数列 递推公式如何导出通项公式
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如何通过数列的递推公式推出通项公式? 请举例说明。
![数列 递推公式如何导出通项公式](/uploads/image/z/18620482-58-2.jpg?t=%E6%95%B0%E5%88%97+%E9%80%92%E6%8E%A8%E5%85%AC%E5%BC%8F%E5%A6%82%E4%BD%95%E5%AF%BC%E5%87%BA%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F)
解题思路: 多种情况
解题过程:
同学你好,如对解答还有疑问,可在答案下方的【 添加讨论】中留言,我收到后会尽快给你答复。感谢你的配合! 祝你学习进步,生活愉快 利用递推关系求数列通项的十三种类型 湖北省巴东县第三高级中学 杨开亮(电话:13997763366) 数列是一种特殊的函数,数列的通项公式可以看作项数
(
的取值范围为正整数或其子集)的函数。数列的通项公式直接表述数列的本质。是给出数列的一种重要方法。递推数列的通项具有很强的逻辑性,是高考考查创新意识、逻辑推理、转化化归能力好素材。近几年的 高考试题和竞赛试题中,几乎都有数列解答题,常常以“可怕”的递推形式出现。 递推是一种重要的 数学思想方法。根据问题的特点建立递推关系,再借助递推关系求解问题的方法之一 ——递推法。凡是可视为正整数为自变量的函数且可建立其递推关系的问题大多可以用此法求解。递推数列的通项有以下几种类型: 题型1.形如
型 (1)若
为常数,即:
,此时数列为等差数列,则
=
. (2)若
为
的函数时,用累加法. ①若
是关于
的一次函数,累加后可转化为等差数列求和; ②若
是关于
的二次函数,累加后可分组求和; ③若
是关于
的指数函数,累加后可转化为等比数列求和; ④若
是关于
的分式函数,累加后可裂项求和。 方法如下: 由
得:
时,
,
,
所以各式相加得
即:
. 为了书写方便,也可用横式来写:
时,
, ![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
=
. 例 1. 已知数列
的首项为1,且
写出数列
的通项公式. 解:由
得:
时,
,
,
叠加得
∴
∴![](http://img.wesiedu.com/upload/9/cf/9cf6ddb242570704cb41a31b09f77ad0.png)
例2. (2003天津文) 已知数列
满足
, 证明
. 证明:由已知得:
=
∴
. 例3.已知数列
满足
,
,求此数列的通项公式. 解:由
得:
时,
,
,
叠加得
∴![](http://img.wesiedu.com/upload/9/cf/9cf6ddb242570704cb41a31b09f77ad0.png)
. 此题也可以用数学归纳法来求解. 例4、(2005年天津理科13) 在数列
中,
且
则
=2600. 题型2.形如
型 (1)当
为常数,即:
(其中q是不为0的常数),此时数列为等比数列,
=
. (2)当
为
的函数时,用累乘法. 由
得
时,
, ![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
=![](http://img.wesiedu.com/upload/4/9e/49e35579d061862d24753b73b9eac3f0.png)
. 例1. 已知
,求数列{an}的通项公式. 解:因为
所以
故
又因为
,即
, 所以由上式可知
,所以
,故由累乘法得
=
所以![](http://img.wesiedu.com/upload/8/60/86081ff1da00a1685ac865b523cfc86b.png)
-1. 评注:本题解题的关键是把原来的递推关系式
转化为
若令
,则问题进一步转化为
形式,进而应用累乘法求出数列的通项公式. 例2. 设
是首项为1的正项数列,且
(
=1,2, 3,…),则它的通项公式是
=________. 解:已知等式可化为:
![](http://img.wesiedu.com/upload/d/bc/dbcea55d9a16ea7d9b0293c844763d96.png)
(
)
(n+1)
, 即
∴
时,
∴
=
=
. 评注:本题是关于
和
的二次齐次式,可以通过因式分解(一般情况时用求根公式)得到
与
的更为明显的关系式,从而求出
. 题型3.形如
型 (1)若
(d为常数),则数列{
}为“等和数列”,它是一个周期数列,周期为2,其通项分奇数项和偶数项来讨论; (2)若f(n)为n的函数(非常数)时,可通过构造转化为
型,通过累加来求出通项;或用逐差法(两式相减)得
,,分奇偶项来分求通项. 例1. 数列{
}满足
,
,求数列{an}的通项公式. 分析 1:构造转化为
型 解法1:令
则
.
