2024學(xué)生版大二輪數(shù)學(xué)新高考提高版（京津瓊魯遼粵冀鄂湘渝閩蘇浙黑吉晉皖云豫新甘貴贛桂）專題三　第2講　數(shù)列求和及其綜合應(yīng)用50

第2講數(shù)列求和及其綜合應(yīng)用[考情分析]1.數(shù)列求和重點(diǎn)考查分組轉(zhuǎn)化、錯(cuò)位相減、裂項(xiàng)相消三種求和方法.2.數(shù)列的綜合問題，一般以等差數(shù)列、等比數(shù)列為背景，與函數(shù)、不等式相結(jié)合，考查最值、范圍以及證明不等式等.3.主要以選擇題、填空題及解答題的形式出現(xiàn)，難度中等．考點(diǎn)一數(shù)列求和核心提煉1．裂項(xiàng)相消法就是把數(shù)列的每一項(xiàng)分解，使得相加后項(xiàng)與項(xiàng)之間能夠相互抵消，但在抵消的過程中，有的是相鄰項(xiàng)抵消，有的是間隔項(xiàng)抵消．常見的裂項(xiàng)方式有：eq\f(1,nn＋k)＝eq\f(1,k)eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,n)－\f(1,n＋k)))；eq\f(1,4n2－1)＝eq\f(1,2)eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,2n－1)－\f(1,2n＋1))).2．錯(cuò)位相減法求和，主要用于求{anbn}的前n項(xiàng)和，其中{an}，{bn}分別為等差數(shù)列和等比數(shù)列．考向1分組轉(zhuǎn)化法例1(2023·棗莊模擬)已知數(shù)列{an}的首項(xiàng)a1＝3，且滿足an＋1＋2an＝2n＋2.(1)證明：{an－2n}為等比數(shù)列；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)已知bn＝eq\b\lc\{\rc\(\a\vs4\al\co1(an，n為奇數(shù)，,log2an，n為偶數(shù)，))Tn為{bn}的前n項(xiàng)和，求T10.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考向2裂項(xiàng)相消法例2(2023·沈陽質(zhì)檢)設(shè)n∈N*，向量eq\o(AB,\s\up6(→))＝(n－1,1)，eq\o(AC,\s\up6(→))＝(n－1,4n－1)，an＝eq\o(AB,\s\up6(→))·eq\o(AC,\s\up6(→)).(1)令bn＝an＋1－an，求證：數(shù)列{bn}為等差數(shù)列；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)求證：eq\f(1,a1)＋eq\f(1,a2)＋…＋eq\f(1,an)<eq\f(3,4).________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考向3錯(cuò)位相減法例3(2023·全國甲卷)記Sn為數(shù)列{an}的前n項(xiàng)和，已知a2＝1,2Sn＝nan.(1)求{an}的通項(xiàng)公式；(2)求數(shù)列eq\b\lc\{\rc\}(\a\vs4\al\co1(\f(an＋1,2n)))的前n項(xiàng)和Tn.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________規(guī)律方法(1)分組轉(zhuǎn)化法求和的關(guān)鍵是將數(shù)列通項(xiàng)轉(zhuǎn)化為若干個(gè)可求和的數(shù)列通項(xiàng)的和或差．(2)裂項(xiàng)相消法的基本思路是將通項(xiàng)拆分，可以產(chǎn)生相互抵消的項(xiàng)．(3)用錯(cuò)位相減法求和時(shí)，應(yīng)注意：①等比數(shù)列的公比為負(fù)數(shù)的情形；②在寫出“Sn”和“qSn”的表達(dá)式時(shí)應(yīng)特別注意將兩式“錯(cuò)項(xiàng)對齊”，以便準(zhǔn)確寫出“Sn－qSn”的表達(dá)式．跟蹤演練1(1)(2023·淮南模擬)已知數(shù)列{an}滿足an＋1－an＝2n，且a1＝1.①求數(shù)列{an}的通項(xiàng)公式；②設(shè)bn＝eq\f(an＋1,anan＋1)，求數(shù)列{bn}的前n項(xiàng)和Tn.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)(2023·浙江省強(qiáng)基聯(lián)盟模擬)已知a1＝1，{an＋1}是公比為2的等比數(shù)列，{bn}為正項(xiàng)數(shù)列，b1＝1，當(dāng)n≥2時(shí)，(2n－3)bn＝(2n－1)bn－1.①求數(shù)列{an}，{bn}的通項(xiàng)公式；②記cn＝an·bn.求數(shù)列{cn}的前n項(xiàng)和Tn.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考點(diǎn)二數(shù)列的綜合問題核心提煉數(shù)列與函數(shù)、不等式，以及數(shù)列新定義的綜合問題，是高考命題的一個(gè)方向，考查邏輯推理、數(shù)學(xué)運(yùn)算、數(shù)學(xué)建模等核心素養(yǎng)．解決此類問題，一是把數(shù)列看成特殊的函數(shù)，利用函數(shù)的圖象、性質(zhì)求解；二是將新數(shù)列問題轉(zhuǎn)化為等差或等比數(shù)列，利用特殊數(shù)列的概念、公式、性質(zhì)，結(jié)合不等式的相關(guān)知識(shí)求解．例4(1)分形的數(shù)學(xué)之美，是以簡單的基本圖形，凝聚擴(kuò)散，重復(fù)累加，以迭代的方式而形成的美麗的圖案．自然界中存在著許多令人震撼的天然分形圖案，如鸚鵡螺的殼、蕨類植物的葉子、孔雀的羽毛、菠蘿等．如圖所示，為正方形經(jīng)過多次自相似迭代形成的分形圖形，且相鄰的兩個(gè)正方形的對應(yīng)邊所成的角為15°.若從外往里最大的正方形邊長為9，則第