2024年數(shù)學(xué)高考大一輪復(fù)習(xí)第六章 §6.6　數(shù)列中的綜合問題-試卷下載-教習(xí)網(wǎng)

§6.6　數(shù)列中的綜合問題考試要求　數(shù)列的綜合運(yùn)算問題以及數(shù)列與函數(shù)、不等式等知識的交匯問題，是歷年高考的熱點(diǎn)內(nèi)容．一般圍繞等差數(shù)列、等比數(shù)列的知識命題，涉及數(shù)列的函數(shù)性質(zhì)、通項(xiàng)公式、前n項(xiàng)和公式等．題型一　等差數(shù)列、等比數(shù)列的綜合運(yùn)算例1 (2023·廈門模擬)已知數(shù)列{an}的前n項(xiàng)和為Sn，且Sn＝n2＋n，遞增的等比數(shù)列{bn}滿足b1＋b4＝18，b2·b3＝32.(1)求數(shù)列{an}，{bn}的通項(xiàng)公式；(2)若cn＝an·bn，n∈N*，求數(shù)列{cn}的前n項(xiàng)和Tn.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華數(shù)列的綜合問題常將等差、等比數(shù)列結(jié)合，兩者相互聯(lián)系、相互轉(zhuǎn)化，解答這類問題的方法：尋找通項(xiàng)公式，利用性質(zhì)進(jìn)行轉(zhuǎn)化．跟蹤訓(xùn)練1 (2022·全國甲卷)記Sn為數(shù)列{an}的前n項(xiàng)和．已知＋n＝2an＋1.(1)證明：{an}是等差數(shù)列；(2)若a4，a7，a9成等比數(shù)列，求Sn的最小值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二　數(shù)列與其他知識的交匯問題命題點(diǎn)1　數(shù)列與不等式的交匯例2 (1)已知數(shù)列{an}滿足a1＋a2＋a3＋…＋an＝n2＋n(n∈N*)，設(shè)數(shù)列{bn}滿足：bn＝，數(shù)列{bn}的前n項(xiàng)和為Tn，若Tn<λ(n∈N*)恒成立，則實(shí)數(shù)λ的取值范圍為(　　)A. B.C. D.聽課記錄：______________________________________________________________________________________________________________________________________(2)已知數(shù)列{an}滿足a1＝，3an，2an＋1，anan＋1成等差數(shù)列．①證明：數(shù)列是等比數(shù)列，并求{an}的通項(xiàng)公式；②記{an}的前n項(xiàng)和為Sn，求證：≤Sn<.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________命題點(diǎn)2　數(shù)列與函數(shù)的交匯例3 (1)(2023·龍巖模擬)已知函數(shù)f(x)＝x3＋4x，記等差數(shù)列{an}的前n項(xiàng)和為Sn，若f(a1＋2)＝100，f(a2 022＋2)＝－100，則S2 022等于(　　)A．－4 044 B．－2 022C．2 022 D．4 044(2)數(shù)列{an}是等差數(shù)列，a1＝1，公差d∈[1，2]，且a4＋λa10 ＋a16＝15，則實(shí)數(shù)λ的最大值為________．聽課記錄：______________________________________________________________________________________________________________________________________思維升華 (1)數(shù)列與不等式的綜合問題及求解策略①判斷數(shù)列問題的一些不等關(guān)系，可以利用數(shù)列的單調(diào)性比較大小或借助數(shù)列對應(yīng)的函數(shù)的單調(diào)性比較大?。?/span>②以數(shù)列為載體，考查不等式恒成立的問題，此類問題可轉(zhuǎn)化為函數(shù)的最值．③考查與數(shù)列有關(guān)的不等式證明問題，此類問題一般采用放縮法進(jìn)行證明，有時(shí)也可通過構(gòu)造函數(shù)進(jìn)行證明．(2)數(shù)列與函數(shù)交匯問題的主要類型及求解策略①已知函數(shù)條件，解決數(shù)列問題，此類問題一般利用函數(shù)的性質(zhì)、圖象研究數(shù)列問題．②已知數(shù)列條件，解決函數(shù)問題，解決此類問題一般要利用數(shù)列的通項(xiàng)公式、前n項(xiàng)和公式、求和方法等對式子化簡變形．跟蹤訓(xùn)練2 (1)設(shè){an}是等比數(shù)列，函數(shù)y＝x2－x－2 023的兩個(gè)零點(diǎn)是a2，a3，則a1a4等于(　　)A．2 023 B．1 C．－1 D．－2 023(2)數(shù)列{an}滿足a1＝1，an＋1＝2an(n∈N*)，Sn為其前n項(xiàng)和．?dāng)?shù)列{bn}為等差數(shù)列，且滿足b1＝a1，b4＝S3.①求數(shù)列{an}，{bn}的通項(xiàng)公式；②設(shè)cn＝，數(shù)列{cn}的前n項(xiàng)和為Tn，證明：≤Tn<.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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