2023高二·全国·专题练习
1 . 等比数列的性质
(1)与项有关的性质
①在等比数列
中,
.
②在等比数列
中,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/315eff4d9c14702305080b015bbad198.png)
____ =_______ .
③在公比为q的等比数列
中,取出项数成等差数列的项
,…,仍可组成一个等比数列,公比是
④m个等比数列,由它们的各对应项之积组成一个新数列,仍然是等比数列,公比是原来每个等比数列对应的公比之积.
⑤若
,
均为等比数列,公比分别为
,则
仍为等比数列,且公比为
;
仍为等比数列,且公比为
;
仍为等比数列,且公比为
.
⑥当
是公比为q(q>0)的正项等比数列时,数列
是等差数列,首项为
,公差为
(2)与和有关的性质
①等比数列连续k项的和仍为等比数列,即
,…,仍为等比数列,且公比为
(q≠-1,或q=-1且k为奇数).
②在等比数列中,若项数为
,则
=q.
③在等比数列中,当
时,
,
④在等比数列中,
(1)与项有关的性质
①在等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4986b0e0de914aacd611a2804783ae6b.png)
②在等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96b0003f4dbb3f01799fd8ddd994219a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/315eff4d9c14702305080b015bbad198.png)
③在公比为q的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf91594ec1f4c704032d1f984b6af310.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11dd06e4370d2acabebb3a53d83759c.png)
④m个等比数列,由它们的各对应项之积组成一个新数列,仍然是等比数列,公比是原来每个等比数列对应的公比之积.
⑤若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1577fb037ee50b0eba547dab2fec0d2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd924364bf52ea7f63cb65e01f28a5a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6268630d5e5288048d32f4aa5c8bc02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ebfe656386b20cb304049095e601f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2767882820f4ba0defde0e412adb747f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae633811d9ab4d4b0affcfb318fe2b36.png)
⑥当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4711944aa361bd9f2ed566b24c6888ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d454255d882c4d20e97d8de506473b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7673a50cd7a32c83d3b18a6c67772489.png)
(2)与和有关的性质
①等比数列连续k项的和仍为等比数列,即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfe0e9720aa0313abe747549f0a84b5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6846435bd60680d61d6d7e0ec5e43974.png)
②在等比数列中,若项数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4bbe7017a70b4fd463aa9a0baf5770b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a9372ffa252abb45a77fadf8ab289c4.png)
③在等比数列中,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30a40c19a154e38a04b213be0acb67ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eec9637a68e7ce07407b195f9b7fdcd.png)
④在等比数列中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5dcceab9609adc8c58c9bd131b632de8.png)
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2023高二·全国·专题练习
2 . 数列的分类
分类标准 | 类型 | 含义 |
按项数 | 有穷数列 | 项数 |
无穷数列 | 项数 | |
按项的 变化趋势 | 递增数列 | 从第2项起,每一项都大于它的前一项的数列,即恒有![]() |
递减数列 | 从第2项起,每一项都小于它的前一项的数列,即恒有![]() | |
常数列 | 各项都相等的数列,即恒有![]() | |
按其他 标准 | 周期数列 | 一般地,对于数列![]() ![]() ![]() |
按其他 标准 | 有界(无界)数列 | 任一项的绝对值都小于某一正数的数列称为有界数列,即![]() |
摆动数列 | 从第2项起,有些项大于它的前一项,有些项小于它的前一项的数列 |
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