名校
1 . 对于数列
,
,…,
,定义变换
,
将数列
变换成数列
,
,…,
,
,记
,
,
.对于数列
,
,…,
与
,
,…,
,定义
.若数列
,
,…,
满足
,则称数列
为
数列.
(1)若
,写出
,并求
;
(2)对于任意给定的正整数
,是否存在
数列
,使得
若存在,写出一个数列
,若不存在,说明理由:
(3)若
数列
满足
,求数列A的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140b9dbcada4ac2e5fe3cc30009bcd67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0feb938cee87cf9157a4a952ff38975.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5544c5129150e22392b5aed8f3cb5ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b4c672fb2e729873a90dea3a16b611d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0e0b2913aa1ce57df5bb9fd5a2d4ee1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac49ab7c8001c209b8611b9ea40d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140b9dbcada4ac2e5fe3cc30009bcd67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f17923637012a75a01f309379c1909c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcadae63fa2ce087a0c4debd022ae7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140b9dbcada4ac2e5fe3cc30009bcd67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0feb938cee87cf9157a4a952ff38975.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/787be362d9efcbea93ae48355093b697.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6f4750cc036bd3dab264a7d9b3c1ab.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5d85aed35cb77a487752e2f08776cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc250de2317c83a904f0ebce5fc2989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb0dd6fc19977ebe444dc4a14a0ff3e5.png)
(2)对于任意给定的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ae0b861522b18be1753acc4474cbc9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6f4750cc036bd3dab264a7d9b3c1ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d43ae10d16ae673584fd2ed30407d1b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6f4750cc036bd3dab264a7d9b3c1ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb701a737654dacb67a0cfe7df10dc1.png)
您最近一年使用:0次
2022-05-05更新
|
1709次组卷
|
8卷引用:2022年新高考北京数学高考真题变式题13-15题
(已下线)2022年新高考北京数学高考真题变式题13-15题(已下线)2022年新高考北京数学高考真题变式题19-21题北京卷专题18数列(解答题)北京市东城区2022届高三二模数学试题北京理工大学附属中学2023届高三上学期10月月考数学试题北京市第三十九中学2022届高三下学期适应性练习(三模)数学试题北京市第一六一中学2023-2024学年高三下学期开学测试数学试卷河南省信阳市新县高级中学2024届高三适应性考试(七)数学试题
解题方法
2 . “角谷猜想”是指一个正整数,如果是奇数就乘以3再加1,如果是偶数就除以2,这样经过若干次这两种运算,最终必进入循环图
.对任意正整数
,按照上述规则实施第
次运算的结果为
,( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a16f78ce0dab1ac8fa6abbd70f2b008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99df67b9c60a70e1543672189d079ea8.png)
A.当![]() ![]() |
B.当![]() ![]() |
C.若![]() ![]() ![]() |
D.当![]() ![]() ![]() |
您最近一年使用:0次
3 . 已知数列
的通项
,如果把数列
的奇数项都去掉,余下的项依次排列构成新数列为
,再把数列
的奇数项又去掉,余下的项依次排列构成新数列为
,如此继续下去,……,那么得到的数列(含原已知数列)的第一项按先后顺序排列,构成的数列记为
,则数列
前10项的和为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e70c380aed49c221eb76c8a3328cfc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c5a325806df1a1c3e7ce609fe99085f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c5a325806df1a1c3e7ce609fe99085f.png)
A.1013 | B.1023 | C.2036 | D.2050 |
您最近一年使用:0次
2023-06-01更新
|
890次组卷
|
6卷引用:专题08 数列
(已下线)专题08 数列(已下线)第六章 数列(测试)(已下线)重难点突破01 数列的综合应用 (十三大题型)-1江西师范大学附属中学2023届高三三模考试数学(理)试题江西师范大学附属中学2023届高三三模考试数学(文)试题天津市滨海新区塘沽第一中学2024届高三上学期第三次月考数学试题
名校
解题方法
4 . 已知
和
是各项均为正整数的无穷数列,若
和
都是递增数列,且
中任意两个不同的项的和不是
中的项,则称
被
屏蔽.已知数列
满足
.