时,
各式相加:
当n为偶数时,
.此时
当n为奇数时,
此时
,所以
. 故 ![](http://img.wesiedu.com/upload/b/e3/be3b4872969c4b6034a026bb80b878c9.png)
解法2:![](http://img.wesiedu.com/upload/d/bc/dbcea55d9a16ea7d9b0293c844763d96.png)
![](http://img.wesiedu.com/upload/e/45/e459cdb71133843255b396b328c8bf54.png)
![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
时,
, 两式相减得:
. ![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
构成以
,为首项,以2为公差的等差数列;
构成以
,为首项,以2为公差的等差数列 ![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
,
. ![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
评注:结果要还原成
的表达式. 例2.(2005年江西卷文22)已知数列
的前n项和Sn满足 Sn-Sn-2=3
求数列
的通项公式. 解:方法一:先考虑偶数项有:
………
同理考虑奇数项有:
………
![](http://img.wesiedu.com/upload/d/dd/ddde63233adbb919c63ac64312d353b4.png)
![](http://img.wesiedu.com/upload/6/80/68093a9c6cadb1b680575962d3beb2ae.png)
![](http://img.wesiedu.com/upload/6/e9/6e988acdb5842db5e92929e2b59a4616.png)
综合可得
方法二:因为
两边同乘以
,可得:
令
所以
………
∴
∴
∴
题型4.形如
型 (1)若
(p为常数),则数列{
}为“等积数列”,它是一个周期数列,周期为2,其通项分奇数项和偶数项来讨论; (2)若
为
的函数(非常数)时,可通过逐差法得
,两式相除后,分奇偶项来分求通项. 例. 已知数列![](http://img.wesiedu.com/upload/5/c5/5c503f11623bfbc62efce9948c8273a6.png)
,求此数列的通项公式. 解:
……
所以
. 所以原数列的奇数项与偶数项分别构成以3为首项
为公比和以
为首项
为公比的等比数列. ∴当
为奇数时,
;当
为偶数时,
∴
题型5.形如
,其中
)型 一、
为常数 (1)若
=1时,数列{
}为等差数列;同题型1. (2)若
=0时,数列{
}为等比数列;同题型2. (3)若
时,数列{
}为线性递推数列,其通项可通过待定系数法构造辅助数列来求. 方法如下:设
, 得
,与题设
比较系数得
,所以
所以有:
因此数列
构成以
为首项,以c为公比的等比数列, 所以 ![](http://img.wesiedu.com/upload/9/6a/96a365d3fe6a9115a4f43c35d665f712.png)
即:
. 规律:将递推关系
化为
,构造成公比为c的等比数列
从而求得通项公式
有时我们从递推关系
中把n换成n-1有
,两式相减有
从而化为公比为c的等比数列
,进而求得通项公式.
,再利用类型(1)即可求得通项公式.我们看到此方法比较复杂. 例1.已知数列
中,
求通项
. 分析:两边直接加上
,构造新的等比数列。 解:由
得
, 所以数列
构成以
为首项,以
为公比的等比数列 所以
,即
. 方法二:由 ![](http://img.wesiedu.com/upload/d/9a/d9a1975842c7dacfad70ddcc7014bd33.png)
![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
时,
两式相减得
![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
, 数列
是以
=
为首项,以c为公比的等比数列. ![](http://img.wesiedu.com/upload/a/44/a44c3fa6fbee01257eb2e5fb2358ac57.png)
![](http://img.wesiedu.com/upload/2/ce/2ce6d991b8aabe37a8480a681fe13dbe.png)
![](http://img.wesiedu.com/upload/7/84/784d428cab948f687b1110553dd67c09.png)
=(
![](http://img.wesiedu.com/upload/a/44/a44c3fa6fbee01257eb2e5fb2358ac57.png)
. 方法三:迭代法 由 递推式
直接迭代得
=
=
=
. 方法四:归纳、猜想、证明. 先计算出
,再猜想出通项
,最后用数学归纳法证明. 注:请用这三种方法来解例题,体会并比较它们的不同. 二、若
(其中
是常数,且
) 方法:找规律,相减构造新数列. 例2.在数列
中,
求通项
. 解:
,
① ![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
时,
, 两式相减得
.令
,则
利用题型5的方法知
即
② 再由累加法可得
. 亦可联立 ① ②解出
. 例3. 在数列
中,
,求通项
. 解:原递推式可化为
比较系数可得:
上式即为
所以
是一个等比数列,首项
,公比为
.
