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大于等于a+b

证明：设m,a,b,n等比数列比为q,则a=mq b=mq² n=mq^3a+b=mq+mq² =m(1+q)q 因为m、x、n成等差数列所以2x=m+n2x=m+n=m+mq^3=m(1+q^3)=m(1+q)(1-q+q² )=m(1+q)[(1-2q+q²+q )]=m(1+q)[(1-q)²+q ]因为(1-q)²+q ≧q所以2x=m(1+q)[(1-q)²+q ]≧m(1+q)q=a+b所以2x]≧a+