2025年高考數(shù)學(xué)復(fù)習(xí)大題題型歸納：專題07 數(shù)列中的構(gòu)造問題（原卷）

專題07數(shù)列中的構(gòu)造問題1．已知數(shù)列an滿足a1=1，a(1)證明：an+1(2)證明：存在兩個(gè)等比數(shù)列bn，cn，使得2．已知數(shù)列an的前n項(xiàng)和為Sn，a1=2，(1)求an(2)設(shè)bn=?1n?3n?1，數(shù)列bn的前n3．已知數(shù)列an滿足a1=3(1)證明數(shù)列l(wèi)nan?1(2)若bn=1an+1an4．已知數(shù)列an的前n項(xiàng)和為Sn(1)an(2)若bn=nan+n，求數(shù)列b5．已知各項(xiàng)均為正數(shù)的數(shù)列{an}滿足a1(1)求證：數(shù)列an(2)求數(shù)列{an}的前n項(xiàng)和S6．設(shè)各項(xiàng)均為正數(shù)的數(shù)列{an}滿足Snan=pn+r（p,r(1)若p=1,r=0，求證：{a(2)若p=13,7．已知數(shù)列an，2an+1(1)求證：數(shù)列1a(2)設(shè)bn=1?an1?a8．已知數(shù)列an的前n項(xiàng)和為Sn=nn+1n∈(1)求數(shù)列an(2)求數(shù)列bn(3)對(duì)于n∈N+，試比較bn+19．已知數(shù)列an有遞推關(guān)系(1)記an=bn+k,若數(shù)列bn的遞推式形如(2)求an10．已知數(shù)列an滿足a1+a3=2a(1)求數(shù)列cn和a(2)求數(shù)列an的前n項(xiàng)和S11．已知Sn為數(shù)列an的前n項(xiàng)和，a1=2，(1)求數(shù)列bn(2)已知cn=?1n+1?bn+1b12．已知數(shù)列an滿足a(1)求an(2)若bn=2n?1，數(shù)列cn滿足c4n?3=b2n?113．設(shè)數(shù)列an的前n項(xiàng)和為Sn，且a1(1)求an(2)若bn=1Sn，求數(shù)列b14．已知數(shù)列an滿足a1=1(1)求證：數(shù)列an(2)若bn=2n+1an+1?an，15．設(shè)數(shù)列an的前n項(xiàng)和為S(1)求數(shù)列an(2)若數(shù)列2n+1anan+1的前m16．已知數(shù)列an滿足a1=1(1)求數(shù)列an(2)若bn=2n?an17．記數(shù)列an的前n項(xiàng)和為Sn，已知a1(1)求an(2)記數(shù)列an的前n項(xiàng)和為Tn，證明：18．已知數(shù)列an的前n項(xiàng)和為Sn，且(1)求證；數(shù)列an(2)求證：k=1n19．已知數(shù)列{an}的前n項(xiàng)和為Sn，滿足Sn＝(1)求數(shù)列{an}(2)設(shè)cn=an?bn20．已知數(shù)列an滿足a1=1，a2=4.有以下三個(gè)條件：①an+1=4an?4an?1（n≥2，n∈N?）；21．若數(shù)列an滿足a1=2(1)證明:an+1(2)設(shè)an的前n項(xiàng)和為Sn,求滿足Sn22．已知數(shù)列an的首項(xiàng)a1=(1)求證：數(shù)列1a(2)若1a1+23．已知數(shù)列an滿足a1=1,(1)證明數(shù)列an+1?2a(2)求數(shù)列an的前n項(xiàng)和S24．已知正項(xiàng)數(shù)列an的前n項(xiàng)和為S①數(shù)列an2的前n項(xiàng)和為②a1③a1=1,a2=從上述三個(gè)條件中任選一個(gè)，完成以下問題：(1)求數(shù)列an(2)設(shè)數(shù)列bn滿足b1=1,bn=a25．已知數(shù)列an中，a1=5且(1)求證：數(shù)列bn(2)從條件①n+bn，②求數(shù)列______的前n項(xiàng)和Tn注：如果選擇多個(gè)條件分別解答，按第一個(gè)解答計(jì)分．26．已知數(shù)列an的前n項(xiàng)的和為Sn且滿足Sn=2an?(1)求出數(shù)列an，b(2)求出數(shù)列an+bn的前27．記Sn是公差不為0的等差數(shù)列an的前n項(xiàng)和，已知a3+3a4=(1)求an(2)證明數(shù)列bn2n(3)求證：對(duì)任意的n∈N?，28．已知數(shù)列an的前n項(xiàng)和為Sn，滿足a1(1)求數(shù)列an(2)求數(shù)列1Sn的前n項(xiàng)和29．設(shè)數(shù)列an滿足a1=2(1)求證：an?n為等比數(shù)列，并求(2)若bn=an?n?n，求數(shù)列30．問題：已知n∈N?，數(shù)列an的前n項(xiàng)和為Sn，是否存在數(shù)列an在①an+1=2(Sn+1+Sn注：如果選擇多個(gè)條件分別解答，按第一個(gè)解答計(jì)分．31．已知數(shù)列an的前n項(xiàng)和為S(1)從①S1=1，②2Sn+1=S(2)在第（1）問的前提下，若bn=an+1a注：如果選擇多種情況分別解答，按第一種解答計(jì)分.32．在數(shù)列an中，a1=5(1)證明：an?1為等比數(shù)列，并求(2)令bn=(?1)n?an33．已知數(shù)列an滿足a(1)求數(shù)列an(2)當(dāng)cn=an+1?an34．設(shè)數(shù)列{an}的前n(1)求數(shù)列{a(2)若bnn=log2an2+1(3)證明：i=135．設(shè)數(shù)列an的前n項(xiàng)和為Sn，a1(1)求數(shù)列an(2)設(shè)bn=4n+36．設(shè)數(shù)列an的前n項(xiàng)和為Sn，且(1)求Sn(2)證明：當(dāng)n≥2時(shí)，2S37．已知數(shù)列an的前n項(xiàng)和為Sn，(1)求數(shù)列an(2)記bn=an?3n38．已知各項(xiàng)都為正數(shù)的數(shù)列{an}滿足an＋2＝2an＋1＋3an.(1)證明：數(shù)列{an＋an＋1}為等比數(shù)列；