已知等比数列{An}满足An>0,n=1,2,…,且A5·A(2n-5)=2^(2n)(n≥3),则n≥1时,log2(A1)+log2(A3)+……+log2[A(2n-1)]=?

问题描述:

已知等比数列{An}满足An>0,n=1,2,…,且A5·A(2n-5)=2^(2n)(n≥3),则n≥1时,log2(A1)+log2(A3)+……+log2[A(2n-1)]=?

A5*A(2n-5)=2^(2n)(n≥3)
A1*Q^4 * A1*Q^(2n-6)=2^(2n)
A1^2*Q^(2n-2)=2^(2n)
An=2^n
所以log2(A1)+log2(A3)+……+log2[A(2n-1)]=1+2+3+……+(2n-1)=[(2n-1)*(1+2n-1)]/2=n*(2n-1)