1.An=logn+1(n+2)=lg(n+2)/lg(n+1)
An-1=logn(n+1)=lg(n+1)/lgn
...
A2=lg4/lg3,A1=lg3/lg2 =>
A1*A2*...*Ak=(lg3/lg2)(lg4/lg3)...[lg(k+1)/lgk]
[lg(k+2)/lg(k+1)]
=lg(k+2)lg2＝log2(k+2)
要使上式为正整数,那么(k+2)为2的正整数次方,
k=2,6,14,...,2^n-2
1 Sk=(2-2)+(2^2-2)+(2^3-2)+...+(2^10-2)
=2(1-2^10)/(1-2)-2*10
=2^11-22=2048-22=2026
2.(1)Sn=2An-3n,S(n-1)=2A(n-1)-3(n-1) =>
An=Sn-S(n-1)=2An-3n-2A(n-1)+3(n-1)
=2(An-An-1)-3 => An=2An-1+3 (n>=2)
Bn=An+3 => Bn-1=An-1+3=1/2An+3/2=1/2(An+3)=1/2Bn
=> Bn为等比数列.
S1=A1=2A1-3 => A1=3 => B1=A1+3=6 =>
Bn=B1*q^(n-1)=6*(1/2)^(n-1)
(2)n*An=n*(Sn+3n)/2=3n*[2^(n-1)+n]/2