高中數(shù)學(xué)必修5數(shù)列復(fù)習(xí)專練提高檢測

1、數(shù)列復(fù)習(xí)2.1 數(shù)列的表示一、概念1、定義：按一定順序排列的一列數(shù)叫做數(shù)列。注意：“有序性”是數(shù)列的基本特征！注意和集合區(qū)分2、表示：一般我們用符號(hào)：表示一個(gè)數(shù)列注意：“”是集合的符號(hào)，但不代表數(shù)列就是集合。3、通項(xiàng)公式：用含n的式子表示數(shù)列中的某項(xiàng)。即注意：通項(xiàng)公式是一種特殊的函數(shù)表示形式（離散型）； 并不是所有的數(shù)列都能寫出通項(xiàng)公式。4、前n項(xiàng)和公式：用含n的式子表示數(shù)列前n項(xiàng)的和。即注意：前n項(xiàng)和公式同樣是一種特殊的函數(shù)表示形式； 前n項(xiàng)和與通項(xiàng)的關(guān)系： ···········&

2、#183;·················································&

3、#183;······例題1：下列敘述正確的是注意：數(shù)列與集合的區(qū)別。A、數(shù)列1、3、5、7和數(shù)列7、5、3、1是同一個(gè)數(shù)列B、同一個(gè)數(shù)字在數(shù)列中可能重復(fù)出現(xiàn)C、數(shù)列的通項(xiàng)公式是定義域?yàn)檎麛?shù)集的函數(shù)D、數(shù)列的通項(xiàng)公式是惟一的5、遞增數(shù)列和遞減數(shù)列遞增數(shù)列都滿足：或遞減數(shù)列都滿足：或·····················&#

4、183;··············································例題2：已知數(shù)列是遞增數(shù)列，且，則實(shí)數(shù)的

5、取值范圍是 。·················································

6、;···················2.2 等差數(shù)列一、概念1、如果一個(gè)數(shù)列從第二項(xiàng)起，每一項(xiàng)與它的前一項(xiàng)的差等于同一個(gè)常數(shù)，那么這個(gè)數(shù)列就叫做等差數(shù)列，這個(gè)常數(shù)叫做 。2、定義法證明數(shù)列是等差數(shù)列若數(shù)列中存在：（d為常數(shù)），則為等差數(shù)列；··············

7、;··················································

8、;····例題1：判斷下列數(shù)列是否等差數(shù)列（1）； （2）；·········································

9、3;··························二、等差數(shù)列的通項(xiàng)公式1、通項(xiàng)公式：2、推導(dǎo)過程：累加法3、等差中項(xiàng)：若a，A，b成等差數(shù)列，則A叫做a與b的等差中項(xiàng)，且注意：通項(xiàng)公式中的“”中，知任意三個(gè)可求另一個(gè)。例題2：已知等差數(shù)列：3，7，11，15則：135，是中的項(xiàng)嗎？注意：檢驗(yàn)一個(gè)數(shù)（式）是否數(shù)列中的一項(xiàng)，只需把這個(gè)

10、數(shù)（式）代入數(shù)列的通項(xiàng)公式中即可。···············································&

11、#183;····················三、等差數(shù)列的簡單性質(zhì)1、若，則2、下標(biāo)為等差數(shù)列的項(xiàng)仍為等差數(shù)列 3、數(shù)列（為常數(shù)）仍為等差數(shù)列4、和均為等差數(shù)列，則也為等差數(shù)列。··················

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13、例題1：已知等差數(shù)列中，則的值是 。例題2：等差數(shù)列中，。求數(shù)列的通項(xiàng)公式。注意：利用等差數(shù)列性質(zhì)轉(zhuǎn)換時(shí)，不要混淆性質(zhì)。例題3：設(shè)數(shù)列、都是等差數(shù)列，且，則的值是 。例題4：等差數(shù)列中，則 ··································&

14、#183;·································四、判斷一個(gè)數(shù)列是否為等差數(shù)列的方法定義法：等差中項(xiàng)：通項(xiàng)法：為n的一次函數(shù)；求和法：·········

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16、·········例題1：已知數(shù)列滿足，令，求證：數(shù)列是等差數(shù)列例題2：已知a，b，c成等差數(shù)列，求證：也成等差數(shù)列。································

17、3;···································2.3 等差數(shù)列前n項(xiàng)和一、前n項(xiàng)和公式1、公式：2、推導(dǎo)：倒序求和（等差專用）3、注意：中，“知三求二”。要根據(jù)已知條件合理選用公式，列方程求解。4、運(yùn)用公式，要注意性質(zhì)“”

18、的運(yùn)用。·················································&#

19、183;··················例題1：此類題目的中心思想是方程思想。（1）已知等差數(shù)列的前5項(xiàng)和為25，第8項(xiàng)是15，求第21項(xiàng)。（2）等差數(shù)列16，12，18，的前幾項(xiàng)和為72？（3）一個(gè)等差數(shù)列第5項(xiàng)為10，前3項(xiàng)和為3，求和。例題2：已知數(shù)列的前n項(xiàng)和，則數(shù)列的通向公式為注意：活用前n項(xiàng)和通項(xiàng)的關(guān)系。例題3：在等差數(shù)列中，求。······

