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数列知识点总结及题型归纳

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数列知识点总结及题型归纳让学习成为一种习惯!数列一、数列的概念(1)数列定义:按一定次序排列的一列数叫做数列;数列中的每个数都叫这个数列的项。记作,在数列第一个位置的项叫第1项(或首项),在第二个位置的叫第2项,……,序号为的项叫第项(也叫通项)记作;数列的一般形式:,,,……,,……,简记作。例:判断下列各组元素能否构成数列(1)a,-3,-1,1,b,5,7,9;(2)2010年各省参加高考的考生人数。(2)通项公式的定义:如果数列的第n项与n之间的关...

数列知识点总结及题型归纳
让学习成为一种习惯!数列一、数列的概念(1)数列定义:按一定次序排列的一列数叫做数列;数列中的每个数都叫这个数列的项。记作,在数列第一个位置的项叫第1项(或首项),在第二个位置的叫第2项,……,序号为的项叫第项(也叫通项)记作;数列的一般形式:,,,……,,……,简记作。例:判断下列各组元素能否构成数列(1)a,-3,-1,1,b,5,7,9;(2)2010年各省参加高考的考生人数。(2)通项公式的定义:如果数列的第n项与n之间的关系可以用一个公式 关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf 示,那么这个公式就叫这个数列的通项公式。例如:①:1,2,3,4,5,…②:…数列①的通项公式是=(EMBEDEquation.DSMT47,),数列②的通项公式是=()。说明:①表示数列,表示数列中的第项,=表示数列的通项公式;②同一个数列的通项公式的形式不一定唯一。例如,==;③不是每个数列都有通项公式。例如,1,1.4,1.41,1.414,……(3)数列的函数特征与图象表示:序号:123456项:456789上面每一项序号与这一项的对应关系可看成是一个序号集合到另一个数集的映射。从函数观点看,数列实质上是定义域为正整数集(或它的有限子集)的函数当自变量从1开始依次取值时对应的一系列函数值……,,…….通常用来代替,其图象是一群孤立点。例:画出数列的图像.(4)数列分类:①按数列项数是有限还是无限分:有穷数列和无穷数列;②按数列项与项之间的大小关系分:单调数列(递增数列、递减数列)、常数列和摆动数列。例:下列的数列,哪些是递增数列、递减数列、常数列、摆动数列?(1)1,2,3,4,5,6,…(2)10,9,8,7,6,5,…(3)1,0,1,0,1,0,…(4)a,a,a,a,a,…(5)数列{}的前项和与通项的关系:例:已知数列的前n项和,求数列的通项公式练习:1.根据数列前4项,写出它的通项公式:(1)1,3,5,7……;(2),,,;(3),,,。(4)9,99,999,9999…(5)7,77,777,7777,…(6)8,88,888,8888…2.数列中,已知(1)写出,,,,;(2)是否是数列中的项?若是,是第几项?3.(2003京春理14,文15)在某报《自测健康状况》的报道中,自测血压结果与相应年龄的统计数据如下表.观察表中数据的特点,用适当的数填入表中空白(_____)内。4、由前几项猜想通项:根据下面的图形及相应的点数,在空格及括号中分别填上适当的图形和数,写出点数的通项公式.5.观察下列各图,并阅读下面的文字,像这样,10条直线相交,交点的个数最多是(),其通项公式为.A.40个B.45个C.50个D.55个二、等差数列 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 型一、等差数列定义:一般地,如果一个数列从第项起,每一项与它的前一项的差等于同一个常数,那么这个数列就叫等差数列,这个常数叫做等差数列的公差,公差通常用字母表示。用递推公式表示为或。例:等差数列,题型二、等差数列的通项公式:;说明:等差数列(通常可称为EMBEDEquation.DSMT4数列)的单调性:EMBEDEquation.DSMT4为递增数列,为常数列,为递减数列。例:1.已知等差数列中,等于()A.15B.30C.31D.642.是首项,公差的等差数列,如果,则序号等于(A)667(B)668(C)669(D)6703.等差数列,则为为(填“递增数列”或“递减数列”)题型三、等差中项的概念:定义:如果,,成等差数列,那么叫做与的等差中项。其中,,成等差数列EMBEDEquation.DSMT4即:()例:1.(06全国I)设是公差为正数的等差数列,若,,则()A.B.C.D.2.设数列是单调递增的等差数列,前三项的和为12,前三项的积为48,则它的首项是()A.1B.2C.4D.8题型四、等差数列的性质:(1)在等差数列中,从第2项起,每一项是它相邻二项的等差中项;(2)在等差数列中,相隔等距离的项组成的数列是等差数列;(3)在等差数列中,对任意,,,EMBEDEquation.DSMT4;(4)在等差数列中,若,,,且,则;题型五、等差数列的前和的求和公式:EMBEDEquation.3。(EMBEDEquation.3EMBEDEquation.3是等差数列)递推公式:例:1.如果等差数列中,,那么(A)14(B)21(C)28(D)352.(2009湖南卷文)设是等差数列的前n项和,已知,,则等于()A.13B.35C.49D.633.(2009全国卷Ⅰ理)设等差数列的前项和为,若,则=4.(2010重庆文)(2)在等差数列中,,则的值为()(A)5(B)6(C)8(D)105.若一个等差数列前3项的和为34,最后3项的和为146,且所有项的和为390,则这个数列有()A.13项B.12项C.11项D.10项6.