第6节 数列中的奇偶项、子数列问题

第6节数列中的奇偶项、子数列问题考试要求1.明确数列奇偶项问题的类型，掌握其解决方法.2.会用化归的思想方法求解子数列问题．考点一奇偶项问题角度1含有(－1)n的类型例1已知bn＝(－1)nn2，求数列{bn}的前n项和Tn.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________角度2已知条件明确的奇偶项问题例2(2023·新高考Ⅱ卷节选)已知an＝2n＋3，bn＝eq\b\lc\{(\a\vs4\al\co1(an－6，n是奇数，,2an，n是偶数.))记Sn，Tn分别为数列{an}，{bn}的前n项和，证明：当n>5时，Tn>Sn.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升1.含有(－1)n的数列求和问题一般采用分组(并项)法求和；2．对于通项公式奇、偶项不同的数列{an}求Sn时，我们可以分别求出奇数项的和与偶数项的和，也可以先求出S2k，再利用S2k－1＝S2k－a2k，求S2k－1.训练1(2024·青岛模拟)已知递增等比数列{an}的前n项和为Sn，且S3＝13，aeq\o\al(2,3)＝3a4，等差数列{bn}满足b1＝a1，b2＝a2－1.(1)求数列{an}和{bn}的通项公式；(2)若cn＝eq\b\lc\{(\a\vs4\al\co1(－an＋1bn，n为奇数，,anbn，n为偶数，))请判断c2n－1＋c2n与a2n的大小关系，并求数列{cn}的前20项和．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________考点二子数列问题角度1公共项例3已知an＝3n－1，bn＝3n＋n，n∈N*.若数列{an}与{bn}中有公共项，即存在k，m∈N*，使得ak＝bm成立．按照从小到大的顺序将这些公共项排列，得到一个新的数列，记作{cn}，求c1＋c2＋…＋cn.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________角度2插项、提项例4已知数列{an}中，an＝eq\b\lc\{(\a\vs4\al\co1(3，n＝1，,2n－1，n≥2，))若保持{an}中各项先后顺序不变，在ak与ak＋1之间插入k个1，使它们和原数列的项构成一个新的数列{bn}，记{bn}的前n项和为Tn，求T100的值(用数字作答)．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升1.两个等差(比)数列的公共项是等差(比)数列，且公差(比)是两等差(比)数列公差(比)的最小公倍数，一个等差与一个等比数列的公共项，则要通过其项数之间的关系来确定．2．数列的插项、提项问题可通过研究前n次的变化探究出一般性规律，从而确定新数列的首项、项数、公差(或公比)、末项等信息．训练2(2024·济南模拟)已知等差数列{an}和等比数列{bn}满足a1＝4，b1＝2，a2＝2b2－1，a3＝b3＋2.(1)求{an}和{bn}的通项公式；(2)数列{an}和{bn}中的所有项分别构成集合A，B，将A∪B的所有元素按从小到大依次排列构成一个新数列{cn}，求数列{cn}的前60项和S60._______________________________________________________________________________________________________________________________________________________________________