新高考數(shù)學二輪復習專題三 數(shù)列第1講 等差（等比）數(shù)列 原卷版

第1講等差（等比）數(shù)列目錄第一部分：知識強化第二部分：重難點題型突破突破一：判斷（證明）等差（等比）數(shù)列突破二：等差（等比）中項突破三：等差（等比）數(shù)列下標和性質突破四：等差（等比）數(shù)列的單調性突破五：等差（等比）數(shù)列奇偶項和突破六：等差（等比）數(shù)列片段和性質突破七：兩個等差數(shù)列前SKIPIF1<0項和比的問題

第三部分：沖刺重難點特訓第一部分：知識強化1、等差中項由三個數(shù)SKIPIF1<0，SKIPIF1<0,SKIPIF1<0組成的等差數(shù)列可以看成是最簡單的等差數(shù)列.這時,SKIPIF1<0叫做SKIPIF1<0與SKIPIF1<0的等差中項.這三個數(shù)滿足關系式SKIPIF1<0.2、等差數(shù)列的單調性①當SKIPIF1<0，等差數(shù)列SKIPIF1<0為遞增數(shù)列②當SKIPIF1<0，等差數(shù)列SKIPIF1<0為遞減數(shù)列③當SKIPIF1<0，等差數(shù)列SKIPIF1<0為常數(shù)列3、等差數(shù)列的四種判斷方法(1)定義法SKIPIF1<0（或者SKIPIF1<0）(SKIPIF1<0是常數(shù))SKIPIF1<0是等差數(shù)列．(2)等差中項法：SKIPIF1<0(SKIPIF1<0)SKIPIF1<0是等差數(shù)列．(3)通項公式：SKIPIF1<0(SKIPIF1<0為常數(shù))SKIPIF1<0是等差數(shù)列．（SKIPIF1<0可以看做關于SKIPIF1<0的一次函數(shù)）(4)前SKIPIF1<0項和公式：SKIPIF1<0(SKIPIF1<0為常數(shù))SKIPIF1<0是等差數(shù)列．(SKIPIF1<0可以看做關于SKIPIF1<0的二次函數(shù)，但是不含常數(shù)項SKIPIF1<0）提醒;證明一個數(shù)列是等差數(shù)列,只能用定義法或等差中項法4、等差數(shù)列前SKIPIF1<0項和性質(1)若數(shù)列SKIPIF1<0是公差為SKIPIF1<0的等差數(shù)列,則數(shù)列SKIPIF1<0也是等差數(shù)列,且公差為SKIPIF1<0(2)設等差數(shù)列SKIPIF1<0的公差為SKIPIF1<0,SKIPIF1<0為其前SKIPIF1<0項和,則SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0,…組成公差為SKIPIF1<0的等差數(shù)列(3)在等差數(shù)列SKIPIF1<0，SKIPIF1<0中，它們的前SKIPIF1<0項和分別記為SKIPIF1<0則SKIPIF1<0(4)若等差數(shù)列SKIPIF1<0的項數(shù)為SKIPIF1<0,則SKIPIF1<0SKIPIF1<0，SKIPIF1<0。(5)若等差數(shù)列SKIPIF1<0的項數(shù)為SKIPIF1<0,則SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<05、等比中項如果SKIPIF1<0，SKIPIF1<0，SKIPIF1<0成等比數(shù)列，那么SKIPIF1<0叫做SKIPIF1<0與SKIPIF1<0的等比中項．即：SKIPIF1<0是SKIPIF1<0與SKIPIF1<0的等比中項?SKIPIF1<0，SKIPIF1<0，SKIPIF1<0成等比數(shù)列?SKIPIF1<0.6、等比數(shù)列的單調性已知等比數(shù)列SKIPIF1<0的首項為SKIPIF1<0，公比為SKIPIF1<01、當SKIPIF1<0或SKIPIF1<0時，等比數(shù)列SKIPIF1<0為遞增數(shù)列；2、當SKIPIF1<0或SKIPIF1<0時，等比數(shù)列SKIPIF1<0為遞減數(shù)列；3、當SKIPIF1<0時，等比數(shù)列SKIPIF1<0為常數(shù)列（SKIPIF1<0）4、當SKIPIF1<0時，等比數(shù)列SKIPIF1<0為擺動數(shù)列.7、等比數(shù)列的判斷（證明）1、定義：SKIPIF1<0（或者SKIPIF1<0）（可判斷，可證明）2、等比中項法：驗證SKIPIF1<0（特別注意SKIPIF1<0）（可判斷，可證明）3、通項公式法：驗證通項是關于SKIPIF1<0的指數(shù)型函數(shù)（只可判斷）8、等比數(shù)列前SKIPIF1<0項和的性質公比為SKIPIF1<0的等比數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0,關于SKIPIF1<0的性質?？嫉挠幸韵滤念?(1)數(shù)列SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0,…組成公比為SKIPIF1<0（SKIPIF1<0）的等比數(shù)列(2)當SKIPIF1<0是偶數(shù)時,SKIPIF1<0當SKIPIF1<0是奇數(shù)時,SKIPIF1<0(3)SKIPIF1<0第二部分：重難點題型突破突破一：判斷（證明）等差（等比）數(shù)列1．（2022·廣東·深圳實驗學校光明部高三期中）“數(shù)列SKIPIF1<0為等差數(shù)列”是“數(shù)列SKIPIF1<0為等比數(shù)列”的（

