高中数学数列压轴题练习(江苏)及详解

-.z.高中数学数列压轴题练习〔〕及详解1.数列是公差为正数的等差数列,其前n项和为,且•,

(Ⅰ)求数列的通项公式;

(Ⅱ)数列满足,①求数列的通项公式;

②是否存在正整数m,,使得,,成等差数列"假设存在,求出m,n的值;假设不存在,请说明理由.解:(I)设数列的公差为d,则由•,,得,

计算得出或(舍去).

;

(Ⅱ)①,,

,

,

即,,,,累加得:,

也符合上式.

故,.

②假设存在正整数m、,使得,,成等差数列,

则又,,,

,即,

化简得:当,即时,,(舍去);

当,即时,,符合题意.

存在正整数,,使得,,成等差数列.解析(Ⅰ)直接由列关于首项和公差的方程组,求解方程组得首项和公差,代入等差数列的通项公式得答案;

(Ⅱ)①把数列的通项公式代入,然后裂项,累加后即可求得数列的通项公式;

②假设存在正整数m、,使得,,成等差数列,则.由此列关于m的方程,求计算得出答案.2.在数列中,,

(1)求证:数列为等比数列;

(2)记,且数列的前n项和为,假设为数列中的最小项,求的取值范围.解:(1)证明:,

又,

,,

故,

是以3为首项,公比为3的等比数列

(2)由(1)知道,,

假设为数列中的最小项,则对有恒成立,

即对恒成立当时,有;

当时,有⇒;

当时,恒成立,

对恒成立.

令,则对恒成立,

在时为单调递增数列.

,即综上,解析(1)由,整理得:.由,,可以知道是以3为首项,公比为3的等比数列;

(2)由(1)求得数列通项公式及前n项和为,由为数列中的最小项,则对有恒成立,分类分别求得当时和当的取值范围,

当时,,利用做差法,根据函数的单调性,即可求得的取值范围.3.在数列中,

,

,

,设为的前n项和.

(1)求证:数列是等差数列;

(2)求;

(3)是否存在正整数p,q,

,使,

,

成等差数列"假设存在,求出p,q,r的值;假设不存在,说明理由.(1)证明:由,,

得到,

则又,

,

数列是以1为首项,以-2为公差的等差数列;

(2)由(1)可以推知:,

所以,,

所以,①,②①-②,得,

,

,

所以

(3)假设存在正整数p,q,,使,,成等差数列.

则,

即因为当时,,

所以数列单调递减.

又,

所以且q至少为2,

所以,①当时,,

又,

所以,等式不成立.

②当时,,

所以所以,

所以,(数列单调递减,解唯一确定).

综上可以知道,p,q,r的值分别是1,2,3.解析(1)把给出的数列递推式,,变形后得到新数列,该数列是以1为首项,以-2为公差的等差数列;

(2)由(1)推出的通项公式,利用错位相减法从而求得求;

(3)根据等差数列的性质得到,从而推知p,q,r的值.4.n为正整数,数列满足,

,设数列满足

(1)求证:数列为等比数列;

(2)假设数列是等差数列,求实数t的值;

(3)假设数列是等差数列,前n项和为,对任意的,均存在,使得成立,求满足条件的所有整数的值.(1)证明:数列满足,,

•,•,

数列为等比数列,其首项为,公比为2;

(2)解:由(1)可得:•,

,数列是等差数列,,

,

计算得出或12.

时,,是关于n的一次函数,因此数列是等差数列.

时,,,不是关于n的一次函数,

因此数列不是等差数列.

综上可得;

(3)解:由(2)得,

对任意的,均存在,使得成立,

即有••,

化简可得,

当,,,对任意的,符合题意;

当,,当时,,

对任意的,不符合题意.

综上可得,当,,对任意的,均存在,

使得成立.解析(1)根据题意整理可得,•,再由等比数列的定义即可得证;

(2)运用等比数列的通项公式和等差数列中项的性质,可得,解方程可得t,对t的值,检验即可得到所求值;

(3)由(2)可得,对任意的,均存在,使得成立,即有••,讨论为偶数和奇数,化简整理,即可得到所求值.5.常数,数列满足,

(1)假设,

,

①求的值;

②求数列的前n项和;

(2)假设数列中存在三项,

,

依次成等差数列,求的取值范围.解:(1)①,

,

,

,

②,,

当时,,

当时,,即从第二项起,数列是以1为首项,以3为公比的等比数列,

数列的前n项和,,

显然当时,上式也成立,

;

(2),

,即单调递增.

(i)当时,有,于是,

,假设数列中存在三项,,依次成等差数列,则有,

即,.因此不成立.因此此时数列中不存在三项,,依次成等差数列.

