【答案】
分析:(1)化簡(jiǎn)數(shù)列的關(guān)系式為:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/0.png)
,通過等差數(shù)列{b
n}恰好使數(shù)列{a
n+b
n}成公比為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/1.png)
的等比數(shù)列,直接得到通項(xiàng)b
n.
(2)結(jié)合(1)求出數(shù)列{a
n-3n+2}的通項(xiàng)a
n-3n+2的表達(dá)式,然后解出通項(xiàng)a
n.
(3)先求出數(shù)列的前n項(xiàng)和,然后利用數(shù)列的極限的運(yùn)算法則直接求
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/2.png)
的值.
解答:解:(1)因?yàn)?img src="http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/3.png">,
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/4.png)
,
所以數(shù)列{a
n-3n+2}以1為首項(xiàng),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/5.png)
為公比的等比數(shù)列,
所以b
n=-3n+2時(shí),等差數(shù)列{b
n}恰好使數(shù)列{a
n+b
n}成公比為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/6.png)
的等比數(shù)列.
(2)由(1)可知數(shù)列{a
n-3n+2}以1為首項(xiàng),
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/7.png)
為公比的等比數(shù)列,
所以a
n-3n+2=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/8.png)
)
n-1,所以a
n=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/9.png)
)
n-1+3n-2
(3)由(2)可知,數(shù)列{a
n}的前n項(xiàng)和為:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/10.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/11.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/12.png)
;
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/13.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/14.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024183708304765594/SYS201310241837083047655018_DA/15.png)
.
點(diǎn)評(píng):本題是中檔題,考查數(shù)列的遞推關(guān)系式,數(shù)列通項(xiàng)公式的求法,數(shù)列前n項(xiàng)和的求法以及數(shù)列的極限的運(yùn)算法則,考查計(jì)算能力,轉(zhuǎn)化思想.