题目内容
用a,b,c,d四个不同字母组成一个含n+1(n∈N+)个字母的字符串,要求由a开始,相邻两个字母不同.例如n=1时,排出的字符串是ab,ac,ad;n=2时排出的字符串是aba,abc,abd,aca,acb,acd,ada,adb,adc,…,如图所示.记这含n+1个字母的所有字符串中,排在最后一个的字母仍是a的字符串的种数为an.(1)试用数学归纳法证明:
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_ST/0.png)
(2)现从a,b,c,d四个字母组成的含n+1(n∈N*,n≥2)个字母的所有字符串中随机抽取一个字符串,字符串最后一个的字母恰好是a的概率为P,求证:
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_ST/1.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_ST/images2.png)
【答案】分析:(1)根据题意,易得n=1时,等式成立,进而假设设n=k时,等式正确,再分析n=k+1时的等式与n=k的等式之间的关系,验证n=k+1时等式仍成立;综合可得证明;
(2)根据题意,易得易知
,分①当n为奇数(n≥3)与②当n为偶数(n≥2)两种情况,分别求得P,综合可得证明.
解答:(1)证明:
(ⅰ)当n=1时,因为a1=0,
,所以等式正确.
(ⅱ)假设n=k时,等式正确,即
,
那么,n=k+1时,因为
,
这说明n=k+1时等式仍正确.
据(ⅰ),(ⅱ)可知,
正确;
(2)解:易知
,
①当n为奇数(n≥3)时,
,
因为3n≥27,所以
,又
,所以
;
②当n为偶数(n≥2)时,
,
因为3n≥9,所以
,又
,所以
.
综上所述,
.
点评:本题考查数学归纳法的运用,注意数学归纳法的步骤,2个步骤必须完整、严密,第二步尤其重要,否则将会影响解题的严密性,甚至得到错误的结论.
(2)根据题意,易得易知
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/0.png)
解答:(1)证明:
(ⅰ)当n=1时,因为a1=0,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/1.png)
(ⅱ)假设n=k时,等式正确,即
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/2.png)
那么,n=k+1时,因为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/3.png)
这说明n=k+1时等式仍正确.
据(ⅰ),(ⅱ)可知,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/4.png)
(2)解:易知
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/5.png)
①当n为奇数(n≥3)时,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/6.png)
因为3n≥27,所以
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/7.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/8.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/9.png)
②当n为偶数(n≥2)时,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/10.png)
因为3n≥9,所以
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/11.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/12.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/13.png)
综上所述,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024180942905708476/SYS201310241809429057084022_DA/14.png)
点评:本题考查数学归纳法的运用,注意数学归纳法的步骤,2个步骤必须完整、严密,第二步尤其重要,否则将会影响解题的严密性,甚至得到错误的结论.
![](http://thumb2018.1010pic.com/images/loading.gif)
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