Question

17题详解。谢谢了！

Answer 1

(1)
a(n+1)+1=3an+3=3(an+1)
[a(n+1)+1]/(an+1)=3，为定值
a1+1=2+1=3
数列{an+1}是以3为首项，3为公比的等比数列
(2)
an+1=3·3ⁿ⁻¹=3ⁿ
bn=log3(an+1)=log3(3ⁿ)=n
1/[bnb(n+2)]=1/[n(n+2)]=½[1/n -1/(n+2)]
Sn=1/(b1b3)+1/(b2b4)+...+1/[bnb(n+2)]
=½[1/1 -1/3 +1/2 -1/4+...+1/n -1/(n+2)]
=½[1+1/2+...+1/n-(1/3+ 1/4+...+1/(n+1)+1/(n+2))]
=½[1+ 1/2 -1/(n+1) -1/(n+2)]
=¾ -1/[2(n+1)] -1/[2(n+2)]
1/[2(n+1)]>0，1/[2(n+2)]>0
¾ -1/[2(n+1)] -1/[2(n+2)]<¾
Sn<¾

Answer 2

a1=2

a(n+1)=3an+2
a(n+1) +1 = 3(an +1)
=>{an +1} 是等比数列, q=3
an +1=3^(n-1).(a1 +1)
an = -1+3^n

bn = log<3> (an +1)
= n

cn = 1/[bn.b(n+2)]
= 1/[n(n+2)]
=(1/2) [ 1/n - 1/(n+2) ]
Sn =c1+c2+...+cn
=(1/2)[ 1 - 1/(n+2) ]
< 1/2
<3/4