新高考數(shù)學(xué)二輪復(fù)習(xí) 數(shù)列重難點提升專題07 數(shù)列求和-錯位相減、裂項相消（原卷版）

專題07數(shù)列求和-錯位相減、裂項相消◆錯位相減法錯位相減法是求解由等差數(shù)列SKIPIF1<0和等比數(shù)列SKIPIF1<0對應(yīng)項之積組成的數(shù)列SKIPIF1<0(即SKIPIF1<0)的前SKIPIF1<0項和的方法.這種方法運算量較大,要重視解題過程的訓(xùn)練.在講等比數(shù)列的時候,我們推導(dǎo)過等比數(shù)列的求和公式,其過程正是利用錯位相減的原理,等比數(shù)列的通項SKIPIF1<0其實可以看成等差數(shù)列通項SKIPIF1<0與等比數(shù)列通項SKIPIF1<0的積.公式秒殺:SKIPIF1<0(錯位相減都可化簡為這種形式,對于求解參數(shù)SKIPIF1<0與SKIPIF1<0,可以采用將前1項和與前2項和代入式中,建立二元一次方程求解.此方法可以快速求解出結(jié)果或者作為檢驗對錯的依據(jù).)【經(jīng)典例題1】設(shè)數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，若SKIPIF1<0．(1)求數(shù)列SKIPIF1<0的通項公式；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0．【經(jīng)典例題2】已知等差數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，數(shù)列SKIPIF1<0為等比數(shù)列，且SKIPIF1<0，SKIPIF1<0．(1)求數(shù)列SKIPIF1<0，SKIPIF1<0的通項公式；(2)若SKIPIF1<0，求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0．【經(jīng)典例題3】已知各項均為正數(shù)的等比數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0．(1)求數(shù)列SKIPIF1<0的通項公式；(2)若SKIPIF1<0，求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0．【練習(xí)1】已知數(shù)列SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項公式；(2)求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0.【練習(xí)2】已知數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，且SKIPIF1<0．(1)求SKIPIF1<0的通項公式；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0．【練習(xí)3】已知數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，且SKIPIF1<0．(1)求SKIPIF1<0的通項公式；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0．【練習(xí)4】已知數(shù)列SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0（SKIPIF1<0）.(1)求證數(shù)列SKIPIF1<0為等差數(shù)列；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0.◆裂項相消法把數(shù)列的通項拆成相鄰兩項之差,在求和時中間的一些項可以相互抵消,從而求得其和.在消項時要注意前面保留第幾項,最后也要保留相對應(yīng)的倒數(shù)幾項.例如消項時保留第一項和第3項,相應(yīng)的也要保留最后一項和倒數(shù)第三項.常見的裂項形式:(1)SKIPIF1<0;(2)SKIPIF1<0;(3)SKIPIF1<0;(4)SKIPIF1<0;(5)SKIPIF1<0;(6)SKIPIF1<0;(7)SKIPIF1<0;(8)SKIPIF1<0(9)SKIPIF1<0(10)SKIPIF1<0.(11)SKIPIF1<0(12)SKIPIF1<0【經(jīng)典例題1】已知正項數(shù)列SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，則數(shù)列SKIPIF1<0的前SKIPIF1<0項和為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【經(jīng)典例題2】數(shù)列SKIPIF1<0的通項公式為SKIPIF1<0，該數(shù)列的前8項和為__________．【經(jīng)典例題3】已知數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，若SKIPIF1<0，則數(shù)列SKIPIF1<0的前SKIPIF1<0項和為________．【練習(xí)1】數(shù)列SKIPIF1<0的前2022項和為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【練習(xí)2】數(shù)列SKIPIF1<0的各項均為正數(shù)，SKIPIF1<0為其前SKIPIF1<0項和，對于任意的SKIPIF1<0，總有SKIPIF1<0，SKIPIF1<0，SKIPIF1<0成等差數(shù)列，又記SKIPIF1<0，數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0______．【練習(xí)3】SKIPIF1<0_______.【練習(xí)4】設(shè)數(shù)列SKIPIF1<0滿足SKIPIF1<0.(1)求SKIPIF1<0的通項公式；(2)求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0.【練習(xí)5】已知數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，且SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項公式；(2)設(shè)SKIPIF1<0，SKIPIF1<0，求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0【練習(xí)6】已知數(shù)列SKIPIF1<0中，SKIPIF1<0．(1)證明：SKIPIF1<0為等比數(shù)列，并求SKIPIF1<0的通項公式；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0．【練習(xí)7】記SKIPIF1<0是公差不為零的等差數(shù)列SKIPIF1<0的前SKIPIF1<0項和，若SKIPIF1<0，SKIPIF1<0是SKIPIF1<0和SKIPIF1<0的等比中項.(1)求數(shù)列SKIPIF1<0的通項公式；(2)記SKIPIF1<0，求數(shù)列SKIPIF1<0的前20項和.【練習(xí)8】已知等差數(shù)列SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，SKIPIF1<0（SKIPIF1<0）．(1)求數(shù)列SKIPIF1<0，SKIPIF1<0的通項公式；(2)數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，求SKIPIF1<0．【練習(xí)9】已知正項數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，且SKIPIF1<0、SKIPIF1<0、SKIPIF1<0成等比數(shù)列，其中SKIPIF1<0．(1)求數(shù)列SKIPIF1<0的通項公式；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0．【練習(xí)10】已知SKIPIF1<0是數(shù)列SKIPIF1<0的前SKIPIF1<0項和，SKIPIF1<0，___________.①SKIPIF1<0，SKIPIF1<0；②數(shù)列SKIPIF1<0為等差數(shù)列，且SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0.從以上兩個條件中任選一個補充在橫線處，并求解：(1)求SKIPIF1<0；(2)設(shè)SKIPIF1<0，求數(shù)列SKIPIF1<0的前SKIPIF1<0項和SKIPIF1<0.【過關(guān)檢測】一、單選題1．SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．數(shù)列SKIPIF1<0的前n項和等于（

