《数列》
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要高一数列的知识梳理,谢谢老师了
解题思路: 数列的知识
解题过程:
数列知识点回顾 第一部分:数列的基本概念 1.理解数列定义的四个要点 ⑴数列中的数是按一定“次序”排列的,在这里,只强调有“次序”,而不强调有“规律”.因此,如果组成两个数列的数相同而次序不同,那么它们就是不同的数列. ⑵在数列中同一个数可以重复出现. ⑶项a
与项数n是两个根本不同的概念. ⑷数列可以看作一个定义域为正整数集(或它的有限子集)的函数当自变量从小到大依次取值时对应的一列函数值,但函数不一定是数列. 2.数列的通项公式 一个数列{ a
}的第n项a
与项数n之间的函数关系,如果用一个公式a
=
来表示,就把这个公式叫做数列{ a
}的通项公式。若给出数列{ a
}的通项公式,则这个数列是已知的。若数列{ a
}的前n项和记为S
,则S
与a
的关系是:a
=
。 第二部分:等差数列 1.等差数列定义的几个特点: ⑴公差是从第一项起,每一项减去它前一项的差(同一常数),即d = a
-a
(n≥2)或d = a
-a
(n
N
). ⑵要证明一个数列是等差数列,必须对任意n
N
,a
-a
= d (n≥2)或d = a
-a
都成立.一般采用的形式为: ① 当n≥2时,有a
-a
= d (d为常数). ②当n
时,有a
-a
= d (d为常数). ③当n≥2时,有a
-a
= a
-a
成立. 若判断数列{ a
}不是等差数列,只需有a
-a
≠a
-a
即可. 2.等差中项 若a、A、b成等差数列,即A=
,则A是a与b的等差中项;若A=
,则a、A、b成等差数列,故A=
是a、A、b成等差数列,的充要条件。由于a
=
,所以,等差数列的每一项都是它前一项与后一项的等差中项。 3.等差数列的基本性质 ⑴公差为d的等差数列,各项同加一数所得数列仍是等差数列,其公差仍为d. ⑵公差为d的等差数列,各项同乘以常数k所得数列仍是等差数列,其公差为kd. ⑶若{ a
}、{ b
}为等差数列,则{ a
±b
}与{ka
+b}(k、b为非零常数)也是等差数列. ⑷对任何m、n
,在等差数列{ a
}中有:a
= a
+ (n-m)d,特别地,当m = 1时,便得等差数列的通项公式,此式较等差数列的通项公式更具有一般性. ⑸、一般地,如果l,k,p,…,m,n,r,…皆为自然数,且l + k + p + … = m + n + r + … (两边的自然数个数相等),那么当{a
}为等差数列时,有:a
+ a
+ a
+ … = a
+ a
+ a
+ … . ⑹公差为d的等差数列,从中取出等距离的项,构成一个新数列,此数列仍是等差数列,其公差为kd( k为取出项数之差). ⑺如果{ a
}是等差数列,公差为d,那么,a
,a
,…,a
、a
也是等差数列,其公差为-d;在等差数列{ a
}中,a
-a
= a
-a
= md .(其中m、k、![](http://img.wesiedu.com/upload/b/08/b08b529aad6b1a5415468859a57aa9d5.png)
![](http://img.wesiedu.com/upload/8/b1/8b101cb57cacf5a9e5d4a0abae2d1e21.png)
) ⑻在等差数列中,从第一项起,每一项(有穷数列末项除外)都是它前后两项的等差中项. ⑼当公差d>0时,等差数列中的数随项数的增大而增大;当d<0时,等差数列中的数随项数的减少而减小;d=0时,等差数列中的数等于一个常数. ⑽设a
,a
,a
为等差数列中的三项,且a
与a
,a
与a
的项距差之比
=
(
≠-1),则a
=
. 4.等差数列前n项和公式S
=
与S
= na
+
的比较 前n项和公式 公式适用范围 相同点 S
=
