Question

已知函数f（x）=log以k为底x的对数（k为常数,k＞0,且k≠1）且数列{f（an）}是首相为4,

Answer 1

(1) f(an)=logk(an)

f(an-1)=logk(an-1)

f(an)-f(an-1)=2

logk(an)-logk(an-1)=2

logk(an/an-1)=2

an/an-1=k^2 k^2为常数

所以 数列{an}是等比数列

logka1=4 a1=k^4

q=k^2

an=k^4*(k^2)^(n-1)=k^(2n+2)

(2) k=根号2

a1=4 q=2 an=2^(n+1)

f(an)=2n+2

bn=2(n+1)*2^(n+1)=4*[(n+1)*2^(n)]

Sn=4[2*2^1+3*2^2+4^2^3+……+(n+1)*2^n]

2Sn=4[ 2*2^2+3*2^3+4*2^4+……+n*2^n+(n+1)*2^(n+1)] 相减

-Sn=4[4+2^2+2^3+2^4+……+2^n-(n+1)*2^(n+1)]

=4[2+2(1-2^n)/(1-2)-(n+1)*2^(n+1)]

Sn=n*2^(n+3)