第6節數列中的奇偶項、子數列問題考試要求1.明確數列奇偶項問題的類型，掌握其解決方法.2.會用化歸的思想方法求解子數列問題．考點一奇偶項問題角度1含有(－1)n的類型例1已知bn＝(－1)nn2，求數列{bn}的前n項和Tn.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________角度2已知條件明確的奇偶項問題例2(2023·新高考Ⅱ卷節選)已知an＝2n＋3，bn＝eq\b\lc\{(\a\vs4\al\co1(an－6，n是奇數，,2an，n是偶數.))記Sn，Tn分別為數列{an}，{bn}的前n項和，證明：當n>5時，Tn>Sn.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升1.含有(－1)n的數列求和問題一般采用分組(并項)法求和；2．對于通項公式奇、偶項不同的數列{an}求Sn時，我們可以分別求出奇數項的和與偶數項的和，也可以先求出S2k，再利用S2k－1＝S2k－a2k，求S2k－1.訓練1(2024·青島模擬)已知遞增等比數列{an}的前n項和為Sn，且S3＝13，aeq\o\al(2,3)＝3a4，等差數列{bn}滿足b1＝a1，b2＝a2－1.(1)求數列{an}和{bn}的通項公式；(2)若cn＝eq\b\lc\{(\a\vs4\al\co1(－an＋1bn，n為奇數，,anbn，n為偶數，))請判斷c2n－1＋c2n與a2n的大小關系，并求數列{cn}的前20項和．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考點二子數列問題角度1公共項例3已知an＝3n－1，bn＝3n＋n，n∈N*.若數列{an}與{bn}中有公共項，即存在k，m∈N*，使得ak＝bm成立．按照從小到大的順序將這些公共項排列，得到一個新的數列，記作{cn}，求c1＋c2＋…＋cn.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________角度2插項、提項例4已知數列{an}中，an＝eq\b\lc\{(\a\vs4\al\co1(3，n＝1，,2n－1，n≥2，))若保持{an}中各項先后順序不變，在ak與ak＋1之間插入k個1，使它們和原數列的項構成一個新的數列{bn}，記{bn}的前n項和為Tn，求T100的值(用數字作答)．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升1.兩個等差(比)數列的公共項是等差(比)數列，且公差(比)是兩等差(比)數列公差(比)的最小公倍數，一個等差與一個等比數列的公共項，則要通過其項數之間的關系來確定．2．數列的插項、提項問題可通過研究前n次的變化探究出一般性規律，從而確定新數列的首項、項數、公差(或公比)、末項等信息．訓練2(2024·濟南模擬)已知等差數列{an}和等比數列{bn}滿足a1＝4，b1＝2，a2＝2b2－1，a3＝b3＋2.(1)求{an}和{bn}的通項公式；(2)數列{an}和{bn}中的所有項分別構成集合A，B，將A∪B的所有元素按從小到大依次排列構成一個新數列{cn}，求數列{cn}的前60項和S60._______________________________________________________________________________________________________________________________________________________________________