2025年高考數(shù)學(xué)復(fù)習(xí)大題題型歸納：第05講 錯(cuò)位相減法求數(shù)列前n項(xiàng)和（原卷）

第05講錯(cuò)位相減法求數(shù)列前n項(xiàng)和例題分析【例1】1．設(shè)Sn為數(shù)列an的前n項(xiàng)和，已知(1)求an(2)求數(shù)列an+12n的前n滿分秘籍(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(duì)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)列求和公式必須注意公比q是否等于1，如果q＝1，應(yīng)用公式Sn＝na1.變式訓(xùn)練【變式1-1】已知數(shù)列an的前n項(xiàng)和為Sn，且(1)求a1,a(2)若bn=an?log2【變式1-2】已知數(shù)列an和bn，a1=2，(1)求證數(shù)列1a(2)求數(shù)列nbn的前n項(xiàng)和【變式1-3】在①Sn+1=2Sn+2，②an+1?an(1)求an(2)若bn=(n+1)?an，求數(shù)列bn注：如果選擇多個(gè)條件分別解答，按第一個(gè)解答計(jì)分．【變式1-4】記正項(xiàng)數(shù)列an的前n項(xiàng)和為Sn，已知點(diǎn)an,6Snn∈N?在函數(shù)f(x)=(x+1)(x+2)(1)求數(shù)列an，b(2)設(shè)cn=anbn+1+【變式1-5】設(shè)an是公比不為1的等比數(shù)列，a1=1，a2為(1)求an(2)求數(shù)列(2n?10)an的前n項(xiàng)和真題專練1．已知an是單調(diào)遞增的等差數(shù)列，其前n項(xiàng)和為Sn.bn是公比為q(1)求an和b(2)設(shè)cn=anbn,n2．已知數(shù)列an的前n項(xiàng)和為Sn,(1)求數(shù)列an(2)設(shè)數(shù)列bn滿足bn=log2an+13．已知數(shù)列{an}的首項(xiàng)為1，前n(1)求數(shù)列{a(2)若bn=(an+1)?2n4．在①a1+a4+a7已知公差不為0的等差數(shù)列an的前n項(xiàng)和為Sn,a7(1)求an(2)求數(shù)列2nan的前n注：如果選擇多個(gè)條件分別解答，按第一個(gè)解答計(jì)分.5．已知數(shù)列an的奇數(shù)項(xiàng)成等差數(shù)列，偶數(shù)項(xiàng)成等比數(shù)列，且公差和公比都是2，若對(duì)滿足m+n≤5的任意正整數(shù)m，n，均有a(1)求數(shù)列an(2)令bn=a2n?1a2n，求數(shù)列6．已知數(shù)列an的前n項(xiàng)的和為Sn，Sn=12n(1)求數(shù)列an(2)設(shè)數(shù)列cn滿足cn=a2n?1，設(shè)cnb7．已知數(shù)列an的前n項(xiàng)和為Sn,且滿足an>0,Sn=(1)求數(shù)列an和b(2)求數(shù)列anbn8．已知Sn是公差不為0的等差數(shù)列an的前n項(xiàng)和，a2是a1，(1)求數(shù)列an(2)已知bn=a2n?1?3a9．在①3Sn+1=4an；②a1=1，已知數(shù)列an的前n項(xiàng)和為S(1)求數(shù)列an(2)若bn=2n+2an，求數(shù)列b注：如果選擇多個(gè)條件分別作答，則按所作第一個(gè)解答計(jì)分．10．已知數(shù)列an是公差為3的等差數(shù)列，數(shù)列bn是公比為2的等比數(shù)列，且滿足a1+a3=(1)證明：c(2)求數(shù)列ancn的前n11．設(shè)正項(xiàng)數(shù)列an的前n項(xiàng)和為Sn，且a1=1，當(dāng)(1)求數(shù)列an(2)設(shè)數(shù)列bn滿足b1=1，且b12．已知等比數(shù)列an的前n項(xiàng)和為Sn(1)求數(shù)列an(2)在an與an+1之間插入n個(gè)數(shù)，使這n+2個(gè)數(shù)組成一個(gè)公差為dn的等差數(shù)列，求數(shù)列1dn13．已知數(shù)列an的前n項(xiàng)和為Sn，且(1)求數(shù)列an(2)設(shè)bn=an?2log14．在數(shù)列an中，a1=1(1)求數(shù)列an(2)若bn=nan，求數(shù)列bn15．已知數(shù)列an和等差數(shù)列bn滿足a1=1，a(1)求數(shù)列an的通項(xiàng)a(2)求數(shù)列nan的前n項(xiàng)和16．已知等比數(shù)列an的公比q>1，若a2+a3+a4=14(1)求數(shù)列an和b(2)若cn=2anan+117．已知數(shù)列an的前n項(xiàng)和為S①數(shù)列an是首項(xiàng)為2的單調(diào)遞減的等比數(shù)列，且8②Sn③a1(1)求數(shù)列an(2)求數(shù)列nan3的前n18．已知數(shù)列an滿足a1=1(1)求數(shù)列an(2)求數(shù)列an的前n項(xiàng)和S19．設(shè)各項(xiàng)都為正數(shù)的數(shù)