题目内容
若正数数列{an}满足![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_ST/0.png)
(1)求Sn;
(2)若
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_ST/1.png)
【答案】分析:(1)令n=1,及an>0,可求a1,由
可得
,即Sn2-Sn-12=1,则可得{Sn2}是以1首项,以1为公差的等差数列,由等差数列的通项可求Sn2,进而可求Sn
(2)由(1)可得
,要判断k≠m是否存在bk=bm,考虑函数
(x≥1)的单调性,结合导数的知识可求
解答:解:(1)令n=1,又an>0,得a1=1.
∵
,即![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/5.png)
∴Sn2-Sn-12=1,S12=a12=1
∴{Sn2}是以1为首项,以1为公差的等差数列
∴Sn2=S12+(n-1)•1=1+n-1=n
∴
.
(2)
,则
考虑函数
(x≥1),则
.
令h(x)=x+1+xlnx(x≥1),则h'(x)=-lnx≤0,∴h(x)在[1,+∞)递减
∵h(1)=2>0,h(2)=3-2ln2>0,h(3)=4-3ln3>0,h(4)=5-4ln4<0
∴x≥4时,h(x)≤h(4)<0,则g'(x)<0,g(x)在[4,+∞)递减;
1≤x≤3时,h(x)≥h(3)>0,则g'(x)>0,g(x)在[1,3]递增.
∴g(1)<g(2)<g(3),g(4)>g(5)>g(6)>…
即lnb1<lnb2<lnb3,lnb4>lnb5>lnb6>…
∴b1<b2<b3,b4>b5>b6>…
∵![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/11.png)
∴b1<b2<b3<b4>b5>b6>…
又b1=1,当n≠1时,bn>1.
∴若存在两项相等,只可能是b2、b3与后面的项相等
又
,∴b2=b8
∵
,∴数列bn中存在唯一相等的两项b2=b8
点评:本题主要考查了利用数列的递推公式求解数列的通项公式,等差数列的通项公式的应用及利用函数的导数判断函数的单调性及数列单调性的应用,属于函数与数列的综合应用的考查.
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/1.png)
(2)由(1)可得
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/3.png)
解答:解:(1)令n=1,又an>0,得a1=1.
∵
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/5.png)
∴Sn2-Sn-12=1,S12=a12=1
∴{Sn2}是以1为首项,以1为公差的等差数列
∴Sn2=S12+(n-1)•1=1+n-1=n
∴
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/6.png)
(2)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/7.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/8.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/9.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/10.png)
令h(x)=x+1+xlnx(x≥1),则h'(x)=-lnx≤0,∴h(x)在[1,+∞)递减
∵h(1)=2>0,h(2)=3-2ln2>0,h(3)=4-3ln3>0,h(4)=5-4ln4<0
∴x≥4时,h(x)≤h(4)<0,则g'(x)<0,g(x)在[4,+∞)递减;
1≤x≤3时,h(x)≥h(3)>0,则g'(x)>0,g(x)在[1,3]递增.
∴g(1)<g(2)<g(3),g(4)>g(5)>g(6)>…
即lnb1<lnb2<lnb3,lnb4>lnb5>lnb6>…
∴b1<b2<b3,b4>b5>b6>…
∵
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/11.png)
∴b1<b2<b3<b4>b5>b6>…
又b1=1,当n≠1时,bn>1.
∴若存在两项相等,只可能是b2、b3与后面的项相等
又
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/12.png)
∵
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131024185151104638100/SYS201310241851511046381064_DA/13.png)
点评:本题主要考查了利用数列的递推公式求解数列的通项公式,等差数列的通项公式的应用及利用函数的导数判断函数的单调性及数列单调性的应用,属于函数与数列的综合应用的考查.
![](http://thumb2018.1010pic.com/images/loading.gif)
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