解:由题
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160521786428.gif)
…….. ①
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522223249.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522238421.gif)
,………②……………………………………………………1f
两式相减,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522254611.gif)
……………………………………………….......3f
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522285815.gif)
………………………………………….4f
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160521708267.gif)
是从第二项起是以公比为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522394223.gif)
的等比数列…………………………..……..5f
通项公式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522176868.gif)
…………………………………………………..7f
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231605225031171.gif)
…………………….…….8f
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522535716.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522550627.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316052261373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522628671.gif)
…………………………………………….....10f
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522644791.gif)
……………………………………….…….13f
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522675265.gif)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522691755.gif)
……………………………………………….…..14f
(2)解法二、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231605225031171.gif)
……………..…….8f
下用数学归纳法证明
I.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522722234.gif)
,左边
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522753444.gif)
=右边
结论成立………………………………………………………………………….…….9f
Ii . 假设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522769489.gif)
时,结论成立,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160522784640.gif)
……...10f
那么…
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231605228151676.gif)
…………………………………………………………………………………………11f
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231605228311343.gif)
………………………...13f
即n=k+1时不等式成立.
由(i)(ii)得结论对n取正整数成立…………………………………………………....14f