1 . (1)问题发现
如下图,△ACB和△DCE均为等边三角形,点A、D、E在同一直线上,连接BE.
填空:①∠AEB的度数为____________;
②线段AD、BE之间的数量关系是_________.
![](https://img.xkw.com/dksih/QBM/2018/9/3/2024255076147200/2040158702346240/STEM/fb5569da4f894cdb8803347e3f1bec07.png?resizew=147)
(2)拓展探究
如下图,△ACB和△DCE均为等腰直角三角形,∠ACB=∠DCE=900, 点A、D、E在同一直线上,CM为△DCE中DE边上的高,连接BE.请判断∠AEB的度数及线段CM、AE、BE之间的数量关系,并说明理由.
![](https://img.xkw.com/dksih/QBM/2018/9/3/2024255076147200/null/STEM/55cc05879d52481fbaf1d6071d2e889a.png?resizew=168)
(3)解决问题
如下图,在正方形ABCD中,CD=
.若点P满足PD=1,且∠BPD=900,请直接写出点A到BP的距离.
如下图,△ACB和△DCE均为等边三角形,点A、D、E在同一直线上,连接BE.
填空:①∠AEB的度数为____________;
②线段AD、BE之间的数量关系是_________.
![](https://img.xkw.com/dksih/QBM/2018/9/3/2024255076147200/2040158702346240/STEM/fb5569da4f894cdb8803347e3f1bec07.png?resizew=147)
(2)拓展探究
如下图,△ACB和△DCE均为等腰直角三角形,∠ACB=∠DCE=900, 点A、D、E在同一直线上,CM为△DCE中DE边上的高,连接BE.请判断∠AEB的度数及线段CM、AE、BE之间的数量关系,并说明理由.
![](https://img.xkw.com/dksih/QBM/2018/9/3/2024255076147200/null/STEM/55cc05879d52481fbaf1d6071d2e889a.png?resizew=168)
(3)解决问题
如下图,在正方形ABCD中,CD=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e051d14fd6a787387995331f5e6d026.png)
![](https://img.xkw.com/dksih/QBM/2018/9/3/2024255076147200/2040158702346240/STEM/d9849043f987404cbe555c44293bfc23.png?resizew=134)
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11-12高三上·江苏泰州·期中
2 . (Ⅰ)试比较
的大小;
(Ⅱ)试比较nn+1与(n+1)n(n∈N+)的大小,根据(Ⅰ)的结果猜测一个一般性结论,并加以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/badb98545fb91dbd88cb37bcba9744a9.png)
(Ⅱ)试比较nn+1与(n+1)n(n∈N+)的大小,根据(Ⅰ)的结果猜测一个一般性结论,并加以证明.
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3 . 记
(
)表示从
起连续
个正整数的和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6791ebc81c5fc5e06e06284d034b5cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b04c7ba0ffd54e60b2829f4440c91ec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c884ce1e9436d39f34f6d3116cb2a140.png)
(1)则
(2)将写成
的形式是
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4 . 古希腊毕达哥拉斯学派的数学家研究过各种多边形数.如三角形数1,3,6,10,…,第
个三角形数为
.记第
个
边形数为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/818ed1b032c481e08f713c7b0dbd65d6.png)
,以下列出了部分
边形数中第
个数的表达式:
三角形数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7db6b941b3ec97fa4e43dd503dc0f60e.png)
正方形数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24af989345fe45f4f3bc8394f64d58b5.png)
五边形数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c352d9cd4eff69ba46e752313e4091d.png)
六边形数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5abe0f1b4b08a0b29e2048e66d95d2.png)
……
可以推测
的表达式,由此计算![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e154bf3322578b064341ece695d9ec3e.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/635f7984672c31bef79c9aab577204cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/818ed1b032c481e08f713c7b0dbd65d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c0902ac3138d1276afdcababfa04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
三角形数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7db6b941b3ec97fa4e43dd503dc0f60e.png)
正方形数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24af989345fe45f4f3bc8394f64d58b5.png)
五边形数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c352d9cd4eff69ba46e752313e4091d.png)
六边形数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5abe0f1b4b08a0b29e2048e66d95d2.png)
……
可以推测
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/818ed1b032c481e08f713c7b0dbd65d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e154bf3322578b064341ece695d9ec3e.png)
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