递推关系中的数列通项

男娘i 2022-06-07 00:27 144阅读 0赞

http://[blog.csdn.net/pipisorry/article/details/78142983][blog.csdn.net_pipisorry_article_details_78142983]

# 普通方法 #

### 叠加法/叠乘法 ###

![Center][]

### **公式法** ###

![Center 1][]

### 阶差法 ###

![Center 2][]

### **待定系数法** ###

![Center 3][]

### **辅助数列法** ###

![Center 4][]

### **归纳、猜想** ###

![Center 5][]

### 倒数法 ###

![Center 6][]

\[[求解数列通项公式的常用方法][Link 1]\][  
][Link 2]

[某小皮][Link 2]

# 特征方程法 #

## （一阶线性递推式） ##

设已知数列![Center 7][]的项满足![Center 8][],其中![Center 9][]求这个数列的通项公式？

特征方程法：针对问题中的递推关系式作出一个方程![Center 10][]称之为特征方程；借助这个特征方程的根快速求解通项公式.

定理1：设上述递推关系式的特征方程的根为![20131225104537484][]，则当![20131225104739031][]时，![Center 7][]为常数列，即![20131225105842796][]，其中![20131225110113250][]是以![20131225110119796][]为公比的等比数列，即![20131225110124078][].

证明：因为![Center 11][]由特征方程得![20131226141558375][]作换元![20131226141603781][]则![20131226141611078][]

当![20131226141615984][]时，![20131226141621984][]，数列是以![20131225110113250][]为![20131225110119796][]公比的等比数列，故当![20131226141627453][]时，![20131226141634187][]，![20131225110113250][]为0数列，故![20131226142005093][]（证毕）

Note: 本质是就是带有不定项的辅助数列法嘛。

### 示例 ###

例1．已知数列![Center 7][]满足：![20131226142109437][]求![Center 7][]。

解：作方程![20131226142225171][]

当![20131226142316390][]时，![20131226142352812][]

数列![20131225110113250][]是以![20131226142506906][]为公比的等比数列.于是![20131226142608562][]

## （二阶线性递推式） ##

定理2：对于由递推公式![20131226143832062][]，![20131226143940343][]给出的数列![Center 7][]，方程![20131226144039734][]，叫做数列![Center 7][]的特征方程。

若![20131226144352250][]是特征方程的两个根，当![20131226144426906][]时，数列![Center 7][]的通项为![20131226144520015][]，其中A，B由![20131226144554671][]决定（即把![20131226144712234][]和![20131226144756531][]，代入![20131226144520015][]，得到关于A、B的方程组）；

当![20131226144909281][]时，数列![Center 7][]的通项为![20131226144959781][]，其中A，B由![20131226144554671][]决定（即把![20131226144712234][]和![20131226144756531][]，代入![20131226144520015][]，得到关于A、B的方程组）。

### 示例    ###

例3：已知数列![Center 7][]满足![20131226145122390][]，求数列![Center 7][]的通项公式。

解法一（待定系数——迭加法）

由![20131226145218625][]，得![20131226145313812][]，且![20131226145347000][]。

则数列![20131226145406328][]是以为![20131226145446484][]首项，![20131226145537437][]为公比的等比数列，于是![20131226145623359][]。把![20131226145701750][]代入，得

![20131226145730968][]，

![20131226145803406][]，

……

![20131226145916968][]

把以上各式相加，得

![20131226150012906][]

![20131226150043750][]。

解法二（特征根法）：

数列![Center 7][]：![20131226150842406][] 的特征方程是：![20131226150412500][]。

,![20131226150458546][]

又由![20131226150842406][]，于是

![20131226150635312][]

故![20131226150717484][]

## (分式递推式) ##

定理3：如果数列![Center 7][]满足下列条件：已知![20131226142936156][]的值且对于![Center 12][]，都有![Center 13][]（其中p、q、r、h均为常数，且![Center 14][]），那么![20131226151614140][]，可作特征方程.

（1）当特征方程有两个相同的根![20131226151650218][]（称作特征根）时，

若![20131226151720234][]则![20131226151739265][]

若![20131226151803343][]，则![20131226151835671][]其中![20131226151915781][]

特别地，当存在![20131226151946828][]使b\_n0 = 0时，无穷数列![Center 7][]不存在.