已知等差数列的前项和为,若7.(2009全国卷Ⅱ理)设等差数列的前项和为,若则8.(98全国)已知数列{bn}是等差数列,b1=1,b1+b2+…+b10=100.(Ⅰ)求数列{bn}的通项bn;9.已知数列是等差数列,,其前10项的和,则其公差等于()C.D.10.(2009陕西卷文)设等差数列的前n项和为,若,则11.(00全国)设{an}为等差数列,Sn为数列{an}的前n项和,已知S7=7,S15=75,Tn为数列{}的前n项和,求Tn。12.等差数列的前项和记为,已知①求通项;②若=242,求13.在等差数列中,(1)已知;(2)已知;(3)已知题型六.对于一个等差数列:(1)若项数为偶数,设共有项,则①偶EMBEDEquation.DSMT4奇;②;(2)若项数为奇数,设共有项,则①奇EMBEDEquation.DSMT4偶;②。题型七.对与一个等差数列,仍成等差数列。例:1.等差数列{an}的前m项和为30,前2m项和为100,则它的前3m项和为()A.130B.170C.210D.2602.一个等差数列前项的和为48,前2项的和为60,则前3项的和为。3.已知等差数列的前10项和为100,前100项和为10,则前110项和为4.设为等差数列的前项和,=5.(06全国II)设Sn是等差数列{an}的前n项和,若=,则=A.B.C.D.题型八.判断或证明一个数列是等差数列的方法:①定义法:EMBEDEquation.3EMBEDEquation.3是等差数列②中项法:EMBEDEquation.3EMBEDEquation.3是等差数列③通项公式法:EMBEDEquation.3EMBEDEquation.3是等差数列④前项和公式法:EMBEDEquation.3EMBEDEquation.3是等差数列例:1.已知数列满足,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断2.已知数列的通项为,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断3.已知一个数列的前n项和,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断4.已知一个数列的前n项和,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断5.已知一个数列满足,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断6.数列满足=8,()①求数列的通项公式;7.(01天津理,2)设Sn是数列{an}的前n项和,且Sn=n2,则{an}是()A.等比数列,但不是等差数列B.等差数列,但不是等比数列C.等差数列,而且也是等比数列D.既非等比数列又非等差数列题型九.数列最值(1),时,有最大值;,时,有最小值;(2)最值的求法:①若已知,的最值可求二次函数的最值;可用二次函数最值的求法();②或者求出中的正、负分界项,即:若已知,则最值时的值()可如下确定或。例:1.等差数列中,,则前项的和最大。2.设等差数列的前项和为,已知①求出公差的范围,②指出中哪一个值最大,并说明理由。3.(02上海)设{an}(n∈N*)是等差数列,Sn是其前n项的和,且S5<S6,S6=S7>S8,则下列结论错误的是()A.d<0B.a7=0C.S9>S5D.S6与S7均为Sn的最大值4.已知数列的通项(),则数列的前30项中最大项和最小项分别是5.已知是等差数列,其中,公差。(1)数列从哪一项开始小于0?(2)求数列前项和的最大值,并求出对应的值.6.已知是各项不为零的等差数列,其中,公差,若,求数列前项和的最大值.7.在等差数列中,,,求的最大值.题型十.利用求通项.1.数列的前项和.(1)试写出数列的前5项;(2)数列是等差数列吗?(3)你能写出数列的通项公式吗?2.已知数列的前项和则3.设数列的前n项和为Sn=2n2,求数列的通项公式;4.已知数列中,前和①求证:数列是等差数列②求数列的通项公式5.(2010安徽文)设数列的前n项和,则的值为()(A)15(B)16(C)49(D)64等比数列等比数列定义一般地,如果一个数列从第二项起,每一项与它的前一项的比等于同一个常数,那么这个数列就叫做等比数列,这个常数叫做等比数列的公比;公比通常用字母表示,即::。一、递推关系与通项公式1.在等比数列中,,则2.在等比数列中,,则3.(07重庆文)在等比数列{an}中,a2=8,a1=64,,则公比q为()(A)2(B)3(C)4(D)84.在等比数列中,,,则=5.在各项都为正数的等比数列中,首项,前三项和为21,则()A33B72C84D189二、等比中项:若三个数成等比数列,则称为的等比中项,且为是成等比数列的必要而不充分条件.例:1.和的等比中项为()2.(2009重庆卷文)设是公差不为0的等差数列,且成等比数列,则的前项和=()A.B.C.D.三、等比数列的基本性质,1.(1)EMBEDEquation.3(2)(3)为等比数列,则下标成等差数列的对应项成等比数列.(4)既是等差数列又是等比数列EMBEDEquation.3是各项不为零的常数列.例:1.在等比数列中,和是方程的两个根,则()2.在等比数列,已知,,则=3.在等比数列中,①求②若4.等比数列的各项为正数,且()A.12B.10C.8D.2+5.(2009广东卷理)已知等比数列满足,且,则当时,()A.B.C.D.2.前项和公式例:1.已知等比数列的首相,公比,则其前n项和2.已知等比数列的首相,公比,当项数n趋近与无穷大时,其前n项和3.设等比数列的前n项和为,已EMBEDEquation.3,求和4.(2006年北京卷)设,则等于()A.B.C.D.5.