已知等比数列an满足an大于0,n=1,2.且a5乘a2n-5=2的2n(n大于等于3）,且a(5)乘a(2n-5)=2^2n(n>=3),且当n>=3时,log2a1+log2a3+…+log2a2n-1=

Question

已知等比数列an满足an大于0,n=1,2.且a5乘a2n-5=2的2n(n大于等于3）,
且a(5)乘a(2n-5)=2^2n(n>=3),且当n>=3时,
log 2 a1+log 2 a3+…+log 2 a2n-1=

Answer 1

a5*a(2n-5)=(a1q^4)[a1q^(2n-6)]=a1^2q^(2n-2)=(an)^2=2^(2n)=(2^n)^2.
因为an>0,所以an=2^n(n>=3).
在a5*a(2n-5)=2^(2n)中,取n=3,则a5*a1=a1*2^5=2^6、a1=2.a2=a1q=4.
所以an=2^n(n为正整数).
log2(a1)+log2(a3)+…+log2[a(2n-1)]=1+3+…+(2n-1)=n^2.