設(shè){an}是由正數(shù)組成的等比數(shù)列,公比q=2,且a1•a2•a3•…•a30=230,那么a3•a6•a9•…•a30等于( )
A.210
B.220
C.216
D.215
【答案】
分析:由等比數(shù)列的通項(xiàng)公式,可得a
1•a
2•a
3=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/0.png)
)
3,同理a
4•a
5•a
6=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/1.png)
)
3,…,a
28•a
29•a
30=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/2.png)
)
3,故原式a
1•a
2•a
3•…•a
30=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/3.png)
)
3=2
30,將q=2代入,即可求出a
3•a
6•a
9•…•a
30的值.
解答:解:∵a
1•a
2•a
3=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/4.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/5.png)
•a
3=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/6.png)
)
3,a
4•a
5•a
6=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/7.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/8.png)
•a
6=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/9.png)
)
3,…,a
28•a
29•a
30=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/10.png)
)
3,
∴a
1•a
2•a
3…a
30=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/11.png)
)
3•(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/12.png)
)
3…(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/13.png)
)
3=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213640879067887/SYS201310232136408790678001_DA/14.png)
)
3=2
30,
又∵q=2,
∴a
3•a
6•a
9••a
30=2
20.
故選B.
點(diǎn)評:本題考查了等比數(shù)列的通項(xiàng)公式,找出已知a
1•a
2•a
3•…•a
30和未知a
3•a
6•a
9•…•a
30的關(guān)系是解題的關(guān)鍵.