用于已知等差数列的首项和末项 都是等差数列的前n项和公式 S
= na
+
用于已知等差数列的首项和公差 5.等差数列前n项和公式S
的基本性质 ⑴数列{ a
}为等差数列的充要条件是:数列{ a
}的前n项和S
可以写成S
= an
+ bn的形式(其中a、b为常数). ⑵在等差数列{ a
}中,当项数为2n (n
N
)时,S
-S
= nd,
=
;当项数为(2n-1) (n
)时,S
-S
= a
,
=
. ⑶若数列{ a
}为等差数列,则S
,S
-S
,S
-S
,…仍然成等差数列,公差为
. ⑷若两个等差数列{ a
}、{ b
}的前n项和分别是S
、T
(n为奇数),则
=
. ⑸在等差数列{ a
}中,S
= a,S
= b (n>m),则S
=
(a-b). ⑹等差数列{a
}中,
是n的一次函数,且点(n,
)均在直线y =
x + (a
-
)上. ⑺记等差数列{a
}的前n项和为S
.①若a
>0,公差d<0,则当a
≥0且a
≤0时,S
最大;②若a
<0 ,公差d>0,则当a
≤0且a
≥0时,S
最小. 第三部分:等比数列 1.正确理解等比数列的含义 ⑴q是指从第2项起每一项与前一项的比,顺序不要错,即q =
(n
)或q =
(n≥2). ⑵由定义可知,等比数列的任意一项都不为0,因而公比q也不为0. ⑶要证明一个数列是等比数列,必须对任意n
,
= q;或
= q (n≥2)都成立. 2.等比中项与等差中项的主要区别 如果G是a与b的等比中项,那么
=
,即G
= ab,G =±
.所以,只要两个同号的数才有等比中项,而且等比中项有两个,它们互为相反数;如果A是a与b的等差中项,那么等差中项A唯一地表示为A=
,其中,a与b没有同号的限制.在这里,等差中项与等比中项既有数量上的差异,又有限制条件的不同. 3.等比数列的基本性质 ⑴公比为q的等比数列,从中取出等距离的项,构成一个新数列,此数列仍是等比数列,其公比为q
( m为等距离的项数之差). ⑵对任何m、n
,在等比数列{ a
}中有:a
= a
· q
,特别地,当m = 1时,便得等比数列的通项公式,此式较等比数列的通项公式更具有普遍性. ⑶一般地,如果t ,k,p,…,m,n,r,…皆为自然数,且t + k,p,…,m + … = m + n + r + … (两边的自然数个数相等),那么当{a
}为等比数列时,有:a
.a
.a
.… = a
.a
.a
.… .. ⑷若{ a
}是公比为q的等比数列,则{| a
|}、{a
}、{ka
}、{
}也是等比数列,其公比分别为| q |}、{q
}、{q}、{
}. ⑸如果{ a
}是等比数列,公比为q,那么,a
,a
,a
,…,a
,…是以q
为公比的等比数列. ⑹如果{ a
}是等比数列,那么对任意在n
,都有a
·a
= a
·q
>0. ⑺两个等比数列各对应项的积组成的数列仍是等比数列,且公比等于这两个数列的公比的积. ⑻当q>1且a
>0或0<q<1且a
<0时,等比数列为递增数列;当a
>0且0<q<1或a
<0且q>1时,等比数列为递减数列;当q = 1时,等比数列为常数列;当q<0时,等比数列为摆动数列. 4.等比数列前n项和公式S
的基本性质 ⑴如果数列{a
}是公比为q 的等比数列,那么,它的前n项和公式是S
=
也就是说,公比为q的等比数列的前n项和公式是q的分段函数的一系列函数值,分段的界限是在q = 1处.因此,使用等比数列的前n项和公式,必须要弄清公比q是可能等于1还是必不等于1,如果q可能等于1,则需分q = 1和q≠1进行讨论. ⑵当已知a
,q,n时,用公式S
=
;当已知a
,q,a
时,用公式S
=
. ⑶若S
是以q为公比的等比数列,则有S
= S
+qS
.⑵ ⑷若数列{ a
}为等比数列,则S
,S
-S
,S
-S
,…仍然成等比数列. ⑸若项数为3n的等比数列(q≠-1)前n项和与前n项积分别为S
与T