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21、············二、等差數(shù)列的性質(zhì)1、等差數(shù)列中，連續(xù)m項(xiàng)的和仍組成等差數(shù)列，即：仍為等差數(shù)列。2、設(shè)數(shù)列的前n項(xiàng)和的公式為，則為等差數(shù)列的充要條件是。3、等差數(shù)列中，當(dāng)n為奇數(shù)時(shí)， ，當(dāng)n為偶數(shù)時(shí)， ······················

22、83;·············································例題1：等差數(shù)列的公差，且，求。例題2：已知等差數(shù)列的

23、前n項(xiàng)和為377，項(xiàng)數(shù)n為奇數(shù)，且前n項(xiàng)和中奇數(shù)項(xiàng)和偶數(shù)項(xiàng)的比是6：7，求中間項(xiàng)。例題3：等差數(shù)列的前4項(xiàng)和為25，后四項(xiàng)和為63，前n項(xiàng)和為286，求n。變式1：（中難）在等差數(shù)列中，求。例題4：（中難）已知等差數(shù)列的前n項(xiàng)和分別為和，若，求·····························

24、3;······································三、裂項(xiàng)相消法求數(shù)列前n項(xiàng)和例題1：求數(shù)列；········

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26、··········2.4 等比數(shù)列一、概念1、如果一個(gè)數(shù)列從第二項(xiàng)起，每一項(xiàng)與它的前一項(xiàng)的比等于同一個(gè)常數(shù)，那么這個(gè)數(shù)列就叫做等比數(shù)列，這個(gè)常數(shù)叫做 。2、定義法證明數(shù)列是等比數(shù)列若數(shù)列中存在：（q為常數(shù)），則為等比數(shù)列；3、注意：等比數(shù)列中公比，任意一項(xiàng)不能等于0···················&#

27、183;················································例題1：判斷下面

28、數(shù)列是否等比數(shù)列：（1）（2）在數(shù)列中已知；（3）常數(shù)列（4）在數(shù)列中，其中。···········································

29、83;························二、等比數(shù)列通項(xiàng)公式1、通項(xiàng)公式： 2、推導(dǎo)過程：累乘法3、等差中項(xiàng)：若a，G，b成等比數(shù)列，則A叫做a與b的等比中項(xiàng)，且注意：通項(xiàng)公式中的“”中，知任意三個(gè)可求另一個(gè)。···········&#

30、183;·················································&#

31、183;······例題1：已知等比數(shù)列，若，求。例題2：已知數(shù)列為等比數(shù)列。若，且，求的值例題3：（整體思想的應(yīng)用）若數(shù)列滿足關(guān)系，求數(shù)列的通項(xiàng)公式。································&#

32、183;···································三、等比數(shù)列的簡單性質(zhì)1、若，則2、下標(biāo)為等比數(shù)列的項(xiàng)也為等比數(shù)列3、數(shù)列（為常數(shù)）仍為等比數(shù)列4、和均為等比數(shù)列，則也為等比數(shù)列。5、若為等比數(shù)列，公比為q，則其奇數(shù)項(xiàng)或

33、偶數(shù)項(xiàng)也能組成等比數(shù)列，公比為q2.···············································

34、·····················例題4：在等差數(shù)列中，若，則有等式成立，類比上述性質(zhì)，相應(yīng)地，在等比數(shù)列中，若，則有等式 成立。例題5：設(shè)為公比的等比數(shù)列，若和是方程的兩根，則 。四、判斷一個(gè)數(shù)列是否為等比數(shù)列的方法定義法：等比中項(xiàng)：通項(xiàng)法： 求和法：··········

35、3;·················································

36、3;·······例題5：數(shù)列的前n項(xiàng)和記為，已知。證明：（1）數(shù)列是等比數(shù)列；（2）例題6：設(shè)數(shù)列的前n項(xiàng)和記為，已知，求證：當(dāng)時(shí)，是等比數(shù)列2.5 等比數(shù)列前n項(xiàng)和一、等比數(shù)列前n項(xiàng)和公式1、公式： 2、“知三求二”3、注意求和時(shí)，討論“1”4、對(duì)等比數(shù)列前n項(xiàng)和：“”的理解。····················

37、3;···············································例題1：求和·

38、3;·················································

39、3;················二、等比數(shù)列前n項(xiàng)和的性質(zhì)1、連續(xù)m項(xiàng)的和仍為等比數(shù)列2、為等比數(shù)列3、若n為奇數(shù)，則奇數(shù)項(xiàng)和偶數(shù)項(xiàng)和×公比；若n為偶數(shù)，則偶數(shù)項(xiàng)和奇數(shù)項(xiàng)和×公比·····················&