（2）当特征方程有两个相异的根![20131226152320046][]、![20131226152323843][]（称作特征根）时，则，

![20131226152328765][]![20131226152334656][]其中![20131226152338484][]

### 示例    ###

例3、已知数列![Center 7][]满足性质：对于![20131226152651031][]且![20131226152654250][]![20131226152840765][]![20131226152843625][]求![Center 7][]的通项公式.

解:依定理作特征方程![20131226152834062][]变形得其根为故特征方程有两个相异的根，使用定理2的第（2）部分，则有

![Center 15][]∴![Center 16][]

∴![Center 17][]即![20131229113700062][]

\[[特征方程法求递推数列的通项公式][Link 3]\]

from: [http://blog.csdn.net/pipisorry/article/details/78142983][blog.csdn.net_pipisorry_article_details_78142983]

ref:

[blog.csdn.net_pipisorry_article_details_78142983]: http://blog.csdn.net/pipisorry/article/details/78142983
[Center]: /images/20220607/6768285f75e74775a81a0c9c878f4ce6.png
[Center 1]: /images/20220607/7cd349b151174be4a2afd1c367f68e71.png
[Center 2]: /images/20220607/c9b3be11925643da897f018bd4f46129.png
[Center 3]: /images/20220607/2c52dce224014cef9886228656ca8c55.png
[Center 4]: /images/20220607/abb16a72100e41dcafec2a34aa9b85f1.png
[Center 5]: /images/20220607/275a7cebd4bf46b2bfcca600312d3802.png
[Center 6]: /images/20220607/4dbf9e2109dd40e9b7618cd66f8530bc.png
[Link 1]: https://wenku.baidu.com/view/b87915f69e3143323968930d.html?mark_pay_doc=0&mark_rec_page=1&mark_rec_position=4&clear_uda_param=1
[Link 2]: http://my.csdn.net/
[Center 7]: /images/20220607/04d31f2b184d401196523bb675bcd2ce.png
[Center 8]: https://img-blog.csdn.net/20131225103850734?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[Center 9]: /images/20220607/c6368fb7511f4273b7411d68035cdb23.png
[Center 10]: /images/20220607/bd224145cbc84d789dfa95ca00a6adc9.png
[20131225104537484]: https://img-blog.csdn.net/20131225104537484
[20131225104739031]: https://img-blog.csdn.net/20131225104739031
[20131225105842796]: https://img-blog.csdn.net/20131225105842796
[20131225110113250]: https://img-blog.csdn.net/20131225110113250
[20131225110119796]: https://img-blog.csdn.net/20131225110119796
[20131225110124078]: /images/20220607/e3071b9a4c264c76b4ffb2597046c53d.png
[Center 11]: https://img-blog.csdn.net/20131226141319953?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[20131226141558375]: https://img-blog.csdn.net/20131226141558375
[20131226141603781]: https://img-blog.csdn.net/20131226141603781
[20131226141611078]: /images/20220607/b69f480d11ba4615926598c401bef072.png
[20131226141615984]: https://img-blog.csdn.net/20131226141615984
[20131226141621984]: https://img-blog.csdn.net/20131226141621984
[20131226141627453]: https://img-blog.csdn.net/20131226141627453
[20131226141634187]: https://img-blog.csdn.net/20131226141634187
[20131226142005093]: /images/20220607/201139a1696e43ee80a080c61143c73d.png
[20131226142109437]: https://img-blog.csdn.net/20131226142109437
[20131226142225171]: /images/20220607/66826506216741ccab5e95b9858bb038.png
[20131226142316390]: https://img-blog.csdn.net/20131226142316390
[20131226142352812]: /images/20220607/5c3f3acdb3774db5b5f2500774fc4bed.png
[20131226142506906]: https://img-blog.csdn.net/20131226142506906
[20131226142608562]: /images/20220607/b4a6cbfb00844360ba9daab4ffe5d75c.png
[20131226143832062]: https://img-blog.csdn.net/20131226143832062
[20131226143940343]: https://img-blog.csdn.net/20131226143940343
[20131226144039734]: https://img-blog.csdn.net/20131226144039734
[20131226144352250]: https://img-blog.csdn.net/20131226144352250
[20131226144426906]: https://img-blog.csdn.net/20131226144426906
[20131226144520015]: /images/20220607/1fe67e20bd6a410a8a4f8d4eeb0a2d2d.png
[20131226144554671]: https://img-blog.csdn.net/20131226144554671
[20131226144712234]: https://img-blog.csdn.net/20131226144712234
