【答案】
分析:(1)設(shè)等差數(shù)列{log
2(a
n-1)}的公差為d.根據(jù)a
1和a
3的值求得d,進(jìn)而根據(jù)等差數(shù)列的通項(xiàng)公式求得數(shù)列{log
2(a
n-1)}的通項(xiàng)公式,進(jìn)而求得a
n.
(2)把(1)中求得的a
n代入
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/0.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/1.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/2.png)
中,進(jìn)而根據(jù)等比數(shù)列的求和公式求得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/3.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/4.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/5.png)
=1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/6.png)
原式得證.
解答:(I)解:設(shè)等差數(shù)列{log
2(a
n-1)}的公差為d.
由a
1=3,a
3=9得2(log
22+d)=log
22+log
28,即d=1.
所以log
2(a
n-1)=1+(n-1)×1=n,即a
n=2
n+1.
(II)證明:因?yàn)?img src="http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/7.png">=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/8.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/9.png)
,
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/10.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/11.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/12.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/13.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/14.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/15.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/16.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/17.png)
=1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023212744522451521/SYS201310232127445224515015_DA/18.png)
<1,
即得證.
點(diǎn)評(píng):本題主要考查了等差數(shù)列的通項(xiàng)公式.屬基礎(chǔ)題.