题目内容
下列图形⑴,⑵,⑶,⑷分别包含有1个,5个,13个,25个互不重叠的边长为1的小正方形,按同样的方式构造后面的图形,则第20个图形所包含的边长为1的小正方形的个数是 ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082320514796211911.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082320514796211911.jpg)
761
由图形可知,(2)中的图形是在(1)中图形基础上增加4个正方形得到,(3)中的图形是在(2)中的图形基础上增加8个正方形得到,(4)中的图形是在(3)中的图形基础上增加12个正方形得到,设第
个图形所包含的正方形个数为
,则有上述规律可得
,故有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148024764.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148071837.png)
……
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148086628.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148118508.png)
上述
个等式相加可得,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232051481491077.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148180657.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232051481961052.png)
所以第20个图形所包含的边长为1的小正方形的个数是
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205147993648.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148008348.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148024764.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148024764.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148071837.png)
……
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148086628.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148118508.png)
上述
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148133353.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232051481491077.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148180657.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232051481961052.png)
所以第20个图形所包含的边长为1的小正方形的个数是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823205148211875.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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