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1 . 斐波那契数列又称兔子数列.1202年,27岁的意大利数学家斐波那契在《算盘书》中从兔子问题得到了斐波那契数列
:1,1,2,3,5,8,13,….斐波那契数列满足
.斐波那契数列也被称为黄金数列,因为随着项数的增加,每一项与前一项的比值会越来越逼近黄金分割的数值
.以斐波那契数列的项为半径依次画四分之一扇形,可以画出斐波那契螺旋线,也成为黄金螺旋线.更有趣的是这样一个完全由自然数构成的数列,其通项公式是用无理数来表示的,其通项公式为
.关于斐波那契数列
,下列说法正确的个数为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/0b43a1c1-8699-470b-9390-1f1ee16f7f1b.png?resizew=196)
①![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ef029d76f18f68c46eeb231e161ea43.png)
②斐波那契数列是递增数列
③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ae7469dc72aef34cf090f3d555382f.png)
④![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33cbac90de5405d13afaa3b56bec0eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/0b43a1c1-8699-470b-9390-1f1ee16f7f1b.png?resizew=196)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ef029d76f18f68c46eeb231e161ea43.png)
②斐波那契数列是递增数列
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ae7469dc72aef34cf090f3d555382f.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33cbac90de5405d13afaa3b56bec0eb5.png)
A.1 | B.2 | C.3 | D.4 |
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2卷引用:北京市北京大学附属中学2021-2022学年高二上学期期中数学试题
2 . 数学中有各式各样富含诗意的曲线,螺旋线就是其中比较特别的一类.螺旋线这个名词来源于希腊文,它的原意是“旋卷”或“缠卷”小明对螺旋线有着浓厚的兴趣,用以下方法画出了如图所示的螺旋线.具体作法是:先作边长为1的正三角形
,分别记射线
,
,
为
,
,
,以
为圆心、
为半径作劣弧
交
于点
;以
为圆心、
为半径作劣弧
交
于点
;以
为圆心、
为半径作劣弧
交
于点
,…,依此规律作下去,就得到了一系列圆弧形成的螺旋线.记劣弧
的长,劣弧
的长,劣弧
的长,…依次为
,
,
…,则
( )
![](https://img.xkw.com/dksih/QBM/2021/4/10/2696703180005376/2696742320873472/STEM/1126a47a85b944a3a66dcc413087f1f3.png?resizew=233)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8772aa893a9c1d40f714cb25701701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cabc3303519ac16fc998913ad9f349c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
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![](https://img.xkw.com/dksih/QBM/2021/4/10/2696703180005376/2696742320873472/STEM/1126a47a85b944a3a66dcc413087f1f3.png?resizew=233)
A.![]() | B.![]() | C.![]() | D.![]() |
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3 . 数学中有各式各样富含诗意的曲线,螺旋线就是其中比较特别的一类.螺旋线这个名词来源于希腊文,它的原意是“旋卷”或“缠卷”.小明对螺旋线有着浓厚的兴趣,用以下方法画出了如图所示的螺旋线.具体作法是:先作边长为1的正三角形ABC,分别记射线AC,BA,CB为l1,l2,l3,以C为圆心、CB为半径作劣弧BC1交l1于点C1;以A为圆心、AC1为半径作劣弧C1A1交l2于点A;以B为圆心、BA1为半径作劣弧A1B1交l3于点B1,…,依此规律作下去,就得到了一系列圆弧形成的螺旋线.记劣弧BC1的长,劣弧C1A1的长,劣弧A1B1的长,…依次为
则
( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/8d44d059-e42a-4c8c-aade-dfbbed6d4600.png?resizew=189)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/500443714512f3293c268ddd0c682c7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a004bc7c90ab090e76a37d3a4c58fb3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/26/8d44d059-e42a-4c8c-aade-dfbbed6d4600.png?resizew=189)
A.30π | B.45π | C.60π | D.65π |
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5卷引用:河南省新乡市部分高中联考2020-2021学年高三下学期理科数学试题
名校
4 . 黄金螺旋线又名鹦鹉螺曲线,是自然界最美的鬼斧神工.就是在一个黄金矩形(宽除以长约等于0.6的矩形)先以宽为边长做一个正方形,然后再在剩下的矩形里面再以其中的宽为边长做一个正方形,以此循环做下去,最后在所形成的每个正方形里面画出1/4圆,把圆弧线顺序连接,得到的这条弧线就是“黄金螺旋曲线了.著名的“蒙娜丽莎”便是符合这个比例,现把每一段黄金螺旋线与其每段所在的正方形所围成的扇形面积设为
,每扇形
的半径设为
满足
,若将
的每一项按照上图方法放进格子里,每一小格子的边长为1,记前
项所占的对应正方形格子的面积之和为
,则下列结论错误的是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67bfb31d3a254fee68fd4dbe986f6fa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e8dc461847c60ee9cc8f476ec56c527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3062d705a57706d9321f42b8f64619d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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5卷引用:专题3.1 复杂数列的通项公式求解问题-玩转压轴题,进军满分之2021高考数学选择题填空题
(已下线)专题3.1 复杂数列的通项公式求解问题-玩转压轴题,进军满分之2021高考数学选择题填空题(已下线)专题10 数列(难点)-2020-2021学年高二数学下学期期末专项复习(北师大版2019选择性必修第一册、第二册)山西省临汾市第一中学2018-2019学年高二下学期期末数学(理)试题安徽省池州市第八中学2020-2021学年高三上学期12月月考理科数学试题(已下线)【讲】专题9 与图表有关的数列问题