
Tn=9/4-[(2n+3)/4*(1/3)^(n-1)]<9/4

Answer 2