数列和式不等式问题是高考中的热点难点问题,往往以压轴题的形式出现,学生普遍感觉束手无策,无章可循.笔者经过研究发现,不少数列和式不等式问题若能合理地利用平均不等式,往往能化难为易,突破难点.例1已知数列{an}满足a1=1,a2n-an+1+3=0.求证:1/a1+2+1/a2+2+…+1/an+2<23.证由平均不等式得an+1=(a2n+1)+2≥2an+2,∴an+1+2≥2(an+2), The problems of series and inequality are the hot and difficult problems in college entrance examination, which often appear in the form of finale questions, and the students generally feel helpless and out of step.The author finds through research that if we can make use of the average inequality , Often can make it difficult for the difficult, breaking the difficult.Example 1 Known series {a} satisfy a1 = 1, a2n-an + 1 +3 = 0. Proof: 1 / a1 +2 +1 / a2 +2 + ... +1 / an + 2 <23. The card is given by the mean inequality an + 1 = (a2n + 1) + 2 ≧ 2an + 2, ∴an + 1 + 2 ≧ 2 (an + 2)