即:
故
. 例4(2008年湖北理22) (满分14分)已知数列
和
满足:
,
其中
为实数,
为正整数. (Ⅰ)对任意实数
,证明数列
不是等比数列; (Ⅱ)试判断数列
是否为等比数列,并证明你的结论; (Ⅲ)设
,
为数列
的前
项和.是否存在实数
,使得对任意正整数
,都有
?若存在,求
的取值范围;若不存在,说明理由. 本小题主要考查等比数列的定义、数列求和、不等式等基础知识和分类讨论的思想,考查综合分析问题的能力和推理认证能力, 证明:(Ⅰ)假设存在一个实数λ,使{an}是等比数列,则有
, 即
矛盾. 所以{an}不是等比数列. (Ⅱ)解:因为bn+1=(
1)n+1[an+1
3(n
1)+21]=(
1)n+1(
an
2n+14) =
(
1)n·(an
3n+21)= ![](http://img.wesiedu.com/upload/d/29/d29d595f616d67ea90394d4883f785e2.png)
bn 又
,所以 当λ=-18,bn=0(n
),此时{bn}不是等比数列: 当λ≠-18时,b1=(λ+18) ≠0,由上可知bn≠0,∴
(n
). 故当λ≠
18时,数列{bn}是以-(λ+18)为首项,-
为公比的等比数列. (Ⅲ)由(Ⅱ)知,当λ=
18,bn = 0,Sn = 0,不满足题目要求. ∴λ≠
18,故知bn=
(λ+18)·(-
)n
1,于是可得 Sn=![](http://img.wesiedu.com/upload/d/29/d29d595f616d67ea90394d4883f785e2.png)
要使a<Sn<b对任意正整数n成立, 即a<![](http://img.wesiedu.com/upload/d/29/d29d595f616d67ea90394d4883f785e2.png)
(λ+18)
[1-(-
)n]
b(
)
① 当n为正奇数时,1<f(n)
∴f(n)的最大值为f(1) =
, f(n)的最小值为f(2) =
, 于是,由①式得
a<
(λ+18)<
当a<b
3a时,由
,不存在实数满足题目要求; 当b>3a存在实数λ,使得对任意正整数n, 都有a<Sn<b,且λ的取值范围是
. 该类型一般用待定系数法构造等比数列求解,也可用递推法(迭代法)求解. 三、若
(其中
是常数,且![](http://img.wesiedu.com/upload/3/48/34847cdf25f3511e001e2fdcaebbc471.png)
0,1) ①若
=1时,即:
,累加即可. ②若
时,即:
, 求通项方法有以下三种方向:i. 两边同除以
. 即:
,令
,则
, 然后类型1,累加求通项. ii.两边同除以
. 即:
, 令
,则可化为
.然后转化为类型5来解, iii.待定系数法: 设
.通过比较系数,求出
,转化为等比数列求通项. 例5.(2003年天津理)设
为常数,且
. 证明:对任意
≥1,
证法1:两边同除以(-2)
,得
令
,则
![](http://img.wesiedu.com/upload/b/5d/b5de2431cec2409a9cc12e0ca75ceb3b.png)
=
=
=
∴![](http://img.wesiedu.com/upload/3/41/3410dc325e631c203a9266bceeae72a1.png)
. 证法2:由
得
. 设
,则b
. 即:
, 所以
是以
为首项,
为公比的等比数列. 则
=![](http://img.wesiedu.com/upload/8/bd/8bd831ed640e3ff96c6b24e39bc5a444.png)
, 即:
, 故
. 评注:本题的关键是两边同除以3
,进而转化为类型5,构造出新的等比数列,从而将求一般数列的通项问题转化为求等比数列的通项问题. 证法3:用待定系数法 设
, 即:
, 比较系数得:
,所以 ![](http://img.wesiedu.com/upload/9/89/989817bc50db7f8ba6bc03823e7c4ae3.png)
所以
, 所以数列
是公比为-2,首项为
的等比数列.
