题目内容
已知数列
满足![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527395836.gif)
(1) 求数列
的通项公式;
(2) 设b
=
(n∈N
,n≥2), b
,求证:b
+b
+……+b
< 3 .
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527380267.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527395836.gif)
(1) 求数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527380267.gif)
(2) 设b
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527427178.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527442836.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527551124.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527739167.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527754127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527770129.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527427178.gif)
(1);(2)同解析;
(1)∵
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527817576.gif)
∴数列{
}是以首项a1+1,公比为2的等比数列,
即![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527848564.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527863566.gif)
(2)b
=
=
=
≤
(n≥2)
∴b
+b
+……+b
=1+![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134528004229.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134528019314.gif)
n=1时,b
="1<3" 成立, 所以b
+b
+……+b
< 3
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527801759.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527817576.gif)
∴数列{
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527832370.gif)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527848564.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527863566.gif)
(2)b
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527427178.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527442836.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527910347.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527941645.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527957474.gif)
∴b
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527754127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527770129.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527427178.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134528004229.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134528019314.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231345280351906.gif)
n=1时,b
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527754127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527754127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527770129.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823134527427178.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目