当时,有.此时于是当时,.从而假设数列中存在三项,,依次成等差数列,则有,

同(i)可以知道:.于是有,,是整数,.于是,即.与矛盾.

故此时数列中不存在三项,,依次成等差数列.

当时,有于是此时数列中存在三项,,依次成等差数列.

综上可得:解析(1)①,可得,同理可得,②,,当时,,当时,,即从第二项起,数列是以1为首项,以3为公比的等比数列,利用等比数列的求和公式即可得出

(2),可得,即单调递增.

(i)当时,有,于是,可得,.利用反证法即可得出不存在.

当时,有.此时.于是当时,.从而.假设存在,同(i)可以知道:.得出矛盾,因此不存在.

当时,有.于是.即可得出结论.6.两个无穷数列和的前n项和分别为,

,

,,对任意的,都有

(1)求数列的通项公式;

(2)假设为等差数列,对任意的,都有.证明:

;

(3)假设为等比数列,

,

,求满足的n值.解:(1)由,得,

即,所以由,,可以知道所以数列是以1为首项,2为公差的等差数列.

故的通项公式为,

(2)证法一:设数列的公差为d,

则,

由(1)知,因为,所以,

即恒成立,

所以,即,

又由,得,

所以所以,得证.

证法二:设的公差为d,假设存在自然数,使得,

则,即,

因为,所以所以,

因为,所以存在,当时,恒成立.

这与"对任意的,都有〞矛盾!

所以,得证.

(3)由(1)知,.因为为等比数列,

且,,

所以是以1为首项,3为公比的等比数列.

所以,则,

因为,所以,所以而,所以,即当,2时,式成立;

当时,设,

则,

所以,

故满足条件的n的值为1和2.解析(1)运用数列的递推式和等差数列的定义和通项公式,即可得到所求;

(2)方法一、设数列的公差为d,求出,.由恒成立思想可得,求出,判断符号即可得证;

方法二、运用反证法证明,设的公差为d,假设存在自然数,使得,推理可得,作差,推出大于0,即可得证;

(3)运用等差数列和等比数列的求和公式,求得,,化简,推出小于3,结合等差数列的通项公式和数列的单调性,即可得到所求值.7.数列,

都是单调递增数列,假设将这两个数列的项按由小到大的顺序排成一列(一样的项视为一项),则得到一个新数列

(1)设数列,

分别为等差、等比数列,假设,

,

,求;

(2)设的首项为1,各项为正整数,

,假设新数列是等差数列,求数列的前n项和;

(3)设是不小于2的正整数),

,是否存在等差数列,使得对任意的,在与之间数列的项数总是假设存在,请给出一个满足题意的等差数列;假设不存在,请说明理由.解:(1)设等差数列的公差为d,等比数列的公比为q,

根据题意得,,计算得出或3,因数列,单调递增,

所以,,

所以,,

所以,因为,,,

(2)设等差数列的公差为d,又,且,

所以,所以因为是中的项,所以设,即当时,计算得出,不满足各项为正整数;当时,,此时,只需取,而等比数列的项都是等差数列,中的项,所以;当时,,此时,只需取,

由,得,是奇数,是正偶数,m有正整数解,

所以等比数列的项都是等差数列中的项,所以综上所述,数列的前n项和,或

(3)存在等差数列,只需首项,公差下证与之间数列的项数为.即证对任意正整数n,都有,

即成立.

由,

所以首项,公差的等差数列符合题意解析(1)设等差数列的公差为d,等比数列的公比为q,根据题意得,,计算得出或3,因数列,单调递增,,,可得,,利用通项公式即可得出.

(2)设等差数列的公差为d,又,且,所以,所以.因为是中的项,所以设,即.当时,计算得出,不满足各项为正整数当时,当时,即可得出.

(3)存在等差数列,只需首项,公差.下证与之间数列的项数为.即证对任意正整数n,都有,作差利用通项公式即可得出.8.对于数列,称(其中,为数列的前k项"波动均值〞.假设对任意的,,都有,则称数列为"趋稳数列〞.

(1)假设数列1,*,2为"趋稳数列〞,求*的取值范围;

(2)假设各项均为正数的等比数列的公比,求证:是"趋稳数列〞;

(3)数列的首项为1,各项均为整数,前k项的和为.且对任意,,都有,试计算:.解:(1)根据题意可得,

即,两边平方可得,

计算得出;

(2)证明:由,设,

因且,

故对任意的,,都有,

,,

因,

,,,,,

,

,

,

即对任意的,,都有,故是"趋稳数列〞;

(3)当时,当时,,

同理,,

因,

,

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