[20131226144756531]: https://img-blog.csdn.net/20131226144756531
[20131226144909281]: https://img-blog.csdn.net/20131226144909281
[20131226144959781]: https://img-blog.csdn.net/20131226144959781
[20131226145122390]: https://img-blog.csdn.net/20131226145122390
[20131226145218625]: https://img-blog.csdn.net/20131226145218625
[20131226145313812]: https://img-blog.csdn.net/20131226145313812
[20131226145347000]: /images/20220607/ca0f6d2e27a84bc28fe3b1effcd78787.png
[20131226145406328]: https://img-blog.csdn.net/20131226145406328
[20131226145446484]: https://img-blog.csdn.net/20131226145446484
[20131226145537437]: https://img-blog.csdn.net/20131226145537437
[20131226145623359]: https://img-blog.csdn.net/20131226145623359
[20131226145701750]: /images/20220607/e857b2871ef84d26ba5ee10d82fb000a.png
[20131226145730968]: /images/20220607/d48ec2a6e2df48eaa6b5b10c7b079d9b.png
[20131226145803406]: /images/20220607/e63736f88e6240d5b87736cec20307bd.png
[20131226145916968]: /images/20220607/03f44f8a33df4bf38e913d6e99a5fe61.png
[20131226150012906]: /images/20220607/94f84bae86d345a3b89438f2d1420d8d.png
[20131226150043750]: /images/20220607/07ffb2112c8b4c4d8618f48c1077e493.png
[20131226150842406]: /images/20220607/84d4ac044c1941b58bc641bef339e860.png
[20131226150412500]: /images/20220607/196b1b65e4854e1dbeb3364747f7504b.png
[20131226150458546]: /images/20220607/dc68e4926d994fd9ab760229e2144b77.png
[20131226150635312]: /images/20220607/d273a8cf687747fb9323c650d90f8da4.png
[20131226150717484]: /images/20220607/3af2e7090c05451aaf0dd152cbca731b.png
[20131226142936156]: https://img-blog.csdn.net/20131226142936156
[Center 12]: https://img-blog.csdn.net/20131226151356625?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[Center 13]: https://img-blog.csdn.net/20131226151440687?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[Center 14]: https://img-blog.csdn.net/20131226151517187?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[20131226151614140]: /images/20220607/66efee52bddf4b8d8e2a553b0734effd.png
[20131226151650218]: /images/20220607/a34b36542d9043d98a59db395ddbb397.png
[20131226151720234]: https://img-blog.csdn.net/20131226151720234
[20131226151739265]: /images/20220607/079de51740974bc7b2b1111ce703b57d.png
[20131226151803343]: https://img-blog.csdn.net/20131226151803343
[20131226151835671]: https://img-blog.csdn.net/20131226151835671
[20131226151915781]: /images/20220607/9d080b8df641498f8b18d6abd10151b6.png
[20131226151946828]: https://img-blog.csdn.net/20131226151946828
[20131226152320046]: https://img-blog.csdn.net/20131226152320046
[20131226152323843]: /images/20220607/93f63dd2517b4f1e8bb8be9d757c4874.png
[20131226152328765]: https://img-blog.csdn.net/20131226152328765
[20131226152334656]: https://img-blog.csdn.net/20131226152334656
[20131226152338484]: /images/20220607/ed8f8483c3b94b3c8bf6335c9c2a0820.png
[20131226152651031]: https://img-blog.csdn.net/20131226152651031
[20131226152654250]: https://img-blog.csdn.net/20131226152654250
[20131226152840765]: https://img-blog.csdn.net/20131226152840765
[20131226152843625]: https://img-blog.csdn.net/20131226152843625
[20131226152834062]: /images/20220607/d5531b8c0da84d8098e82dfe1a800c64.png
[Center 15]: https://img-blog.csdn.net/20131229113531062?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[Center 16]: /images/20220607/56f0b184e7164d7e800ef4502e56d8ca.png
[Center 17]: https://img-blog.csdn.net/20131229113649500?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvaGFuY2hlbmd4aQ==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[20131229113700062]: /images/20220607/9e1a5696beb84abaa9615722a324a7c9.png
[Link 3]: https://wenku.baidu.com/view/606a05a1b0717fd5360cdc91.html