即
. 方法4:本题也可用数学归纳法证. (i)
当
时,由已知
,等式成立; ( ii)假设当
等式成立,则
那么
也就是说,当![](http://img.wesiedu.com/upload/4/d5/4d5c2b37ddd0ed25710920bcc93bbdf7.png)
时,等式也成立. 根据(i)和(ii),可知等式对任何
,成立. 四、若
(其中r、q是常数,且![](http://img.wesiedu.com/upload/3/48/34847cdf25f3511e001e2fdcaebbc471.png)
0,1) 对
型递推关系一般可以将递推关系式两边同除以
,得
后再用递推法或构造法求解. 规律:
类型共同的规律为:两边同除以
,累加求和,只是求和的方法不同. 题型6.形如
型 例1、(2006年全国Ⅰ理22) 设数列
的前
项的和
,
(Ⅰ)求首项
与通项
; (Ⅱ)设
,
,证明:
解: (Ⅰ)由 Sn=
-×2n+1+,
=1,2,3,… , ① 得
所以
, 再由①有 Sn-1=
-×2
+,
=2,3,4,… 将①和②相减得:
=Sn-Sn-1= (
-
)-×(2
+1-2
),
=2,3, … 整理得:
+2
= 4(
+2
-1),
=2,3, … , 因而数列{
+2
}是首项为
+2 = 4 ,公比为4的等比数列, 即 :
+2
=4×4
-1= 4
,
=1,2,3, …, 因而
=4
-2
,
=1,2,3, …, (Ⅱ)将
=4
-2
代入①得
= ×(4
-2
)-×2
+1 + = ×(2
+1-1)(2
+1-2) = ×(2
+1-1)(2
-1)
=
= ×
= ×(
) 所以,
= ![](http://img.wesiedu.com/upload/8/95/895d6862a089bc01977191369a101a31.png)
-
) = ×( -
) < 题型7.形如
型 (1)
即
, 采用取倒数法. 例1. 已知数列
中,
,
,求通项公式
. 解:取倒数:![](http://img.wesiedu.com/upload/f/c1/fc11e416c847438fd0ab68340c822de3.png)
例2.(2006年湖北理科)已知不等式
为大于2的整数,
表示不超过
的最大整数. 设数列
的各项为正,且满足
(Ⅰ)证明
分析:本题看似是不等式问题,实质就是求通项问题. 证:∵当
即
于是有
所有不等式两边相加可得
由已知不等式知,当n≥3时有,
∵
评注:本题结合不等式的性质,从两边取倒数入手,再通过裂项求和即可证得. (2).形如
型 方法:不动点法: 我们设
,由方程
求得二根x,y,由
有
同理
,两式相除有
,从而得
,再解出
即可. 例1. 设数列
满足
,求
的通项公式. 分析:此类问题常用参数法化等比数列求解. 解:对等式两端同时加参数t,得:
, 令
, 解之得t=1,-2 代入
得
,
, 相除得
,即{
}是首项为
, 公比为
的等比数列,
=
, 解得
. 方法2:![](http://img.wesiedu.com/upload/7/34/73458b52c9ff4a6542a4baff5a0326a0.png)
, 两边取倒数得
, 令b
,则b![](http://img.wesiedu.com/upload/7/41/741f5b2f72b5a768193ac9bde243835c.png)
,
转化为类型5来求. 题型8.形如
(其中
,
为常数)型 (1)当
+
=1时 用转化法 例1.数列
中,若
,且满足
,求
. 解:把
变形为
. 则数列
是以
为首项,3为公比的等比数列,则
利用类型6的方法可得
. (2)当
时 用待定系数法. 例2. 已知数列
满足
,且
,且满足,求
. 解:令
,即
,与已知
比较,则有
,故
或
下面我们取其中一组
来运算,即有
, 则数列
是以
为首项,3为公比的等比数列,故
,即
,利用题型 4 的方法, 可得
. 评注:形如
的递推数列,我们通常采用两次类型(5)的方法来求解,但这种方法比较复杂,我们采用特征根的方法:设方程