）A．充分不必要條件 B．必要不充分條件 C．充要條件 D．既不充分也不必要條件2．（2022·山東省莒南第一中學高三期中）“數(shù)列SKIPIF1<0為等比數(shù)列”是“數(shù)列SKIPIF1<0為等差數(shù)列”的（

）A．充分不必要條件 B．必要不充分條件C．充要條件 D．既不充分也不必要條件3．（2022·陜西·榆林市第十中學高一期末）已知等比數(shù)列SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，則（

）A．數(shù)列SKIPIF1<0是等差等列 B．數(shù)列SKIPIF1<0是等差數(shù)列C．數(shù)列SKIPIF1<0是遞減數(shù)列 D．數(shù)列SKIPIF1<0是遞增數(shù)列4．（2022·北京·人大附中高三開學考試）若數(shù)列SKIPIF1<0滿足SKIPIF1<0，則“SKIPIF1<0，SKIPIF1<0，SKIPIF1<0”是“SKIPIF1<0為等比數(shù)列”的（

）A．充分而不必要條件 B．必要而不充分條件 C．充分必要條件 D．既不充分也不必要條件5．（2022·全國·高三專題練習）數(shù)列SKIPIF1<0中，“SKIPIF1<0，SKIPIF1<0”是“SKIPIF1<0是公比為2的等比數(shù)列”的（

）A．充分而不必要條件 B．必要而不充分條件C．充分必要條件 D．既不充分也不必要條件6．（2022·江西省萬載中學高一階段練習（文））若數(shù)列{an}的前n項和Sn=an-1(a∈R，且a≠0)，則此數(shù)列是（

）A．等差數(shù)列B．等比數(shù)列C．等差數(shù)列或等比數(shù)列D．既不是等差數(shù)列，也不是等比數(shù)列突破二：等差（等比）中項1．（2022·廣西河池·模擬預測（文））已知SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0是SKIPIF1<0與SKIPIF1<0的等差中項，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·湖北黃岡·高三階段練習）已知正項等比數(shù)列SKIPIF1<0滿足SKIPIF1<0，若SKIPIF1<0是SKIPIF1<0和SKIPIF1<0的等差中項，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·山西·高三期中）已知數(shù)列SKIPIF1<0是等差數(shù)列，且SKIPIF1<0．若SKIPIF1<0是SKIPIF1<0和SKIPIF1<0的等差中項，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．（2022·全國·模擬預測）已知正實數(shù)b是實數(shù)a和實數(shù)c的等差中項，且SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0成等比數(shù)列，則SKIPIF1<0______．5．（2022·山西臨汾·高三階段練習）已知SKIPIF1<0，若SKIPIF1<0是SKIPIF1<0與SKIPIF1<0的等比中項，則SKIPIF1<0的最小值為__________．6．（2022·天津河東·高二期末）設各項均為正數(shù)的等差數(shù)列SKIPIF1<0的前n（SKIPIF1<0）項和為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0是SKIPIF1<0與SKIPIF1<0的等比中項，則數(shù)列SKIPIF1<0的公差d為______．突破三：等差（等比）數(shù)列下標和性質1．（2022·全國·高三專題練習）已知等差數(shù)列SKIPIF1<0滿足SKIPIF1<0（SKIPIF1<0，SKIPIF1<0），則SKIPIF1<0_____．2．（2022·河南·宜陽縣第一高級中學高二階段練習（理））已知數(shù)列SKIPIF1<0為等差數(shù)列，其前SKIPIF1<0項和為SKIPIF1<0，則SKIPIF1<0___________.3．（2022·陜西·長安一中高一階段練習）設SKIPIF1<0為公比SKIPIF1<0的等比數(shù)列，若SKIPIF1<0和SKIPIF1<0是方程SKIPIF1<0的兩根，則SKIPIF1<0___________.4．（2022·福建省福州第八中學高三階段練習）在正項等比數(shù)列SKIPIF1<0中，若SKIPIF1<0，則SKIPIF1<0______．5．（2022·安徽省臨泉第一中學高二期末）已知數(shù)列SKIPIF1<0是等差數(shù)列，數(shù)列SKIPIF1<0是等比數(shù)列，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0___________.6．（2022·全國·高二課時練習）等比數(shù)列SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0是方程SKIPIF1<0的兩根，則SKIPIF1<0的值為___________.7．（2022·全國·高三專題練習（文））已知數(shù)列SKIPIF1<0是等比數(shù)列，數(shù)列SKIPIF1<0是等差數(shù)列，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0___________.突破四：等差（等比）數(shù)列的單調性1．（2022·陜西·渭南市瑞泉中學高二階段練習）在等差數(shù)列SKIPIF1<0中，SKIPIF1<0記SKIPIF1<0，則數(shù)列SKIPIF1<0（