,次n项和与次n项积分别为S
与T
,最后n项和与n项积分别为S
与T
,则S
,S
,S
成等比数列,T
,T
,T
亦成等比数列. 二、难点突破 1.并不是所有的数列都有通项公式,一个数列有通项公式在形式上也不一定唯一.已知一个数列的前几项,这个数列的通项公式更不是唯一的. 2.等差(比)数列的定义中有两个要点:一是“从第2项起”,二是“每一项与它前一项的差(比)等于同一个常数”.这里的“从第2项起”是为了使每一项与它前面一项都确实存在,而“同一个常数”则是保证至少含有3项.所以,一个数列是等差(比)数列的必要非充分条件是这个数列至少含有3项. 3.数列的表示方法应注意的两个问题:⑴{ a
}与a
是不同的,前者表示数列a
,a
,…,a
,…,而后者仅表示这个数列的第n项;⑵数列a
,a
,…,a
,…,与集合{ a
,a
,…,a
,…,}不同,差别有两点:数列是一列有序排布的数,而集合是一个有确定范围的整体;数列的项有明确的顺序性,而集合的元素间没有顺序性. 4.注意设元的技巧时,等比数列的奇数个项与偶数个项有区别,即: ⑴对连续奇数个项的等比数列,若已知其积为S,则通常设…,aq
, aq
, a,aq,aq
,…; ⑵对连续偶数个项同号的等比数列,若已知其积为S,则通常设…,aq
, aq
, aq,aq
,…. 5.一个数列为等比数列的必要条件是该数列各项均不为0,因此,在研究等比数列时,要注意a
≠0,因为当a
= 0时,虽有a
= a
· a
成立,但{a
}不是等比数列,即“b
= a · c”是a、b、 c成等比数列的必要非充分条件;对比等差数列{a
},“2b = a + c”是a、b、 c成等差数列的充要条件,这一点同学们要分清. 6.由等比数列定义知,等比数列各项均不为0,因此,判断一数列是否成等比数列,首先要注意特殊情况“0”.等比数列的前n项和公式蕴含着分类讨论思想,需分分q = 1和q≠1进行分类讨论,在具体运用公式时,常常因考虑不周而出错. 数列基础知识定时练习题 (满分为100分+附加题20分,共120分;定时练习时间120分钟) 一、选择题(本大题共15小题,每小题3分,共45分.在每小题给出的四个选项中,只有一项是符合题目要求的) 1.下列四个数中,哪一个是数列{
}中的一项 ( ) (A)380 (B)39 (C)35 (D)23 2.在等差数列
中,公差
,
,则
的值为( ) (A)40 (B)45 (C)50 (D)55 3.一套共7册的书计划每2年出一册,若各册书的出版年份数之和为13979,则出齐这套书的年份是( ) (A)1997 (B)1999 (C)2001 (D)2003 4.一个项数是偶数的等比数列,它的偶数项的和是奇数项和的2倍,又它的首项为1,且中间两项的和为24,则此等比数列的项数为( ) (A)12 (B)10 (C)8 (D)6 5.已知1是
与
的等比中项,又是
与
的等差中项,则
的值是( ) (A)1或
(B)1或
(C)1或
(D)1或
6.首项为-24的等差数列,从第10项开始为正,则公差
的取值范围是( ) (A)
(B)
(C)
≤
(D)
≤3 7.如果-1,a,b,c,-9成等比数列,那么( ) (A)b=3,ac=9 (B)b=-3,ac=9 (C)b=3,ac=-9 (D)b=-3,ac=-9 8.在等差数列{a
}中,已知a
=2,a
+a
=13,则a
+a
+a
等于( ) A.40 B.42 C.43 D.45 9.已知某等差数列共有10项,其奇数项之和为15,偶数项之和为30,则其公差为( ) A.5 B.4 C. 3 D. 2 10.若互不相等的实数
成等差数列,
成等比数列,且
,则
( ) A.4 B.2 C.-2 D.-4 11.在等比数列{an}中,a1=1,a10=3,则a2 a3 a4 a5 a6 a7 a8 a9 = ( ) A. 81 B. 27
C.
