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+..+log2a(2n-1)=

且a5*a(2n-5)=2^2n,n≥3

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