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2021年高中数学必修五知识点总结.pdf

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1、高中数学必修五知识点总结解直角三角形.2 数列.5 不等式.11|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 1 页，共 14 页解三角形复习知识点一、知识点总结【正弦定理】1正弦定理:2sinsinsinabcRABC(R为三角形外接圆的半径).2.正弦定理的一些变式：sinsinsini a b cABC；sin,sin,sin22abiiABCRR2cR；2sin,2 sin,2 siniiiaRA bRB bRC；（4）RCBAcba2sinsinsin3两类正弦定理解三角形的问题：（1）已知两角和任意一边，求其他的两边及一角.（2）已

2、知两边和其中一边的对角，求其他边角.（可能有一解，两解，无解）【余弦定理】1余弦定理：2222222222cos2cos2cosabcbcAbacacBcbabaC2.推论：222222222cos2cos2cos2bcaAbcacbBacbacCab.设a、b、c是C的角、C的对边，则：若222abc，则90C；若222abc，则90C；若222abc，则90C3.两类余弦定理解三角形的问题：（1）已知三边求三角.（2）已知两边和他们的夹角，求第三边和其他两角.【面积公式】已知三角形的三边为a,b,c,|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载

3、.第 2 页，共 14 页文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1

4、J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J

5、7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:

6、CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9

7、A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8

8、 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3

9、N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J71.111sin()222aSahabCr abc（其中r为三角形内切圆半径）2.设)(21cbap,)()(cpbpappS【三角形中的常见

10、结论】（1）CBA(2)sin()sin,ABC cos()cos,ABCtan()tan,ABC2cos2sinCBA,2sin2cosCBA;AAAcossin22sin,（3）若CBAcbaCBAsinsinsin若CBAsinsinsincbaCBA（大边对大角，小边对小角）（4）三角形中两边之和大于第三边，两边之差小于第三边（5）三角形中最大角大于等于60，最小角小于等于60(6)锐角三角形三内角都是锐角三内角的余弦值为正值任两角和都是钝角任意两边的平方和大于第三边的平方.钝角三角形最大角是钝角最大角的余弦值为负值（7）ABC中，A,B,C 成等差数列的充要条件是60B.(8)ABC

11、为正三角形的充要条件是A,B,C 成等差数列，且a,b,c成等比数列.二、题型汇总题型 1【判定三角形形状】判断三角形的类型（1）利用三角形的边角关系判断三角形的形状:判定三角形形状时，可利用正余弦定理实现边角转化，统一成边的形式或角的形式.（2）在ABC中，由余弦定理可知:222222222是直角ABC 是直角三角形是钝角ABC 是钝角三角形是锐角abcAabcAabcAABC 是锐角三角形（注意：是锐角AABC 是锐角三角形）(3)若BA2sin2sin，则 A=B或2BA.例 1.在ABC中，Abccos2，且abcbacba3)(，试判断ABC形状.|精.|品.|可.|编.|辑.|学.

12、|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 3 页，共 14 页文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE1

13、0T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H

14、4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE

15、3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A

16、9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ

17、4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10

18、T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7题型 2【解三角形及求面积】一般地，把三角形的三个角A,B,C 和它们的对边a,b,c叫

19、做三角形的元素.已知三角形的几个元素求其他元素的过程叫做解三角形.例 2.在ABC中，1a，3b，030A，求的值例 3.在ABC中，内角CBA,对边的边长分别是cba,，已知2c，3C（）若ABC的面积等于3，求ba,；（）若AABC2sin2)(sinsin，求ABC的面积题型 3【证明等式成立】证明等式成立的方法：（1）左右，（2）右左，（3）左右互相推.例 4.已知ABC中，角CBA,的对边分别为cba,，求证：BcCbacoscos.题型 4【解三角形在实际中的应用】仰角俯角方向角方位角视角例 5如图所示，货轮在海上以40km/h 的速度沿着方位角（从指北方向顺时针转到目标方向线的水

20、平转角）为140 的方向航行，为了确定船位，船在B 点观测灯塔A 的方位角为 110，航行半小时到达C点观测灯塔A 的方位角是65，则货轮到达C点时，与灯塔 A 的距离是多少？数列知识点1.等差数列的定义与性质|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 4 页，共 14 页文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9

21、A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8

22、 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3

23、N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3

24、 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1

25、J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J

26、7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:

27、CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7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2nS

28、anbn（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，|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第

29、5 页，共 14 页文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10

30、T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档

31、编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE1

32、0T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H

33、4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE

34、3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A

35、9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7naSS偶奇，1nnSS偶奇.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，

36、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解1n时，112 152a，114a2n时，1212111121 5222nnaaan得：122nna，12nna，114(1)2(2)nnnan练习数列na满足111543nnnSSaa，求na注意到11nnnaSS，代入得14

37、nnSS；又14S，nS是等比数列，4nnS2n时，113 4nnnnaSS|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 6 页，共 14 页文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J

38、10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7

39、文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:C

40、E10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A

41、9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8

42、HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N

43、9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3

44、ZQ4U1J10T8J7（2）叠乘法如：数列na中，1131nnanaan，求na解321211212 3nnaaanaaan，11naan又13a，3nan.（3）等差型递推公式由110()nnaaf naa，求na，用迭加法2n时，21321(2)(3)()nnaafaafaaf n 两边相加得1(2)(3)()naafff n0(2)(3)()naafff n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

45、为11dacc，为公比的等比数列1111nnddaaccc，1111nnddaaccc（5）倒数法如：11212nnnaaaa，求na由已知得：1211122nnnnaaaa，11112nnaa1na为等差数列，111a，公差为12，11111122nnna，|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 7 页，共 14 页文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O

46、3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U

47、1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8

48、J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码

49、:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T

50、9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z8 HE3H3N9A9O3 ZQ4U1J10T8J7文档编码:CE10T9A9H4Z

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