D. 243 12. 在等比数列
中,
,前
项和为
,若数列
也是等比数列,则
等于( ) (A)
(B)
(C)
(D)
【点评】本题考查了等比数列的定义和求和公式,着重考查了运算能力。 13.设
是公差为正数的等差数列,若
,
,则
( ) A.
B.
C.
D.
14.设
是等差数列
的前
项和,若
,则
( ) A.
B.
C.
D.
15.设Sn是等差数列{an}的前n项和,若=,则 = ( ) (A) (B) (C) (D) 二、填空题(本大题共5小题,每小题3分,共15分.把答案填在题中横线上) 1.在数列
中,
,且
,则
. 2.等比数列
的前三项为
,
,
,则
3. 若数列
满足:
,2,3….则
. 4.设
为等差数列
的前n项和,
=14,S10-
=30,则S9= . 5.在数列
中,若
,
,则该数列的通项
。 三、解答题(本大题共4小题,每小题10分,共40分) 1.已知
为等比数列,
,求
的通项式。 2.设等比数列
的前n项和为
,
3. 已知正项数列{an},其前n项和Sn满足10Sn=an2+5an+6且a1,a3,a15成等比数列,求数列{an}的通项an . 4.数列
的前
项和记为
(Ⅰ)求
的通项公式; (Ⅱ)等差数列
的各项为正,其前
项和为
,且
,又
成等比数列,求
本小题主要考察等差数列、等比数列的基础知识,以及推理能力与运算能力。满分12分。 四、附加题(20分) 某校有教职员工150人,为了丰富教工的课余生活,每天定时开放健身房和娱乐室。据调查统计,每次去健身房的人有10%下次去娱乐室,而在娱乐室的人有20%下次去健身房.请问,随着时间的推移,去健身房的人数能否趋于稳定? 1. A 2.B 3.D 4.C 5.D 6.D 7.B 解:由等比数列的性质可得ac=(-1)×(-9)=9,b×b=9且b与奇数项的符号相同,故b=-3,选B 8.B 解:在等差数列
中,已知
∴ d=3,a5=14,
=3a5=42,选B. 9.C 解:
,故选C. 10. D 解:由互不相等的实数
成等差数列可设a=b-d,c=b+d,由
可得b=2,所以a=2-d,c=2+d,又
成等比数列可得d=6,所以a=-4,选D 11.A 解:因为数列{an}是等比数列,且a1=1,a10=3,所以a2a3a4a5a6a7a8a9= (a2a9)(a3a8)(a4a7)(a5a6)=(a1a10)4=34=81,故选A 12.C 【解析】因数列
为等比,则
,因数列
也是等比数列, 则
即
,所以
,故选择答案C。 13.B 【解析】
是公差为正数的等差数列,若
,
,则
,
,∴ d=3,
,![](http://img.wesiedu.com/upload/b/7b/b7b908e4d75955779da9066fdead44cf.png)
,选B. 14. D 【解析】
是等差数列
的前
项和,若
∴ ![](http://img.wesiedu.com/upload/8/a7/8a7f59826e06b326d34b1b32edf19d51.png)
,选D. 15.A 解析:由等差数列的求和公式可得
且
所以
,故选A 二、填空题 1. 99 2.
