【答案】
分析:(Ⅰ)根据题意,有a
1=C
2m+33m•A
m-21,由二项式系数的性质,可得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/0.png)
,解可得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/1.png)
;即m=3,写出
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/2.png)
的展开式中的通项的第二项,即可得公比;
(Ⅱ)由(Ⅰ)的结论,可得a
1与公比,可得等比数列的通项为a
n=x
n-1,分x=1与x≠1两种情况讨论,分别求出S
n,综合可得答案;
(Ⅲ)分x=1与x≠1两种情况讨论,当x=1时,S
n=n,A
n=C
n1+2C
n2+3C
n3+…+nC
nn,倒序相加可得2A
n=n(C
n+C
n1+C
n2+…+C
nn),由二项式定理可得2A
n=n•2
n,化简可得A
n=n•2
n-1,当x≠1时,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/3.png)
,代入可得A
n的表达式,综合可得答案.
解答:解:(Ⅰ)∵a
1=C
2m+33m•A
m-21∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/4.png)
,解可得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/5.png)
;
∴m=3,
由
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/6.png)
的展开式中的通项公式知q=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/7.png)
,
(Ⅱ)由(Ⅰ)可得,a
1=C
2m+33m•A
m-21=C
66•A
11=1,其公比为x,
则a
n=x
n-1,
当x=1时,a
n=1,S
n=1+1+…+1=n,
当x≠1时,S
n=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/8.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/9.png)
,
则
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/10.png)
;
(Ⅲ)当x=1时,S
n=n,A
n=C
n1+2C
n2+3C
n3+…+nC
nn=0C
n+C
n1+2C
n2+3C
n3+…+nC
nn①
又∵A
n=nC
nn+(n-1)C
nn-1+(n-2)C
nn-2+…+C
n1+0C
n,②
①+②可得:2A
n=n(C
n+C
n1+C
n2+…+C
nn)=n•2
n,
∴A
n=n•2
n-1,
当x≠1时,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/11.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/12.png)
则
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101230148687821217/SYS201311012301486878212020_DA/13.png)
.
点评:本题考查等比数列的求和、二项式定理的应用;注意对等比数列求和时,讨论公比是否为1.