【答案】
分析:(1)用解方程或根與系數(shù)的關(guān)系表示a
2k-1,a
2k,k賦值即可.
(2)由S
2n=(a
1+a
2)+…+(a
2n-1+a
2n)可分組求和.
(3)T
n復(fù)雜,常用放縮法,但較難.
解答:解:(I)解:方程x
2-(3k+2
k)x+3k•2
k=0的兩個(gè)根為x
1=3k,x
2=2
k,
當(dāng)k=1時(shí),x
1=3,x
2=2,所以a
1=2;
當(dāng)k=2時(shí),x
1=6,x
2=4,所以a
3=4;
當(dāng)k=3時(shí),x
1=9,x
2=8,所以a
5=8時(shí);
當(dāng)k=4時(shí),x
1=12,x
2=16,所以a
7=12.
(II)解:S
2n=a
1+a
2++a
2n=(3+6++3n)+(2+2
2++2
n)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/0.png)
.
(III)證明:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/1.png)
,
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/2.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/3.png)
.當(dāng)n≥3時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/4.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/5.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/7.png)
,
同時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/8.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/9.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/10.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/11.png)
.
綜上,當(dāng)n∈N*時(shí),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214519055849662/SYS201310232145190558496020_DA/12.png)
.
點(diǎn)評(píng):本題主要考查等差、等比數(shù)列的基本知識(shí),考查運(yùn)算及推理能力.本題屬難題,一般要求做(1),(2)即可,讓學(xué)生掌握常見(jiàn)方法,對(duì)(3)不做要求.