3. 解:数列
满足:
,2,3…,该数列为公比为2的等比数列,∴ ![](http://img.wesiedu.com/upload/a/1e/a1e3b60eb8263bd0ab87f917b49e9aac.png)
. 4.解:设等差数列
的首项为a1,公差为d,由题意得
,联立解得a1=2,d=1,所以S9=
5.解:由
可得数列
为公差为2的等差数列,又
,所以
2n-1 三、解答题 1.解: 设等比数列{an}的公比为q, 则q≠0, a2= = , a4=a3q=2q 所以 + 2q= , 解得q1= , q2= 3, 当q1=, a1=18.所以 an=18×()n-1= = 2×33-n. 当q=3时, a1= , 所以an=×3n-1=2×3n-3. 2.解:设
的公比为q,由
,所以得
…①
……②由①、②式得整理得
解得
所以 q=2或q=-2 将q=2代入①式得
,所以
将q=-2代入①式得
,所以
3.解析:解: ∵10Sn=an2+5an+6, ① ∴10a1=a12+5a1+6,解之得a1=2或a1=3. 又10Sn-1=an-12+5an-1+6(n≥2),② 由①-②得 10an=(an2-an-12)+6(an-an-1),即(an+an-1)(an-an-1-5)=0 ∵an+an-1>0 , ∴an-an-1=5 (n≥2). 当a1=3时,a3=13,a15=73. a1, a3,a15不成等比数列∴a1≠3; 当a1=2时,a3=12, a15=72, 有a32=a1a15 , ∴a1=2, ∴an=5n-3. 附加题 解: 引入字母,转化为递归数列模型. 设第n次去健身房的人数为an,去娱乐室的人数为bn,则
.
.
,于是
即
.
.故随着时间的推移,去健身房的人数稳定在100人左右. 4.解:(Ⅰ)由
可得
,两式相减得
又
∴
故
是首项为
,公比为
得等比数列 ∴
(Ⅱ)设
的公差为
由
得,可得
,可得
故可设
又
由题意可得
解得
∵等差数列
的各项为正,∴
∴
∴
最终答案:略
解题过程:
数列知识点回顾 第一部分:数列的基本概念 1.理解数列定义的四个要点 ⑴数列中的数是按一定“次序”排列的,在这里,只强调有“次序”,而不强调有“规律”.因此,如果组成两个数列的数相同而次序不同,那么它们就是不同的数列. ⑵在数列中同一个数可以重复出现. ⑶项a
![](http://img.wesiedu.com/upload/9/93/9938b153f90a40187d352aab3f727dfd.png)
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![](http://img.wesiedu.com/upload/8/4d/84d2038f769c0b22b13f1dda853538d9.png)
![](http://img.wesiedu.com/upload/a/cd/acd20703b5e26f23998f67cf5ef72a8b.png)
![](http://img.wesiedu.com/upload/8/58/8582580e94fad0562a9caf94b1b2f66c.png)
![](http://img.wesiedu.com/upload/b/bf/bbf5a94dfcb4ce3361d74bf87e59837c.png)
![](http://img.wesiedu.com/upload/f/2a/f2ae52bcd48431ddd6089864acc90c07.png)
![](http://img.wesiedu.com/upload/f/ff/fff25a021414fb787d1e9673a2668e3b.png)
![](http://img.wesiedu.com/upload/6/7d/67d48b5155bb2ab85b41396dae4873bb.png)
![](http://img.wesiedu.com/upload/b/83/b837b94db3e1d9b70bd2984fe2967f68.png)
![](http://img.wesiedu.com/upload/8/64/864d4141a86a222acaa0a0bb1061a408.png)
![](http://img.wesiedu.com/upload/5/b4/5b48e28e38a1ba38b9a22a5f79934cd2.png)
![](http://img.wesiedu.com/upload/9/70/970ec7c62456192ea1d11371a7a7694f.png)
![](http://img.wesiedu.com/upload/3/6e/36e4cf7a2b7cff818490f075f198044d.png)
![](http://img.wesiedu.com/upload/2/cf/2cfa6f869687ea9655a83162106c5333.png)
![](http://img.wesiedu.com/upload/0/54/054dd4c9bd5aa3af4648b5a885a83225.png)
![](http://img.wesiedu.com/upload/d/b5/db51a7f37372e3126328c4984a9cec8d.png)
![](http://img.wesiedu.com/upload/f/43/f431d2c31ec684133f221f11b179fae6.png)
