已知m＞0,n＞0且m,x,n成等差数列,m,a,b,n成等比数列,求证2x≥a＋b

题目

已知m＞0,n＞0且m,x,n成等差数列,m,a,b,n成等比数列,求证2x≥a＋b

答案

证明：m,x,n成等差数列,∴2x=m+nm,a,b,n成等比数列 ∴ab=mn,设a=mq,b=mq²,c=mq³,q>0∴ 2x-(a+b)= (m+n)-(a+b)=(1+q³)-(q+q²)=(1+q)(1-q+q²)-q(1+q)=(1+q)(1-2q+q²)=(1+q)(1-q)²...