(完整word版)高中数学数列知识点总结(2),推荐文档.pdf

数列基础知识点和方法归纳1 1.等差数列的定义与性质定义：1nnaad（d为常数），11naand等差中项：xAy，成等差数列2Axy前 n项和11122nnaann nSnad性质：na是等差数列（1）若 mnpq，则mnpqaaaa；（2）数列12212,nnnaaa仍为等差数列，232nnnnnSSSSS，仍为等差数列，公差为dn2；（3）若三个成等差数列，可设为adaad，（4）若nnab，是等差数列，且前 n项和分别为nnST，则2121mmmmaSbT（5）na为等差数列2nSanbn（ab，为常数，是关于 n 的常数项为 0 的二次函数）nS的最值可求二次函数2nSanbn的最值；或者求出na中的正、负分界项，即：当100ad，解不等式组100nnaa可得nS达到最大值时的 n值.当100ad，由100nnaa可得nS达到最小值时的 n值.(6)项数为偶数n2 的等差数列na，有),)()()(11122212为中间两项nnnnnnnaaaanaanaanSndSS奇偶，1nnaaSS偶奇.（7）项数为奇数12n的等差数列na，有)()12(12为中间项nnnaanS，naSS偶奇，1nnSS偶奇.数列基础知识点和方法归纳2 2.等比数列的定义与性质定义：1nnaqa（q为常数，0q），11nnaa q.等比中项：xGy、成等比数列2Gxy，或 Gxy.前 n项和：11(1)1(1)1nnna qSaqqq（要注意！）性质：na是等比数列（1）若 mnpq，则mnpqaaaa（2）232nnnnnSSSSS，仍为等比数列,公比为nq.注意：由nS求na时应注意什么？1n时，11aS；2n时，1nnnaSS.3求数列通项公式的常用方法（1）求差（商）法如：数列na，12211125222nnaaan，求na（2）叠乘法如：数列na中，1131nnanaan，求na（3）等差型递推公式由110()nnaaf naa，求na，用迭加法文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2数列基础知识点和方法归纳3 练习数列na中，111132nnnaaan，求na（1312nna）（4）等比型递推公式1nnacad（cd、为常数，010ccd，）可转化为等比数列，设111nnnnaxc axacacx令(1)cxd，1dxc，1ndac是首项为11dacc，为公比的等比数列1111nnddaaccc，1111nnddaaccc（5）倒数法如：11212nnnaaaa，求na附：公 式 法、利 用1(2)1(1)nnSSnS nna、累 加 法、累 乘 法.构 造 等 差 或 等 比1nnapaq或1()nnapaf n、待定系数法、对数变换法、迭代法、数学归纳法、换元法)4.求数列前 n 项和的常用方法(1)裂项法把数列各项拆成两项或多项之和，使之出现成对互为相反数的项.如：na是公差为d的等差数列，求111nkkka a（2）错位相减法若na为等差数列，nb为等比数列，求数列n na b（差比数列）前 n项和，可由nnSqS，求nS，其中 q为nb的公比.文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2数列基础知识点和方法归纳4 如：2311234nnSxxxnx23412341nnnx Sxxxxnxnx2111nnnx Sxxxnx1x时，2111nnnxnxSxx，1x时，11232nn nSn（3）倒序相加法把数列的各项顺序倒写，再与原来顺序的数列相加.121121nnnnnnSaaaaSaaaa相加12112nnnnSaaaaaa练习已知22()1xf xx，则111(1)(2)(3)(4)234fffffff(附：a.用倒序相加法求数列的前n 项和如果一个数列 an，与首末项等距的两项之和等于首末两项之和，可采用把正着写与倒着写的两个和式相加，就得到一个常数列的和，这一求和方法称为倒序相加法。我们在学知识时，不但要知其果，更要索其因，知识的得出过程是知识的源头，也是研究同一类知识的工具，例如：等差数列前n 项和公式的推导，用的就是“倒序相加法”。b.用公式法求数列的前n 项和对等差数列、等比数列，求前n 项和 Sn可直接用等差、等比数列的前n 项和公式进行求解。运用公式求解的注意事项：首先要注意公式的应用范围，确定公式适用于这个数列之后，再计算。c.用裂项相消法求数列的前n 项和裂项相消法是将数列的一项拆成两项或多项，使得前后项相抵消，留下有限项，从而求出数列的前n 项和。d.用错位相减法求数列的前n 项和错位相减法是一种常用的数列求和方法，应用于等比数列与等差数列相乘的形式。即若在数列 an bn中，an成等差数列，bn 成等比数列，在和式的两边同乘以公比，再与原式错位相减整理后即可以求出前 n 项和。e.用迭加法求数列的前n 项和迭加法主要应用于数列 an 满足 an+1=an+f(n)，其中 f(n)是等差数列或等比数列的条件下，可把这个式子变成 an+1-an=f(n)，代入各项，得到一系列式子，把所有的式子加到一起，经过整理，可求出an，从而求出 Sn。f.用分组求和法求数列的前n 项和所谓分组求和法就是对一类既不是等差数列，也不是等比数列的数列，若将这类数列适当拆开，可文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2数列基础知识点和方法归纳5 分为几个等差、等比或常见的数列，然后分别求和，再将其合并。g.用构造法求数列的前n 项和所谓构造法就是先根据数列的结构及特征进行分析，找出数列的通项的特征，构造出我们熟知的基本数列的通项的特征形式，从而求出数列的前n 项和。文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2文档编码:CC3V8W3W1M6 HG1X7Y5V1J3 ZO10O4Z3U1N2