![](http://img.wesiedu.com/upload/3/54/3547f6599135bd87523281bdb5015597.png)
![](http://img.wesiedu.com/upload/f/96/f96b2a0a6fb05f66b0f8efe655c63806.png)
![](http://img.wesiedu.com/upload/f/22/f221676a5c6a56f1cf0cd8a9ea097444.png)
![](http://img.wesiedu.com/upload/2/d1/2d1f6f976403a3f16b23c3b315cad9bf.png)
![](http://img.wesiedu.com/upload/a/ba/aba36a7283c833e6c7e6d3af2d1a614a.png)
![](http://img.wesiedu.com/upload/2/37/237df09cc327dbc45a0892247ef59ed9.png)
![](http://img.wesiedu.com/upload/a/86/a867fffda3d5d541bffa25a1a8d5d0ac.png)
![](http://img.wesiedu.com/upload/f/40/f408fc6927bd256465f9399490bb68ce.png)
![](http://img.wesiedu.com/upload/5/4d/54de216377ac6a321e6a63b93271a0a7.png)
![](http://img.wesiedu.com/upload/e/a9/ea98570c1809d456756303cdbfbd8ad8.png)
![](http://img.wesiedu.com/upload/b/44/b448d5f01dc57114d18def306dfed987.png)
![](http://img.wesiedu.com/upload/1/93/193eab242d61fcff07c38adb7775c93a.png)
![](http://img.wesiedu.com/upload/2/e7/2e79c5231305d482f37c42e7faa89727.png)
![](http://img.wesiedu.com/upload/b/4a/b4a2b09a6a8161d04b792caff91b034c.png)
![](http://img.wesiedu.com/upload/8/e1/8e109426fad1a9d2c864cfae6473b1ac.png)
![](http://img.wesiedu.com/upload/a/1b/a1b94d7ee717dd3669e436ae43a2e97f.png)
![](http://img.wesiedu.com/upload/0/af/0aff5ee10668b22f7610c3a3ccd12f55.png)
![](http://img.wesiedu.com/upload/a/d0/ad02d45e9fb06ce7506077d3e26ee26d.png)
![](http://img.wesiedu.com/upload/3/48/3489bbd9a6bf7d796a5feee90d79c839.png)
![](http://img.wesiedu.com/upload/a/11/a112ae9e1bec72c2e3c070d954673720.png)
![](http://img.wesiedu.com/upload/8/a1/8a1fadeb0b12e4f6707403bc402a1b7b.png)
![](http://img.wesiedu.com/upload/8/36/83644636a3ef278171c43e493b595d9b.png)
![](http://img.wesiedu.com/upload/f/59/f592850775bd348be4f9513032f56297.png)
![](http://img.wesiedu.com/upload/9/4e/94e7812fcfd6b58c21ed3cb4b0ec6aa0.png)
![](http://img.wesiedu.com/upload/7/38/7385f23ab4c600f393f3f665c442b47b.png)
![](http://img.wesiedu.com/upload/0/0e/00e0b9481e0de7b40677926e5022f4d0.png)
![](http://img.wesiedu.com/upload/a/d0/ad02d45e9fb06ce7506077d3e26ee26d.png)
![](http://img.wesiedu.com/upload/3/56/356619a33d35bdc9b917d86dfa2cea15.png)
![](http://img.wesiedu.com/upload/8/33/833a380a546c315c254368fb09e87b4a.png)
![](http://img.wesiedu.com/upload/9/2d/92d5481ee869b53f19ad0c34b313b19a.png)
最终答案:略