在正项等比数列{an}中,公比q≠1

∵等比数列{an}的各项均是正数,公比q≠1
∴a5×a7=a3×a9,且a3≠a9
∴P=1/2(log1/2 a5+log1/2 a7)=1/2log1/2*(a5×a7)=log1/2 √(a5×a7)=log1/2 √(a3×a9)
又∵(a3+a9)/2>√(a3+a9),
【上式的原因：a²+b²≥2ab,则(a+b)²-2ab≥2ab,即(a+b)²≥4ab ∴[(a+b)/2]²≥ab】
∴P=log1/2 √(a3+a9)>log1/2 (a3+a9）/2]=Q
∴P>Q

P=(1/2)[log(1/2) a5+log(1/2) a7]
=(1/2)[log(1/2) (a5a7)]
=log(1/2) √(a5a7)
{an}为等比数列
a5a7=a3a9
P=log(1/2) √(a3a9)
Q=log(1/2) (a3+a9)/2
令f(x)=log(1/2) x
P=f[√(a3a9)]
Q=f[(a3+a9)/2]
(a3+a9)/2>√a3a9>0
P>Q