的二根为
,设
,再利用
的值求得p,q的值即可. 题型9. 形如
(其中p,r为常数)型 (1)![](http://img.wesiedu.com/upload/7/56/7565d09221fab1cc8af77dbb38346cf0.png)
0,
用对数法. 例1. 设正项数列
满足
,
(
2).求数列
的通项公式. 解:两边取对数得:
,
, 设
,则
, ∴
是以2为公比的等比数列,
∴
,
,
, ∴
例2.数列
中,
,
(
2),求数列
的通项公式. 答案:
(2)![](http://img.wesiedu.com/upload/7/56/7565d09221fab1cc8af77dbb38346cf0.png)
0时 用迭代法. 例3(2005年江西理21)已知数列
的各项都是正数,且满足:
(1)证明
(2)求数列
的通项公式
. 解:(1)方法一 用数学归纳法证明: 1°当n=1时,
∴
,命题正确. 2°假设n=k时有
则
而
又
∴
时命题正确. 由1°、2°知,对一切
时有
方法二:用数学归纳法证明: 1°当
=1时,
∴
; 2°假设n=k时有
成立, 令
,
在[0,2]上单调递增,所以由假设 有:
即
也即当
=
+1时
成立, 所以对一切
(2)下面来求数列的通项:
所以
, 又
=-1,所以
.
. 解法2:本题用归纳-猜想-证明,也很简捷,请试一试. 解法3:设c
,则c
,转化为上面类型(1)来解. 题型10. 形如
型 (1)若
是常数,同题型1. (2)若
是一次式同题型1 (3)若
是二次式。 例1.(2006年陕西理20)已知正项数列
,其前n项和S
满足
成等比数列,且10 S
=
,求数列
的通项公式
. 解:∵10 S
=
① ∴![](http://img.wesiedu.com/upload/5/63/563d2406b567717f0b8142b01b497724.png)
又10 S![](http://img.wesiedu.com/upload/f/14/f14a16c9cd26724b60b3bdb579943f6b.png)
=
(
2), ② ① - ②,得
, 即
. ∵
∴
. 当
.此时
不成等比数列,∴
. 当
.此时有
.∴
. ∴
. 评注:该题用
即
的关系,
. 消去
,也可用
的方法求出
. 例2.(2007年重庆理科21)已知各项均为正数的数列
的前
项和
满足
,且
,
. (Ⅰ)求
的通项公式; (Ⅱ)设数列
满足
,并记
为
的前
项和, 求证:
. 解:(I)解由
,解得
或
, 由假设
,因此
, 又由
, 得
,即
或
, 因
,故
不成立,舍去. 因此
,从而
是公差为
,首项为
的等差数列, 故
的通项为
. (II)证法一:由
可解得
; 从而
. 因此
. 令
, 则
. 因
,故
. 特别地
,从而
. 即
. 证法二:同证法一求得
及
, 由二项式定理知,当
时,不等式
成立. 由此不等式有
证法三:同证法一求得
及
. 令
,
. 因
. 因此
. 从而![](http://img.wesiedu.com/upload/3/46/34618cea42de536a1f53cecd4977505e.png)
. 证法四:同证法一求得
及
. 下面用数学归纳法证明:
. 当
时,
,
, 因此
,结论成立. 假设结论当
时成立,即
. 则当
时,
; 因
.故
. 从而
.这就是说,当
时结论也成立. 综上
对任何
成立. 例3.(2008年全国理科2)设函数
.数列
满足
,
. (Ⅰ)证明:函数
在区间
是增函数; (Ⅱ)证明:
; (Ⅲ)设
,整数
.证明:
. 解:(Ⅰ)证明:
,
故函数
在区间(0,1)上是增函数. (Ⅱ)证明:(用数学归纳法)(i)当n=1时,
,
,
. 由函数
在区间
是增函数,且函数
在
处连续,则
在区间
是增函数,
,即
成立; (ⅱ)假设当
时,
成立,即
那么当
时,由
在区间
是增函数,
得:
.而
,则
,
,也就是说当
时,
也成立; 根据(ⅰ)、(ⅱ)可得对任意的正整数
,
恒成立. (Ⅲ)证明:由
.
