已知數(shù)列{an},a1=2a+1(a≠-1的常數(shù)),an=2an-1+n2-4n+2(n≥2,n∈N?),數(shù)列{bn}的首項,b1=a,bn=an+n2(n≥2,n∈N?).
(1)證明:{bn}從第2項起是以2為公比的等比數(shù)列并求{bn}通項公式;
(2)設(shè)Sn為數(shù)列{bn}的前n項和,且{Sn}是等比數(shù)列,求實數(shù)a的值;
(3)當(dāng)a>0時,求數(shù)列{an}的最小項.
【答案】
分析:(1)由題意可得,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/0.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/1.png)
(n≥2)及b
2=a
2+4=4a+4,可證{b
n}從第2項起的等比數(shù)列,結(jié)合等比數(shù)列的通項公式可求;
(2)由(1)可求S
n,結(jié)合{S
n}是等比數(shù)列,及等比數(shù)列的特點可求a;
(3)由n≥2時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/2.png)
,可求a
n=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/3.png)
,可得數(shù)列{a
n}的項為2a+1,4a,8a-1,16a,32a+7,顯然最小項是前三項中的一項,結(jié)合a的范圍可求最小項.
解答:解:由題意可得,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/5.png)
(n≥2)
b
2=a
2+4=4a+4,
∵a≠-1,b
2≠0,即{b
n}從第2項起是以2為公比的等比數(shù)列
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/6.png)
=(a+1)•2
n(n≥2)
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/7.png)
(2)由(1)求得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/8.png)
∵{S
n}是等比數(shù)列,
∴3a+4=0,即
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/9.png)
.
(3)由已知當(dāng)n≥2時,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/10.png)
,
∴a
n=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/11.png)
所以數(shù)列{a
n}為2a+1,4a,8a-1,16a,32a+7,顯然最小項是前三項中的一項.
當(dāng)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/12.png)
時,最小項為8a-1;
當(dāng)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/13.png)
時,最小項為4a;
當(dāng)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/14.png)
時,最小項為2a+1.
當(dāng)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/15.png)
時,最小項為4a或8a-1
當(dāng)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024191419544384537/SYS201310241914195443845019_DA/16.png)
時,最小項為4a或2a+1;
點評:本題主要考查了等比數(shù)列的定義在數(shù)列中應(yīng)用,數(shù)列的遞推公式在數(shù)列的通項求解中的應(yīng)用,屬于數(shù)列知識的綜合應(yīng)用