2024年數(shù)學(xué)高考大一輪復(fù)習(xí)第六章 §6.5　數(shù)列求和-試卷下載-教習(xí)網(wǎng)

§6.5　數(shù)列求和考試要求　1.熟練掌握等差、等比數(shù)列的前n項(xiàng)和公式.2.掌握非等差數(shù)列、非等比數(shù)列求和的幾種常用方法．知識梳理數(shù)列求和的幾種常用方法1．公式法直接利用等差數(shù)列、等比數(shù)列的前n項(xiàng)和公式求和．(1)等差數(shù)列的前n項(xiàng)和公式：Sn＝____________＝__________________.(2)等比數(shù)列的前n項(xiàng)和公式：Sn＝2．分組求和法與并項(xiàng)求和法(1)分組求和法若一個(gè)數(shù)列是由若干個(gè)等差數(shù)列或等比數(shù)列或可求和的數(shù)列組成，則求和時(shí)可用分組求和法，分別求和后相加減．(2)并項(xiàng)求和法一個(gè)數(shù)列的前n項(xiàng)和中，可兩兩結(jié)合求解，則稱之為并項(xiàng)求和．形如an＝(－1)nf(n)類型，可采用兩項(xiàng)合并求解．3．錯(cuò)位相減法如果一個(gè)數(shù)列的各項(xiàng)是由一個(gè)等差數(shù)列和一個(gè)等比數(shù)列的對應(yīng)項(xiàng)之積構(gòu)成的，那么這個(gè)數(shù)列的前n項(xiàng)和即可用此法來求，如等比數(shù)列的前n項(xiàng)和公式就是用此法推導(dǎo)的．4．裂項(xiàng)相消法把數(shù)列的通項(xiàng)拆成兩項(xiàng)之差，在求和時(shí)中間的一些項(xiàng)可以相互抵消，從而求得其和．常見的裂項(xiàng)技巧(1)＝－.(2)＝.(3)＝.(4)＝－.(5)＝.常用結(jié)論常用求和公式(1)1＋2＋3＋4＋…＋n＝.(2)1＋3＋5＋7＋…＋(2n－1)＝n2.(3)12＋22＋32＋…＋n2＝.(4)13＋23＋33＋…＋n3＝2.思考辨析判斷下列結(jié)論是否正確(請?jiān)诶ㄌ栔写?/span>“√”或“×”)(1)如果數(shù)列{an}為等比數(shù)列，且公比不等于1，則其前n項(xiàng)和Sn＝.(　　)(2)求Sn＝a＋2a2＋3a3＋…＋nan時(shí)，只要把上式等號兩邊同時(shí)乘a即可根據(jù)錯(cuò)位相減法求得．(　　)(3)已知等差數(shù)列{an}的公差為d，則有＝. (　　)(4)sin21°＋sin22°＋sin23°＋…＋sin288°＋sin289°＝44.5.(　　)教材改編題1．已知函數(shù)f(n)＝且an＝f(n)＋f(n＋1)，則a1＋a2＋a3＋…＋a100等于(　　)A．0 B．100 C．－100 D．10 2002．?dāng)?shù)列{an}的前n項(xiàng)和為Sn.若an＝，則S5等于(　　)A．1 B. C. D.3．Sn＝＋＋＋…＋等于(　　)A. B.C. D. 題型一　分組求和與并項(xiàng)求和例1 (2023·菏澤模擬)已知數(shù)列{an}中，a1＝1，它的前n項(xiàng)和Sn滿足2Sn＋an＋1＝2n＋1－1.(1)證明：數(shù)列為等比數(shù)列；(2)求S1＋S2＋S3＋…＋S2n.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________延伸探究在本例(2)中，如何求S1＋S2＋S3＋…＋Sn?________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 (1)若數(shù)列{cn}的通項(xiàng)公式為cn＝an±bn，且{an}，{bn}為等差或等比數(shù)列，可采用分組求和法求數(shù)列{cn}的前n項(xiàng)和．(2)若數(shù)列{cn}的通項(xiàng)公式為cn＝其中數(shù)列{an}，{bn}是等比數(shù)列或等差數(shù)列，可采用分組求和法求{cn}的前n項(xiàng)和．跟蹤訓(xùn)練1 記數(shù)列{an}的前n項(xiàng)和為Sn，已知Sn＝2an－2n＋1.(1)求數(shù)列{an}的通項(xiàng)公式；(2)記bn＝(－1)n·log2，求數(shù)列{bn}的前n項(xiàng)和Tn.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二　錯(cuò)位相減法求和例2 (12分)(2021·全國乙卷)設(shè){an}是首項(xiàng)為1的等比數(shù)列，數(shù)列{bn}滿足bn＝.已知a1，3a2，9a3成等差數(shù)列．(1)求{an}和{bn}的通項(xiàng)公式； [切入點(diǎn)：設(shè)基本量q](2)記Sn和Tn分別為{an}和{bn}的前n項(xiàng)和．證明：Tn<.[關(guān)鍵點(diǎn)：bn＝n·n]思維升華 (1)如果數(shù)列{an}是等差數(shù)列，{bn}是等比數(shù)列，求數(shù)列{an·bn}的前n項(xiàng)和時(shí)，常采用錯(cuò)位相減法．(2)錯(cuò)位相減法求和時(shí)，應(yīng)注意：①在寫出“Sn”與“qSn”的表達(dá)式時(shí)應(yīng)特別注意將兩式“錯(cuò)項(xiàng)對齊”，以便于下一步準(zhǔn)確地寫出“Sn－qSn”的表達(dá)式．②應(yīng)用等比數(shù)列求和公式時(shí)必須注意公比q是否等于1，如果q＝1，應(yīng)用公式Sn＝na1.跟蹤訓(xùn)練2 (2021·浙江)已知數(shù)列{an}的前n項(xiàng)和為Sn，a1＝－，且4Sn＋1＝3Sn－9(n∈N*)．(1)求數(shù)列{an}的通項(xiàng)公式；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)設(shè)數(shù)列{bn}滿足3bn＋(n－4)an＝0(n∈N*)，記{bn}的前n項(xiàng)和為Tn.若Tn≤λbn，對任意n∈N*恒成立，求實(shí)數(shù)λ的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型三　裂項(xiàng)相消法求和例3 (2022·新高考全國Ⅰ)記Sn為數(shù)列{an}的前n項(xiàng)和，已知a1＝1，是公差為的等差數(shù)列．(1)求{an}的通項(xiàng)公式；(2)證明：＋＋…＋<2.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華裂項(xiàng)相消法的原則及規(guī)律(1)裂項(xiàng)原則一般是前面裂幾項(xiàng)，后面就裂幾項(xiàng)，直到發(fā)現(xiàn)被消去項(xiàng)的規(guī)律為止．(2)消項(xiàng)規(guī)律消項(xiàng)后前面剩幾項(xiàng)，后面就剩幾項(xiàng)，前面剩第幾項(xiàng)，后面就剩倒數(shù)第幾項(xiàng)．跟蹤訓(xùn)練3 (2022·湛江模擬)已知數(shù)列{an}是等比數(shù)列，且8a3＝a6，a2＋a5＝36.(1)求數(shù)列{an}的通項(xiàng)公式；(2)設(shè)bn＝，求數(shù)列{bn}的前n項(xiàng)和Tn，并證明：Tn<.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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