(1996全国文,21)设等比数列{an}的前n项和为Sn,若S3+S6=2S9,求数列的公比q;6.设等比数列的公比为q,前n项和为S​n,若Sn+1,S​n,Sn+2成等差数列,则q的值为.3.若数列是等比数列,是其前n项的和,,那么,,成等比数列.如下图所示:例:1.(2009辽宁卷理)设等比数列{}的前n项和为,若=3,则=A.2B.C.D.32.一个等比数列前项的和为48,前2项的和为60,则前3项的和为()A.83B.108C.75D.633.已知数列是等比数列,且4.等比数列的判定法(1)定义法:EMBEDEquation.3为等比数列;(2)中项法:EMBEDEquation.3为等比数列;(3)通项公式法:EMBEDEquation.3为等比数列;(4)前项和法:EMBEDEquation.3为等比数列。EMBEDEquation.3为等比数列。例:1.已知数列的通项为,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断2.已知数列满足,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断3.已知一个数列的前n项和,则数列为()A.等差数列B.等比数列C.既不是等差数列也不是等比数列D.无法判断5.利用求通项.例:1.(2005北京卷)数列{an}的前n项和为Sn,且a1=1,,n=1,2,3,……,求a2,a3,a4的值及数列{an}的通项公式.2.(2005山东卷)已知数列的首项前项和为,且,证明数列是等比数列.四、求数列通项公式方法(1).公式法(定义法)根据等差数列、等比数列的定义求通项例:1已知等差数列满足:,求;2.已知数列满足,求数列的通项公式;3.数列满足=8,(),求数列的通项公式;4.已知数列满足,求数列的通项公式;5.设数列满足且,求的通项公式6.已知数列满足,求数列的通项公式。7.等比数列的各项均为正数,且,,求数列的通项公式8.已知数列满足,求数列的通项公式;9.已知数列满足(),求数列的通项公式;10.已知数列满足且(),求数列的通项公式;11.已知数列满足且(),求数列的通项公式;12.数列已知数列满足则数列的通项公式=(2)累加法1、累加法适用于:若EMBEDEquation.DSMT4,则两边分别相加得例:1.已知数列满足,求数列的通项公式。2.已知数列满足,求数列的通项公式。3.已知数列满足,求数列的通项公式。4.设数列满足,,求数列的通项公式(3)累乘法适用于:若,则两边分别相乘得,例:1.已知数列满足,求数列的通项公式。2.已知数列满足,,求。3.已知,,求。(4)待定系数法适用于解题基本步骤:1、确定2、设等比数列,公比为3、列出关系式4、比较系数求,5、解得数列的通项公式6、解得数列的通项公式例:1.已知数列中,,求数列的通项公式。2.(2006,重庆,文,14)在数列中,若,则该数列的通项EMBEDEquation.DSMT4_______________3.(2006.福建.理22.本小题满分14分)已知数列满足求数列的通项公式;4.已知数列满足,求数列的通项公式。解:设5.已知数列满足,求数列的通项公式。解:设6.已知数列中,,,求7.已知数列满足,求数列的通项公式。解:设8.已知数列满足,求数列的通项公式。递推公式为(其中p,q均为常数)。先把原递推公式转化为其中s,t满足9.已知数列满足,求数列的通项公式。(5)递推公式中既有分析:把已知关系通过转化为数列或的递推关系,然后采用相应的方法求解。1.(2005北京卷)数列{an}的前n项和为Sn,且a1=1,,n=1,2,3,……,求a2,a3,a4的值及数列{an}的通项公式.2.(2005山东卷)已知数列的首项前项和为,且,证明数列是等比数列.3.已知数列中,前和①求证:数列是等差数列②求数列的通项公式4.已知数列的各项均为正数,且前n项和满足,且成等比数列,求数列的通项公式。(6)根据条件找与项关系例1.已知数列中,,若,求数列的通项公式2.(2009全国卷Ⅰ理)在数列中,(I)设,求数列的通项公式(7)倒数变换法适用于分式关系的递推公式,分子只有一项例:1.已知数列满足,求数列的通项公式。(8)对无穷递推数列消项得到第与项的关系例:1.(2004年全国I第15题,原题是填空题)已知数列满足,求的通项公式。2.设数列满足,.求数列的通项;(9)、迭代法例:1.已知数列满足,求数列的通项公式。解:因为,所以又,所以数列的通项公式为。(10)、变性转化法1、对数变换法适用于指数关系的递推公式例:已知数列满足,,求数列的通项公式。解:因为,所以。两边取常用对数得2、换元法适用于含根式的递推关系例:已知数列满足,求数列的通项公式。解:令,则五、数列求和1.直接用等差、等比数列的求和公式求和。公比含字母时一定要讨论(理)无穷递缩等比数列时,例:1.已知等差数列满足EMBEDEquation.3,求前项和2.等差数列{an}中,a1=1,a3+a5=14,其前n项和Sn=100,则n=(  )A.9B.10C.11D.123.已知等比数列满足EMBEDEquation.3,求前项和4.设,则等于()A.B.C.D.2.错位相减法求和:如:例:1.求和2.求和:3.设是等差数列,是各项都为正数的等比数列,且,,(Ⅰ)求,的通项公式;(Ⅱ)求数列的前n项和.3.裂项相消法求和:把数列的通项拆成两项之差、正负相消剩下首尾若干项。常见拆项:数列是等差数列,数列的前项和例:1.数列的前项和为,若,则等于( )A.1B.C.D.2.已知数列的通项公式为,求前项的和;3.已知数列的通项公式为,求前项的和.4.已知数列的通项公式为=,设,求.5.求。6.已知,数列是首项为a,公比也为a的等比数列,令,求数列的前项和。4.倒序相加法求和例:1.