已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708574481.png)
是正數(shù)組成的數(shù)列,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708590371.png)
,且點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708605943.png)
在函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708621515.png)
的圖象上.
(Ⅰ)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708574481.png)
的通項公式;
(Ⅱ)若數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708652479.png)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708668389.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708683671.png)
,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708699169.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708714624.png)
.
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708730466.png)
;(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708714624.png)
.
試題分析:(Ⅰ)將點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708605943.png)
代入到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708621515.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708792508.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708808504.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708590371.png)
,所以數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708574481.png)
是以1為首項,公差為1的等差數(shù)列.故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708855744.png)
.
(Ⅱ)因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708683671.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708886600.png)
,利用迭加法求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708902537.png)
,再作差比較
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240257089171171.png)
,化簡得出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240257089331119.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708948573.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708964495.png)
,所以得證.
試題解析:(Ⅰ)由已知得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708792508.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708808504.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708590371.png)
,
所以數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708574481.png)
是以1為首項,公差為1的等差數(shù)列.故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708855744.png)
.
(Ⅱ)由(Ⅰ)知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708730466.png)
從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708886600.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240257090891145.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025709104657.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025709120622.png)
.
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240257091361208.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240257091511140.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708948573.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708964495.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824025708714624.png)
.
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