等比数列{an}各项均为正数,a1+a2+a3+a4+a5+a6=1,1/a1+1/a2+1/a3+1/a4+1/a5+1/a6=10,则a1*a2*a3*a4*a5*a6=?

题目

等比数列{an}各项均为正数,a1+a2+a3+a4+a5+a6=1,1/a1+1/a2+1/a3+1/a4+1/a5+1/a6=10,则a1*a2*a3*a4*a5*a6=?

答案

a1+a2+a3+a4+a5+a6=1,得出 a1(1-q^6)/1-q=1同理1/a1+1/a2+1/a3+1/a4+1/a5+1/a6=10得到1/a1*(1-1/q^6)/(1-1/q)=10接触a1^2*q^5=1/10而a1*a2*a3*a4*a5*a6=a1^6*q^15=(a1^2*q^5)^3=1/1000