可得: ![](http://img.wesiedu.com/upload/a/4b/a4bf7200e57c3cf59548dabce258cf70.png)
. 1、若存在某
满足
,则
. 2、若对任意
都有
,则:
![](http://img.wesiedu.com/upload/c/69/c69315f094f82b65f376448790bb3005.png)
成立. 例4.已知数列
中,
且
,求数列
的通项公式. 解:由已知
得
, 化简有
,由类型(1)有
, 又
得
,所以
,又
,
, 则
. 题型11. 形如
型 例1.(2008年湖南理科)(本小题满分12分) 数列
(Ⅰ)求
并求数列
的通项公式; (Ⅱ)设
证明:当
解 (Ⅰ)因为
一般地,当
时,
=
,即
所以数列
是首项为1、公差为1的等差数列,因此
当
时,
所以数列
是首项为2、公比为2的等比数列,因此
故数列
的通项公式为
(Ⅱ)由(Ⅰ)知,
①
② ①-②得,![](http://img.wesiedu.com/upload/b/2c/b2ce2cd09f05f7bc71bf9a8e7da8a6db.png)
所以
要证明当
时,
成立,只需证明当
时,
成立. 证法一(1)当n = 6时,
成立. (2)假设当
时不等式成立,即
则当n =k+1时,
由(1)、(2)所述,当
6时,
,即当![](http://img.wesiedu.com/upload/3/48/34847cdf25f3511e001e2fdcaebbc471.png)
6时,
证法二 令
,则
所以当
时,
.因此当
时,
. 于是当
时,
.综上所述,当
时,
. 题型12. 形如
型 例1.(2008年重庆理科22)设各项均为正数的数列{
}满足
. (Ⅰ)若
,求
,并猜想
的值(不需证明); (Ⅱ)记
对
2恒成立,求
的值及数列{bn}的通项公式. 解:(Ⅰ)因![](http://img.wesiedu.com/upload/2/ff/2ff0bf975160af710b23435481194dd4.png)
, ![](http://img.wesiedu.com/upload/3/91/391dc4ff13306574d212145f5beb3a1d.png)
由此有
, 故猜想
的通项为
.∴
(Ⅱ)令
. 由题设知x1=1且
, ①
② 因②式对n=2成立,有![](http://img.wesiedu.com/upload/0/ca/0ca149b6050dff596edf37f8477e4286.png)
③ 下面用反证法证明: 由①得
. 因此数列
是首项为
,公比为
的等比数列.故
④ 又由①知
因此是
是首项为
,公比为-2的等比数列,所以
⑤ 由④-⑤得
⑥ 对n求和得
⑦ 由题设知
即不等式22k+1<
,对
恒成立.但这是不可能的,矛盾. 因此x2![](http://img.wesiedu.com/upload/1/92/19297cf4ac5f3b217100463edce7069e.png)
,结合③式知
2 =
, 因此
代入⑦式得Sn = 2-
(
), 所以bn =
=
(
) 题型13. 形如
= 0型 例1.(2008年天津理科22)在数列
与
中,
,数列
的前
项和
满足
,
为
与
的等比中项,
. (Ⅰ)求
的值; (Ⅱ)求数列
与
的通项公式; (Ⅲ)设
.证明
. 本小题主要考查等差数列的概念、通项公式及前
项和公式、等比数列的概念、等比中项、不等式证明、数学归纳等基础知识,考查运算能力和推理论证能力及分类讨论的思想方法.满分14分 解:(Ⅰ)由题设有
,
,解得
.由题设又有
,
,解得
. (Ⅱ)解法一:由题设
,
,
,及
,
,进一步可得
,
,
,
, 猜想
,
,
. 先证
,
. 当
时,
,等式成立.当
时用数学归纳法证明如下: (1当
时,
,等式成立. (2)假设
时等式成立,即
,
. 由题设,
1
2 ①的两边分别减去②的两边,整理得
,从而
. 这就是说,当
时等式也成立.根据(1)和(2)可知,等式
对任何的