）A．有最大項，有最小項 B．有最大項，無最小項C．無最大項，有最小項 D．無最大項，無最小項2．（2022·全國·高三專題練習）已知數(shù)列SKIPIF1<0是首項為SKIPIF1<0，公差為1的等差數(shù)列，數(shù)列SKIPIF1<0滿足SKIPIF1<0若對任意的SKIPIF1<0，都有SKIPIF1<0成立，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·全國·高三專題練習）設等比數(shù)列SKIPIF1<0的公比為SKIPIF1<0，其前SKIPIF1<0項的積為SKIPIF1<0，并且滿足條件SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則使SKIPIF1<0成立的最大自然數(shù)SKIPIF1<0的值為（

）A．9 B．10C．18 D．194．（2022·安徽·高三開學考試）設正項等比數(shù)列SKIPIF1<0的前SKIPIF1<0項乘積為SKIPIF1<0，已知SKIPIF1<0，則SKIPIF1<0的（

）A．最大值為32 B．最大值為1024C．最小值為SKIPIF1<0 D．最小值為SKIPIF1<0突破五：等差（等比）數(shù)列奇偶項和1．（2022·全國·高三專題練習）已知等差數(shù)列SKIPIF1<0共有SKIPIF1<0項，其中奇數(shù)項之和為290，偶數(shù)項之和為261，則SKIPIF1<0的值為（

）．A．30 B．29 C．28 D．272．（2022·全國·高三專題練習）已知等差數(shù)列SKIPIF1<0共有SKIPIF1<0項，若數(shù)列SKIPIF1<0中奇數(shù)項的和為SKIPIF1<0，偶數(shù)項的和為SKIPIF1<0，SKIPIF1<0，則公差SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·全國·高三專題練習）已知等差數(shù)列SKIPIF1<0的公差為4，項數(shù)為偶數(shù)，所有奇數(shù)項的和為15，所有偶數(shù)項的和為55，則這個數(shù)列的項數(shù)為A．10 B．20 C．30 D．404．（2020·全國·高二課時練習）一個項數(shù)為偶數(shù)的等比數(shù)列，它的偶數(shù)項和是奇數(shù)項和的2倍，又它的首項為1，且中間兩項的和為24，則此等比數(shù)列的項數(shù)為（

）A．6 B．8 C．10 D．125．（2022·全國·高三專題練習）已知一個等比數(shù)列首項為SKIPIF1<0，項數(shù)是偶數(shù)，其奇數(shù)項之和為SKIPIF1<0，偶數(shù)項之和為SKIPIF1<0，則這個數(shù)列的項數(shù)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022·全國·高二課時練習）等比數(shù)列SKIPIF1<0共有SKIPIF1<0項，其中SKIPIF1<0，偶數(shù)項和為84，奇數(shù)項和為170，則SKIPIF1<0（

