高考真题-理科数学-2012 (21)(本小题满分13分) 2 数列{x}满足:x?0,x??x?x?c(n?N) 1n?1nn n (I)证明:数列{x}是单调递减数列的充分必要条件是c?0 n (II)求c的取值范围,使数列{x}是单调递增数列。 n 【解析】(I)必要条件 2 当时,x??x?x?c?x?数列是单调递减数列 c?0{x} n?1nnn n 充分条件 22 数列{x}是单调递减数列?x?x??x?x?c?c?x?0 12111 n 得:数列{x}是单调递减数列的充分必要条件是c?0 n (II)由(I)得:C?0 ①当c?0时,a?a?0,不合题意 n1 2 ②当时,x?c?x,x??c?2c?x?c?0?c?1 c?0 2132 22 x?x?c?x?0?x?c?1?0?x?x?c n?1nnn1n 22 x?x??(x?x)?(x?x)??(x?x)(x?x?1) n?2n?1n?1nn?1nn?1nn?1n 11 当c?时,x?c??x?x?1?0?x?x与x?x同号, nnn?1n?2n?1n?1n 42 由x?x?c?0?x?x?0?x?x 21n?2nn?1n 2 limx?lim(?x?x?c)?limx?c n?1nnn n??n??n?? 11 当c?时,存在N,使x??x?x?1?x?x与x?x异号 NNN?1N?2N?1N?1N 42 与数列是单调递减数列矛盾 {x} n 1 得:当0?c?时,数列{x}是单调递增数列. n 4 福麟:Http://1793465064.360doc.com/ 1111100000 |
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