設(shè)函數(shù)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_ST/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_ST/1.png)
,數(shù)列{a
n}滿
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_ST/2.png)
,則數(shù)列{a
n}的前n項和S
n等于
.
【答案】
分析:首先根據(jù)題干條件求出a
1的值,然后根據(jù)f(1)=n
2•a
n,得到a
1+a
2+a
3+…+a
n=n
2•a
n,最后根據(jù)當(dāng)n≥2時,a
n=S
n-S
n-1=n
2•a
n-(n-1)
2•a
n-1求出數(shù)列{a
n}的通項
解答:解:∵函數(shù)f(x)=a
1+a
2x+a
3x
2+…+a
nx
n-1,
∴f(0)=a
1=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/0.png)
,f(1)=a
+a
1+…+a
n∵f(1)=n
2•a
n,
∴S
n=a
1+a
2+a
3+…+a
n=n
2•a
n,
又∵a
n=S
n-S
n-1=n
2•a
n-(n-1)
2•a
n-1,
∴(n
2-1)a
n=(n-1)
2•a
n-1(n≥2),
則
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/1.png)
利用疊乘可得,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/2.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/3.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/4.png)
×…×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/5.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/6.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/7.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/8.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/9.png)
×…×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/10.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/11.png)
,
∴a
n=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/12.png)
,
故答案為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103102011354123778/SYS201311031020113541237014_DA/13.png)
.
點評:本題主要考查數(shù)列遞推式的應(yīng)用,解答本題的關(guān)鍵是由(n
2-1)a
n=(n-1)
2•a
n-1,利用疊乘法求解通項公式,此題難度一般.
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