已知數(shù)列{a
n}的首項(xiàng)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_ST/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_ST/1.png)
,n=1,2,….
(Ⅰ)求{a
n}的通項(xiàng)公式;
(Ⅱ)證明:對(duì)任意的x>0,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_ST/2.png)
,n=1,2,…;
(Ⅲ)證明:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_ST/3.png)
.
【答案】
分析:(Ⅰ)由題設(shè)條件知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/0.png)
,再由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/1.png)
,知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/2.png)
是以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/3.png)
為首項(xiàng),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/4.png)
為公比的等比數(shù)列.由此可知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/5.png)
.
(Ⅱ)由題意知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/6.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/7.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/8.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/9.png)
≤a
n,所以對(duì)任意的x>0,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/10.png)
,n=1,2,….
(Ⅲ)由題意知,對(duì)任意的x>0,有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/11.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/12.png)
.由此入手能夠求出
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/13.png)
.
解答:解:(Ⅰ)∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/14.png)
,∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/15.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/16.png)
,
又
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/17.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/18.png)
是以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/19.png)
為首項(xiàng),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/20.png)
為公比的等比數(shù)列.
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/21.png)
,∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/22.png)
.
(Ⅱ)由(Ⅰ)知
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/23.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/24.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/25.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/26.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/27.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/28.png)
≤a
n,
∴原不等式成立.
(Ⅲ)由(Ⅱ)知,對(duì)任意的x>0,有
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/29.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/30.png)
.
∴取
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/31.png)
,
則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024182105713914843/SYS201310241821057139148020_DA/32.png)
.∴原不等式成立.
點(diǎn)評(píng):本題考查數(shù)列的性質(zhì)和應(yīng)用,難度較大,解題時(shí)要注意挖掘題設(shè)中的隱含條件.
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