设函数F（x）=logmX,数列{an}是公比为m的等比数列设函数F（x）=log以m为底X,数列{an}是公比为m的等比数列,若F（a2*a4*...a2006）=8,则F（a1的方）+F（a2的方）+...+F（a2006的方)的值等于?

Question

设函数F（x）=logmX,数列{an}是公比为m的等比数列
设函数F（x）=log以m为底X,数列{an}是公比为m的等比数列,若F（a2*a4*...a2006）=8,则F（a1的方）+F（a2的方）+...+F（a2006的方)的值等于?

Answer 1

F（a2*a4*...a2006）=8
F(a1m)+F(a1m^3)+.F(a1m^2005)
=[F(a1)+F(m)]+[F(a1)+F(m^3)]+.[F(a1)+F(m^2005)]
=1003F(a1)+(1+3+5+.2005)
=1003F(a1)+(1+2005)*1003/2
=1003F(a1)+1003^2=8
∴1003F(a1)=8-1003^2
F（a1的方）+F（a2的方）+...+F（a2006的方)
=2[F(a1)+F(a2)+.F(a2006)]
=2{F(a1)+[F(a1)+F(m)]+[F(a1)+F(m^2)]+.[F(a1)+F(m^2005)]}
=2[2006F(a1)+(1+2005)2005/2]
=2[2006F(a1)+1003*2005]
=2[2*(8-1003^2)+1003*2005]
=2[16-2006*1003+2005*1003]
=2(16-1003)
=-1974