求2.求证:3.设数列是公差为,且首项为的等差数列,求和:综合练习:1.设数列满足且(1)求的通项公式(2)设记,证明:2.等比数列的各项均为正数,且,(1)求数列的通项公式(2)设,求数列的前n项和3.已知等差数列满足,.(1)求数列的通项公式及(2)求数列的前n项和4.已知两个等比数列,,满足,,,(1)若求数列的通项公式(2)若数列唯一,求的值5.设数列满足,(1)求数列的通项公式(2)令,求数列的前n项和6.已知a1=2,点(an,an+1)在函数f(x)=x2+2x的图象上,其中=1,2,3,…(1)证明数列{lg(1+an)}是等比数列;(2)设Tn=(1+a1)(1+a2)…(1+an),求Tn及数列{an}的通项;(3)记bn=,求{bn}数列的前项和Sn,并证明Sn+=1.7.已知等差数列满足:,的前n项和(1)求及(2)令(),求数列前n项和8.已知数列中,前和①求证:数列是等差数列②求数列的通项公式③设数列的前项和为,是否存在实数,使得对一切正整数都成立?若存在,求的最小值,若不存在,试说明理由。9.数列满足=8,(),(Ⅰ)求数列的通项公式;(Ⅱ)设,是否存在最大的整数m,使得任意的n均有总成立?若存在,求出m;若不存在,请说明理由.(1)(4)(7)()()2条直线相交,最多有1个交点3条直线相交,最多有3个交点4条直线相交,最多有6个交点1_1234568145.unknown_1234568273.unknown_1234568401.unknown_1234568465.unknown_1234568497.unknown_1234568529.unknown_1234568545.unknown_1234568553.unknown_1234568561.unknown_1234568565.unknown_1234568569.unknown_1234568571.unknown_1234568572.unknown_1234568573.unknown_1234568570.unknown_1234568567.unknown_1234568568.unknown_1234568566.unknown_1234568563.unknown_1234568564.unknown_1234568562.unknown_1234568557.unknown_1234568559.unknown_1234568560.unknown_1234568558.unknown_1234568555.unknown_1234568556.unknown_1234568554.unknown_1234568549.unknown_1234568551.unknown_1234568552.unknown_1234568550.unknown_1234568547.unknown_1234568548.unknown_1234568546.unknown_1234568537.unknown_1234568541.unknown_1234568543.unknown_1234568544.unknown_1234568542.unknown_1234568539.unknown_1234568540.unknown_1234568538.unknown_1234568533.unknown_1234568535.unknown_1234568536.unknown_1234568534.unknown_1234568531.unknown_1234568532.unknown_1234568530.unknown_1234568513.unknown_1234568521.unknown_1234568525.unknown_1234568527.unknown_1234568528.unknown_1234568526.unknown_1234568523.unknown_1234568524.unknown_1234568522.unknown_1234568517.unknown_1234568519.unknown_1234568520.unknown_1234568518.unknown_1234568515.unknown_1234568516.unknown_1234568514.unknown_1234568505.unknown_1234568509.unknown_1234568511.unknown_1234568512.unknown_1234568510.unknown_1234568507.unknown_1234568508.unknown_1234568506.unknown_1234568501.unknown_1234568503.unknown_1234568504.unknown_1234568502.unknown_1234568499.unknown_1234568500.unknown_1234568498.unknown_1234568481.unknown_1234568489.unknown_1234568493.unknown_1234568495.unknown_1234568496.unknown_1234568494.unknown_1234568491.unknown_1234568492.unknown_1234568490.unknown_1234568485.unknown_1234568487.unknown_1234568488.unknown_1234568486.unknown_1234568483.unknown_1234568484.unknown_1234568482.unknown_1234568473.unknown_1234568477