）．A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．已知等比數(shù)列{an}的前n項和為Sn，若S3=7，S6=63，則數(shù)列{nan}的前n項和為（

）A．-3+(n+1)×2n B．3+(n+1)×2nC．1+(n+1)×2n D．1+(n-1)×2n4．已知等差數(shù)列SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則數(shù)列SKIPIF1<0的前8項和為（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，SKIPIF1<0．記SKIPIF1<0，數(shù)列的前SKIPIF1<0項和為SKIPIF1<0，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．已知數(shù)列滿足SKIPIF1<0，設(shè)SKIPIF1<0，則數(shù)列SKIPIF1<0的前2022項和為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知數(shù)列SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．2021 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．等差數(shù)列SKIPIF1<0中，SKIPIF1<0，設(shè)SKIPIF1<0，則數(shù)列SKIPIF1<0的前61項和為（

）A．SKIPIF1<0 B．7 C．SKIPIF1<0 D．89．設(shè)數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，則（

）A．25<S100<25.5 B．25.5<S100<26C．26<S100<27 D．27<S100<27.510．已知數(shù)列SKIPIF1<0滿足SKIPIF1<0，則數(shù)列SKIPIF1<0的前5項和為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．已知數(shù)列SKIPIF1<0的首項SKIPIF1<0，且滿足SKIPIF1<0，記數(shù)列SKIPIF1<0的前n項和為SKIPIF1<0，若對于任意SKIPIF1<0，不等式SKIPIF1<0恒成立，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．在數(shù)列SKIPIF1<0中，SKIPIF1<0，其前SKIPIF1<0項和SKIPIF1<0滿足SKIPIF1<0，若對任意SKIPIF1<0總有SKIPIF1<0恒成立，則實數(shù)SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題13．已知正項數(shù)列{an}滿足a1＝2且an+12﹣2an2﹣anan+1＝0，令bn＝（n+2）an，則數(shù)列{bn}的前8項的和等于__．14．已知數(shù)列{an}的前n項和為Sn，且Sn＝2an﹣2，則數(shù)列{SKIPIF1<0}的前n項和Tn＝__．15．將SKIPIF1<0（SKIPIF1<0）的展開式中SKIPIF1<0的系數(shù)記為SKIPIF1<0，則SKIPIF1<0__________．16．數(shù)列SKIPIF1<0的前項n和為SKIPIF1<0，滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0______．三、解答題17．已知數(shù)列SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0.(1)求證：數(shù)列SKIPIF1<0為等差數(shù)列；(2)求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0.18．已知正項數(shù)列SKIPIF1<0的前SKIPIF1<0項和為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項公式；(2)若SKIPIF1<0，求數(shù)列SKIPIF1<0的前n項和SKIPIF1<0.19．已