Question

已知等比数列{an}满足an＞0,n=1,2等,且a5*a(2n-5)=2^2n,则当n≥1时log2a1+log2a3

Answer 1

当 n = 3,a5 * a1 = 2^6

即 a1^2 q^4 = 2^6－－－－（1）

当 n = 4 ,a5 * a3 = 2^8

即 a1^2 * q^6 = 2^8－－－－（2）

解得 q = 2,q = -2（因为an＞0,舍去）

a1 = 2,a3 = 2^3,a5 = 2^5.a(2n-1) = 2^(2n-1)

log2 a1 + log2 a3 +..+ log2 a(2n-1)

=log2 2 + log2 2^3 +..+ log2 2^(2n-1)

=1 + 3 +.+ (2n-1)

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

=n^2