利用图形来表示数量或数量关系,也可以利用数量或数量关系来描述图形特征或图形之间的关系,这种思想方法称为数形结合.我们刚学过的《从面积到乘法公式》就很好地体现了这一思想方法,你能利用数形结合的思想解决下列问题吗?
如图,一个边长为1的正方形,依次取正方形的
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754216662.png)
,根据图示我们可以知道:第一次取走
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754231256.png)
后还剩
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754231256.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754231256.png)
=1﹣
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754231256.png)
;前两次取走
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754231256.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754294255.png)
后还剩
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754294255.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754231256.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754341254.png)
=1﹣
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754341254.png)
;前三次取走
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754372254.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754341254.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754387256.png)
后还剩
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754387256.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754372254.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754341254.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754387256.png)
=1﹣
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754387256.png)
;…前n次取走后,还剩
_________ ,即
_________ =
_________ .
利用上述计算:
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754481764.png)
=
_________ .
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823023754497854.png)
=
_________ .
(3)2﹣2
2﹣2
3﹣2
4﹣2
5﹣2
6﹣…﹣2
2011+2
2012(本题写出解题过程)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230237545121514.png)