.unknown_1234568479.unknown_1234568480.unknown_1234568478.unknown_1234568475.unknown_1234568476.unknown_1234568474.unknown_1234568469.unknown_1234568471.unknown_1234568472.unknown_1234568470.unknown_1234568467.unknown_1234568468.unknown_1234568466.unknown_1234568433.unknown_1234568449.unknown_1234568457.unknown_1234568461.unknown_1234568463.unknown_1234568464.unknown_1234568462.unknown_1234568459.unknown_1234568460.unknown_1234568458.unknown_1234568453.unknown_1234568455.unknown_1234568456.unknown_1234568454.unknown_1234568451.unknown_1234568452.unknown_1234568450.unknown_1234568441.unknown_1234568445.unknown_1234568447.unknown_1234568448.unknown_1234568446.unknown_1234568443.unknown_1234568444.unknown_1234568442.unknown_1234568437.unknown_1234568439.unknown_1234568440.unknown_1234568438.unknown_1234568435.unknown_1234568436.unknown_1234568434.unknown_1234568417.unknown_1234568425.unknown_1234568429.unknown_1234568431.unknown_1234568432.unknown_1234568430.unknown_1234568427.unknown_1234568428.unknown_1234568426.unknown_1234568421.unknown_1234568423.unknown_1234568424.unknown_1234568422.unknown_1234568419.unknown_1234568420.unknown_1234568418.unknown_1234568409.unknown_1234568413.unknown_1234568415.unknown_1234568416.unknown_1234568414.unknown_1234568411.unknown_1234568412.unknown_1234568410.unknown_1234568405.unknown_1234568407.unknown_1234568408.unknown_1234568406.unknown_1234568403.unknown_1234568404.unknown_1234568402.unknown_1234568337.unknown_1234568369.unknown_1234568385.unknown_1234568393.unknown_1234568397.unknown_1234568399.unknown_1234568400.unknown_1234568398.unknown_1234568395.unknown_1234568396.unknown_1234568394.unknown_1234568389.unknown_1234568391.unknown_1234568392.unknown_1234568390.unknown_1234568387.unknown_1234568388.unknown_1234568386.unknown_1234568377.unknown_1234568381.unknown_1234568383.unknown_1234568384.unknown_1234568382.unknown_1234568379.unknown_1234568380.unknown_1234568378.unknown_1234568373.unknown_1234568375.unknown_1234568376.unknown_1234568374.unknown_1234568371.unknown_1234568372.unknown_1234568370.unknown_1234568353.unknown_1234568361.unknown_1234568365.unknown_1234568367.unknown_1234568368.unknown_1234568366.unknown_1234568363.unknown_1234568364.unknown_1234568362.unknown_1234568357.unknown_1234568359.unknown_1234568360.unknown_1234568358.unknown_1234568355.unknown_1234568356.unknown_1234568354.unknown_1234568345.unknown_1234568349.unk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