递推法证明不等式.doc

顺藤摸瓜,顺水推舟——递推法证明数列不等式众所周知, 证明不等式的方法多种多样, 技巧层出不穷, 今天笔者试图给出一个较为巧妙的好方法——递推法,此法在证明一些较为复杂的代数不等式时显得特别有效,请看例 1.( 20 12 年第 11 届中国女子竞赛)设 n 为正整数, 1 2 , , , n a a a ???为非负数,求证; 1 2 3 1 1 1 2 1 1 2 1 2 3 1 2 11 1 (1 )(1 ) (1 )(1 )(1 ) (1 )(1 ) (1 ) nn a a a a a a a a a a a a a a a a ????? ? ??????? ???????????. 证明:因为 11 1 1 a a a ? ?? ?,所以 1 2 3 i 1 1 2 i 1 2 3 i 1 1 2 i-1 i 1 2 3 i 1 i 1 2 i-1 i 1 2 3 i 1 1 2 3 i 1 i 1 2 i-1 1 2 i (1 )(1 ) (1 ) 1= (1 )(1 ) (1 ) 1 = - (1 )(1 ) (1 ) (1 ) = (1 )(1 ) (1 ) (1 )(1 ) (1 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a ???? ????? ?????????? ??????????? ????????? ????? ???????????(1) -1 i )(1 ) a?所以 1 2 3 1 1 1 2 1 1 2 1 2 3 1 2 1 2 3 1 2 1 1 (1 )(1 ) (1 )(1 )(1 ) (1 )(1 ) (1 ) =1 1 (1 )(1 ) (1 ) nn nn a a a a a a a a a a a a a a a a a a a a a a a ????? ? ?????? ??????????????? ?? ?????即结论获得证明。例 2.( 2006 中国数学奥林匹克) 实数列?? na 满足: k kka aaa??????2 1,2 1 11),3,2,1(??k 证明:)1 1()1 1 )(1 1(1)(2 21 2121???????????????????????????????? n nn nnaaan aaaaaa n 分析:从题目