1 . 在平面直角坐标系中,动点由坐标原点出发,先向右移动1个单位达到点
,然后沿原方向逆时针旋转90°的方向,移动
个单位达到点
,若照此无限继续下去,每次都沿逆时针方向旋转90°,移动上次移动距离的一半,求此动点的极限位置.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e92b8041e98e4f435acdbeb983efbe46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a9e83a7565752489c9aa4f20b3ad73f.png)
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解题方法
2 . 无限循环小数可以通过等比数列法转化为分数.如![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a608bf95339a27cf4366bcc5d99acac6.png)
;应用上述方法转化
(
,
为互质整数),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2236f581b11a0bb63addd062529acee8.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a608bf95339a27cf4366bcc5d99acac6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a619c1d5651e32a70cfdeb205fcb56d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c26c147133b19c8be6a1895764ac4fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2236f581b11a0bb63addd062529acee8.png)
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3 . 若一个等比数列
有无穷多项,并且它的公比
满足
,称
为无穷递缩等比数列,规定:无穷递缩等比数列
,
,
,…,
,…所有项的和
,
.《庄子·天下篇》中写道“一尺之棰,日取其半,万世不竭”,其中隐含了关系:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeacc72725cf111c7e2ff46e722866de.png)
__________ ,类似可以将一个无限循环小数表示为分数:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40518320bea37d0b8ec1f6b9d55cb393.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89589a95648bf664f093a995ced97cfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b2e6db493ca4e8efd9722ee21125689.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6de975efced10e4d380b272f0122bd2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03b3cf0a2d049745b7d405a5c6a5118d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/881231f6c7cadb33580954af32fa1cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeacc72725cf111c7e2ff46e722866de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40518320bea37d0b8ec1f6b9d55cb393.png)
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