）A．3 B．4 C．7 D．9突破六：等差（等比）數(shù)列片段和性質1．（2022·全國·高三專題練習）已知等差數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．13 C．-13 D．-182．（2022·全國·高二課時練習）等差數(shù)列SKIPIF1<0中其前n項和為SKIPIF1<0,SKIPIF1<0則SKIPIF1<0為．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·全國·高二課時練習）已知等差數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022·寧夏·吳忠中學高二期中（理））設等差數(shù)列的前n項和為SKIPIF1<0，則SKIPIF1<0=.5．（2022·四川南充·三模（理））若等比數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0_____．6．（2022·新疆維吾爾自治區(qū)喀什第二中學高三階段練習）已知等比數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0________.7．（2022·廣東·潮州市湘橋區(qū)南春中學高二階段練習）已知SKIPIF1<0為等比數(shù)列SKIPIF1<0的前n項和，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0_____________.8．（2022·全國·高二課時練習）一個等比數(shù)列的前SKIPIF1<0項和為10，前SKIPIF1<0項和為30，則前SKIPIF1<0項和為_____________.9．（2022·全國·高二課時練習）已知數(shù)列是等比數(shù)列，其前SKIPIF1<0項和為SKIPIF1<0.若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0___________.突破七：兩個等差數(shù)列前SKIPIF1<0項和比的問題1．（2022·云南昭通·高三期末（理））等差數(shù)列SKIPIF1<0的前n項和分別為SKIPIF1<0，則SKIPIF1<0的公差為___________.2．（2022·上?！じ呷龑ｎ}練習）已知數(shù)列SKIPIF1<0、SKIPIF1<0均為正項等比數(shù)列，SKIPIF1<0、SKIPIF1<0分別為數(shù)列SKIPIF1<0、SKIPIF1<0的前SKIPIF1<0項積，且SKIPIF1<0，則SKIPIF1<0的值為___________.3．（2022·天津·南開中學高二期末）設等差數(shù)列SKIPIF1<0，SKIPIF1<0的前SKIPIF1<0項和分別為SKIPIF1<0，SKIPIF1<0，若對任意自然數(shù)SKIPIF1<0都有SKIPIF1<0，則SKIPIF1<0的值為______.4．（2022·上海·高二課時練習）已知兩個等差數(shù)列SKIPIF1<0和SKIPIF1<0的前SKIPIF1<0項和的比SKIPIF1<0，則它們相應的第SKIPIF1<0項的比SKIPIF1<0______．5．（2022·四川·達州市第一中學校高一階段練習）已知等差數(shù)列SKIPIF1<0，SKIPIF1<0的前SKIPIF1<0項和分別為SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0______.6．（2022·全國·高二課時練習）等差數(shù)列SKIPIF1<0，SKIPIF1<0的前SKIPIF1<0項和分別為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0______.7．（2022·福建·莆田第五中學高三期中）已知SKIPIF1<0、SKIPIF1<0分別是等差數(shù)列SKIPIF1<0、SKIPIF1<0的前SKIPIF1<0項的和，且SKIPIF1<0.則SKIPIF1<0______.第三部分：沖刺重難點特訓一、單選題1．（2022·全國·模擬預測）設SKIPIF1<0為等差數(shù)列SKIPIF1<0的前SKIPIF1<0項和，且SKIPIF1<0，都有SKIPIF1<0.若SKIPIF1<0，則（

）A．SKIPIF1<0的最小值是SKIPIF1<0 B．SKIPIF1<0的最小值是SKIPIF1<0C．SKIPIF1<0的最大值是SKIPIF1<0 D．SKIPIF1<0的最大值是SKIPIF1<02．（2022·甘肅·高臺縣第一中學模擬預測（理））已知SKIPIF1<0是各項不全為零的等差數(shù)列，前n項和是SKIPIF1<0，且SKIPIF1<0，若SKIPIF1<0，則正整數(shù)m＝（

）A．2020 B．2019 C．2018 D．20173．（2022·浙江臺州·模擬預測）已知數(shù)列SKIPIF1<0滿足：SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0（

）A．1 B．2 C．3 D．20224．（2022·甘肅·高臺縣第一中學模擬預測（文））已知正項等比數(shù)列SKIPIF1<0滿足SKIPIF1<0（其中SKIPIF1<0），則SKIPIF1<0的最小值為（

）．A．6 B．16 C．SKIPIF1<0 D．25．（2022·全國·模擬預測）已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是SKIPIF1<0與SKIPIF1<0的等比中項，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．（2022·黑龍江·哈爾濱市第一二二中學校三模（文））公比為q的等比數(shù)列SKIPIF1<0，其前n項和為SKIPIF1<0，前n項積為SKIPIF1<0，滿足SKIPIF1<0．則下列結論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0的最大值為SKIPIF1<0C．SKIPIF1<0的最大值為SKIPIF1<0 D．SKIPIF1<07．（2022·安徽·蕪湖一中模擬預測）已知正項等比數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，前n項積為SKIPIF1<0，滿足SKIPIF1<0，則SKIPIF1<0的最小值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2022·四川廣安·模擬預測（文））已知數(shù)列SKIPIF1<0為等比數(shù)列，若SKIPIF1<0，SKIPIF1<0為函數(shù)SKIPIF1<0的兩個零點，則SKIPIF1<0（

）A．10 B．12 C．32 D．33二、多選題9．（2022·全國·模擬預測）在數(shù)列SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0，使得SKIPIF1<0D．SKIPIF1<0，都有SKIPIF1<010．（2022·湖北省仙桃中學模擬預測）已知SKIPIF1<0為數(shù)列SKIPIF1<0的前SKIPIF1<0項之和，且滿足SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<