成立. 综上所述,等式
对任何的
都成立
再用数学归纳法证明
,
. (1)当
时,
,等式成立. (2)假设当
时等式成立,即
,那么
. 这就是说,当
时等式也成立.根据(1)和(2)可知,等式
对任何的
都成立. 解法二:由题设
1
2 ①的两边分别减去②的两边,整理得
,
.所以
,
,……
,
. 将以上各式左右两端分别相乘,得
, 由(Ⅰ)并化简得
,
. 止式对
也成立. 由题设有
,所以
, 即
,
. 令
,则
,即
. 由
得
,
.所以
,即
,
. 解法三:由题设有
,
, 所以
,
,……
,
. 将以上各式左右两端分别相乘,得
,化简得
,
. 由(Ⅰ),上式对
也成立.所以
,
. 上式对
时也成立. 以下同解法二,可得
,
. (Ⅲ)证明:
. 当
,
时,
. 注意到
,
. 当
,
时,
当
,
时,
. 当
,
时,
. 所以
. 从而
时,有
总之,当
时有
,即
.
最终答案:略
解题过程:
同学你好,如对解答还有疑问,可在答案下方的【 添加讨论】中留言,我收到后会尽快给你答复。感谢你的配合! 祝你学习进步,生活愉快 利用递推关系求数列通项的十三种类型 湖北省巴东县第三高级中学 杨开亮(电话:13997763366) 数列是一种特殊的函数,数列的通项公式可以看作项数
![](http://img.wesiedu.com/upload/3/48/34847cdf25f3511e001e2fdcaebbc471.png)
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![](http://img.wesiedu.com/upload/5/bf/5bf8f728ec30189d4735a578c74a9316.png)
![](http://img.wesiedu.com/upload/d/bb/dbba6a5d858a0470864687f272eb8aa8.png)
![](http://img.wesiedu.com/upload/c/1c/c1c4fb89fa7d04a726d3198222f29de2.png)
![](http://img.wesiedu.com/upload/5/bf/5bf8f728ec30189d4735a578c74a9316.png)
![](http://img.wesiedu.com/upload/6/1e/61efc39231287ad4aa3dde77f84381ac.png)
![](http://img.wesiedu.com/upload/a/b4/ab490e0499523778520c3a7a995028da.png)
![](http://img.wesiedu.com/upload/5/bf/5bf8f728ec30189d4735a578c74a9316.png)
![](http://img.wesiedu.com/upload/6/f0/6f02f6a6cd279ac25e278cf88dbe6da7.png)
![](http://img.wesiedu.com/upload/e/2e/e2e71840bad1baa17396414fa6a6539d.png)
![](http://img.wesiedu.com/upload/a/57/a57b76dece4919ce15353160ea96c4ac.png)
![](http://img.wesiedu.com/upload/0/b9/0b95d2bdcdcd75e8cb226470916e6e6d.png)
![](http://img.wesiedu.com/upload/a/57/a57b76dece4919ce15353160ea96c4ac.png)
![](http://img.wesiedu.com/upload/5/6e/56ed100797bee2d6417068765b6feda2.png)
![](http://img.wesiedu.com/upload/4/5f/45fe0580bd3b704781158947983e849a.png)
最终答案:略