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2022年数列、数学归纳法及数列极限的复习 .pdf

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1、数列、数学归纳法及数列极限的复习一、基本概念1 数列：按一定次序排列的一列数.按项数是否有穷可分为有穷数列和无穷数列.按相邻两项的大小关系可分为递增数列、递减数列、常数数列和摆动数列.数列可以看成一类特殊的函数，其定义域为正整数集或其子集.数列na的第n项na与n的关系式)(nfan叫做数列na的通项公式.注意：不是每个数列都有通项公式！根据数列的前几项不能确定这个数列，但是可以写出它的一个或几个通项公式！有些数列有单调性、周期性、最值.2 数列的递推公式：给出数列na的初始项(第一项或前几项)，再给出相邻两项或几项的关系，这样的关系式称为数列na的递推公式.如：daaaann 101，nnq

2、aaaa101，nnnaaaaa12211,1等.3 等差数列若daann 1(d为常数)，则数列na是等差数列.公差为d的等差数列na的通项公式为dnaan)1(1.(该公式可整理为)(1dadnan)设na是等差数列，若),(2都是正整数ktsqpktsqp，则kqpnmaaaaa2.注意：nmnmaaa一般不成立！设na是等差数列，则na的前n项和nS的公式为dnnnaaanSnn2)1(2)(11.(该公式可整理为ndandSn)2(212).若等差数列na的公差0d，则na是递增数列.若0d，则na是常数数列.若0d，则na是递减数列.若bAa,成等差数列，则2baA.任意两个实数都

3、有唯一的等差中项.判断na是等差数列的方法：(1)利用定义daann 1(d为常数).(2)根据nnnnaaaa112.(3)根据qpnan.(4)根据bnanSn2.注意：证明数列na是等差数列时通常用定义！4 等比数列若qaann 1(q为常数)，则数列na是等比数列.公比为q的等比数列na的通项公式为mnmnnqaqaa11.设na是等比数列，若),(2都是正整数ktsqpktsqp，则2kqpnmaaaaa.设na是等比数列，则na的前n项和nS的公式为)1(1)1(1)1(111qqqaqqaaqnaSnnn.等比数列na的公比0q.若bGa,成等比数列，则abG.两个实数不一定有等

4、比中项，若有等比中项，则有两个！判断na是等比数列的方法：(1)利用定义qaann 1(q为常数).(2)根据nnnnaaaa112.(3)根据nnbqa.(4)根据bbqSnn.注意：证明数列na是等比数列时通常用定义！5 数学归纳法对于关于正整数n的很多命题可以用数学归纳法解决.它是用有限的步骤完成无限的递推！第一步，证明0nn时命题成立(这是递推的基础).第二步，假设),(0nkNkkn命题成立.由此证明1kn时命题也成立(这是递推的依据).根据这两步，就可以无限递推下去，因而对于任意的正整数0nn，命题都成立.数学归纳法的第二步必须用假设的结论！6 数列的极限几个基本数列的极限：)(0

5、1limNmnmn.当1x时，0limnnx.CCnlim(C为常数).运算法则：若BbAannnnlim,lim，则(1)BAbabannnnnnnlimlim)(lim.(2)BAbabannnnnnnlimlim)(lim.文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7

6、HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S

7、4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7

8、ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N

9、10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J1

10、0文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:

11、CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K

12、1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10(3)BAbabannnnnnnlimlimlim(0B).(4)ACaCaCnnnnlim)(lim(C为常数).数列极限的四则运算法则只适用于有限项.如果是无限项，应该先化简.设数列na为等比数列，首项为1a，公比为q，na的前n项和为nS.如果10q，那么qaaaaaaaSnnnnn1)(limlim12121公比为q的无穷等比数列na的各项和存在的充要条件是10q.二、常用结论1 若数列na、nb是等差数列，则npa、1nnaa、nnqbpa也是等差数列.nac(1,0 cc)是等比数列.2 若等差数列na的前n项和为nS.

13、则kkkkkSSSSS232,也成等差数列.3 设等差数列na的首项为1a，公差为d，其前n项和为nS.若0d，则nS有最小值.若0d，则nS有最大值.4 数列na是等比数列，则npa、)(Nkakn、1nnaa也是等比数列.若数列na是等比数列，且0na，则nCalog(1,0 cc)是等差数列.5 若等比数列na的前n项和为nS.则kkkkkSSSSS232,也成等比数列.6 若等差数列na的前n项和为nS.则nSn成等差数列.若等比数列na的前n项的乘积为nT.则nnT成等比数列.7 若km,都是正整数，则)()()(0lim1010kmkmbakmnbnbbnanaakmkkmmn不存

14、在.8 若na是等差数列，公差为d，前n项和为nS.则na的奇数项,1231naaa成等差数列，公差为d2.na的偶数项,242naaa成等差数列，公差为d2.文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6

15、K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7

16、 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6

17、S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7

18、 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9

19、N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J

20、10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J101121

21、1231)1(2)(1(nnnanaanaaa.1222422)(nnnnaaanaaa.1111224321)12()1(nnnnnannaanaaaaaa.nnnnaaanaaa2)(1211231.ndaaaaaaaaaaaannnn)()()()()(122341212312429 若na和nb都是等差数列，它们的前n项和分别为nS和nT.则1212nnnnTSba.三、常见的与数列na的前n项和nS有关的问题)2()1(11nSSnSannn四、常见的分类讨论问题1 若数列na是等差数列，且na有正有负，求na的前n项和.如：数列na的通项公式为nan219，求na的前n项和nT.2

22、若数列na满足),2(3),12(14NkknNkknnann，求na的前n项和nS.3 若等比数列na的首项是1a，公比为q，其前n项和为nS，1nnnSST，求nnTlim.五、求数列na的最大项、最小项问题0001nnaa，如：32922nnan.若0na，1111nnaa.如：nnna)109)(1(.六、数列的求和方法1 公式法文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编

23、码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O

24、6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X

25、7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B

26、6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B

27、7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M

28、9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1

29、J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10数列na的通项公式为12nna，求其前n项和为nS.数列na的通项公式为nann231，求其前n项和为nS.2 倒序相加法设221)(xxf，利用课本中推导等差数列前n项和的公式的方法，可求得)50()49()1()0()48()49(ffffff的值为 _.3 错位相减法数列na的通项公式为13)12(nnna，求其前n项和为nS.数列na的通项公式为nnna52，求其前n项和为n

30、S.4 裂项相消法数列na的通项公式为)1(1nnan，求其前n项和为nS.数列na的通项公式为1412nan，求其前n项和为nS.文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4

31、P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 Z

32、D4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N1

33、0L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10

34、文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:C

35、G3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10文档编码:CG3O6K1D2X7 HQ8B6S4P4B7 ZD4M9N10L1J10

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