【答案】
分析:(Ⅰ)由題設條件可知a
2=1+2a
1=3,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/0.png)
,a
4=1+2a
2=7,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/1.png)
.
(Ⅱ)由題意知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/2.png)
,又
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/3.png)
,所以b
n+1=2b
n.再由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/4.png)
可知b
n=2
n.
(Ⅲ)對任意的m≥2,k∈N
*,在數(shù)列{a
n}中,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/5.png)
這連續(xù)的2
m項就構成一個等差數(shù)列.再用分析法進行證明.
解答:解:(Ⅰ)因為a
1=1,所以a
2=1+2a
1=3,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/6.png)
,a
4=1+2a
2=7,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/7.png)
(3分)
(Ⅱ)由題意,對于任意的正整數(shù)n,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/8.png)
,
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/9.png)
(4分)
又
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/10.png)
所以b
n+1=2b
n(6分)
又
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/11.png)
(7分)
所以{b
n}是首項為2,公比為2的等比數(shù)列,所以b
n=2
n(8分)
(Ⅲ)存在.事實上,對任意的m≥2,k∈N
*,在數(shù)列{a
n}中,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/12.png)
這連續(xù)的2
m項就構成一個等差數(shù)列(10分)
我們先來證明:
“對任意的n≥2,n∈N
*,k∈(0,2
n-1),k∈N
*,有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/13.png)
”
由(II)得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/14.png)
,所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/15.png)
.
當k為奇數(shù)時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/16.png)
當k為偶數(shù)時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/17.png)
記
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/18.png)
因此要證
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/19.png)
,只需證明
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/20.png)
,
其中k
1∈(0,2
n-2),k
1∈N
*(這是因為若
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/21.png)
,則當
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/22.png)
時,則k一定是奇數(shù),
有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/23.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/24.png)
;
當
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/25.png)
時,則k一定是偶數(shù),有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/26.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/27.png)
)
如此遞推,要證
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/28.png)
,只要證明
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/29.png)
,
其中
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/30.png)
,k
2∈(0,2
n-3),k
2∈N
*如此遞推下去,我們只需證明
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/31.png)
,k
n-2∈(0,2
1),k
n-2∈N
*即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/32.png)
,即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/33.png)
,由(I)可得,
所以對n≥2,n∈N
*,k∈(0,2
n-1),k∈N
*,有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/34.png)
,
對任意的m≥2,m∈N
*,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/35.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/36.png)
,
其中i∈(0,2
m-1),i∈N
*,
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/37.png)
又
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/38.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/39.png)
,所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/40.png)
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/41.png)
這連續(xù)的2
m項,
是首項為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/42.png)
,公差為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/43.png)
的等差數(shù)列(13分)
說明:當m
2>m
1(其中m
1≥2,m
1∈N
*,m
2∈N
*)時,
因為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/44.png)
構成一個項數(shù)為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/45.png)
的等差數(shù)列,
所以從這個數(shù)列中任取連續(xù)的
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/46.png)
項,也是一個項數(shù)為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/47.png)
,公差為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180913611089437/SYS201310241809136110894019_DA/48.png)
的等差數(shù)列.
點評:本題考查數(shù)列性質的綜合應用,具有一定的難度,解題時要認真審題,注意培養(yǎng)計算能力.