已知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918297481.png)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918313297.png)
項和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918329416.png)
記
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329183601293.png)
(1)若數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918297481.png)
是首項與公差均為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918375208.png)
的等差數(shù)列,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918391622.png)
;
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918407562.png)
且數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918422719.png)
均是公比為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918422248.png)
的等比數(shù)列,
求證:對任意正整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918313297.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918453588.png)
試題分析:(1)根據(jù)等差數(shù)列的通項公式和前n項和公式,求出a
n,S
n,然后代入f(n)中,整理即可求解.
(2)根據(jù)等比數(shù)列的通項公式求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918469517.png)
的表達(dá)式,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918485347.png)
,再求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918500531.png)
,代入f(n)中,整理得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918500983.png)
,然后證
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918500983.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918531241.png)
0即可.
試題解析:(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918547235.png)
數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918297481.png)
是首項與公差均為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918375208.png)
的等差數(shù)列, 1分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918594195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918594609.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329186091121.png)
3分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329186251054.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329186411402.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918656760.png)
5分
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918672690.png)
6分
(2)由題意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918594609.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918703797.png)
7分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918719769.png)
8分
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918734509.png)
9分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918594609.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918765990.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329186251054.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329187811486.png)
10分
(證法一)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918797341.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918812498.png)
; 11分
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918828425.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329188431171.png)
, 12分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329188592174.png)
13分
故對任意正整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918313297.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918890577.png)
14分
(證法二)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918594609.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918906650.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329189211652.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329189371208.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918968911.png)
11分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329189841017.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240329189842467.png)
,
數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032918999575.png)
是遞增數(shù)列. 12分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032919031921.png)
13分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824032919046895.png)
14分
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