(1)求数列
的通项公式;
(2)若
为首项与公比均为
的等比数列,求数列
的前
项和
,并判断
能否被
屏蔽,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc67e4349da07ad777054019a302113d.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24314c37a77e5e3190b54ec9b3298009.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78da089772e95d1cdf435451d7d98b7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
您最近一年使用:0次
2023-06-06更新
|
788次组卷
|
4卷引用:专题11 数列前n项和的求法 微点10 数列前n项和的求法综合训练
(已下线)专题11 数列前n项和的求法 微点10 数列前n项和的求法综合训练河南省创新发展联盟大联考2023届高三预测数学(理科)试题2023届河南省创新发展联盟大联考仿真模拟预测数学(文科)试题河南省南阳市第一中学校2024届高三上学期第六次月考数学试题
5 . 若数列
从第二项起,每一项与前一项的差构成等差数列,则称数列
为二阶等差数列.某数学小组在数学探究课上,用剪刀沿直线剪一圆形纸片,将剪
刀最多可以将圆形纸片分成的块数记为
,经实际操作可得
,
,
,
,…,根据这一规律,得到二阶等差数列
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0562c7a41342ff15e2bd3887d7201d86.png)
________ ;若将圆形纸片最多分成1276块,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfc321599521a98661ed719cc82ca87c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab7d59ce066c8f0b346719003f8e28f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1196be612370e6db1275c1f087d010dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc5a47eb206ee12c7f65ea26fc26e90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0562c7a41342ff15e2bd3887d7201d86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
您最近一年使用:0次
2023-04-19更新
|
785次组卷
|
4卷引用:押新高考第16题 数列性质及其应用
(已下线)押新高考第16题 数列性质及其应用(已下线)江苏省八市2023届高三二模数学试题变式题11-16河北省邯郸市2023届高三二模数学试题广东省广州市从化区从化中学2023届考前仿真模拟3数学试题
6 . 若数列
满足:存在等比数列
,使得集合
元素个数不大于![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
,则称数列
具有
性质.如数列
,存在等比数列
,使得集合
,则数列
具有
性质.若数列
满足
,
,记数列
的前
项和为
.证明:
(1)数列
为等比数列;
(2)数列
具有
性质.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a75786d2d4c9abbb9ddea89c3dd1e39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eb60c1a8dba48239b667dcf1235dd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46bf6ded2f869744c6c50785f974aa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41ad4f005f5def1e6c0d54610692c03d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05d8a4d9c52ac3520d5b64c62499ecf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0b64100360c766baccac7c39255d8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bea0dd7e474bcd04db2544427ba0488.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1bae03ee4ac75dacfb026290e4207dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aa128888d2031f4c634be7f954238ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b24baee90fd506c597bd9bd7293e90a.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d013861990cf331c82eb453416ae31bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bea0dd7e474bcd04db2544427ba0488.png)
您最近一年使用:0次
2024-01-13更新
|
831次组卷
|
4卷引用:压轴题数列新定义题(九省联考第19题模式)练
(已下线)压轴题数列新定义题(九省联考第19题模式)练(已下线)模型1 用综合法快解新情境背景下的数列创新题模型(高中数学模型大归纳)湖北省武汉市江岸区2024届高三上学期1月调考数学试题河南省郑州市宇华实验学校2024届高三下学期第二次模拟考试数学试题
7 . 设集合
是一个非空数集,对任意
,定义
,称
为集合
的一个度量,称集合
为一个对于度量
而言的度量空间,该度量空间记为
.
定义1:若
是度量空间
上的一个函数,且存在
,使得对任意
,均有:
,则称
是度量空间
上的一个“压缩函数”.
定义2:记无穷数列
为
,若
是度量空间
上的数列,且对任意正实数
,都存在一个正整数
,使得对任意正整数
,均有
,则称
是度量空间
上的一个“基本数列”.
(1)设
,证明:
是度量空间
上的一个“压缩函数”;
(2)已知
是度量空间
上的一个压缩函数,且
,定义
,
,证明:
为度量空间
上的一个“基本数列”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9661053f3ef4cfa926e5d5fd5c6555f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00a1f35848a78a4f00c21500e2610e21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171102a883b22fe6ca578efc8926f5b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171102a883b22fe6ca578efc8926f5b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c975849cb845a1eb30e25d6bb0782075.png)
定义1:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/974cd5eed14d5002f6155dced3e62432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c975849cb845a1eb30e25d6bb0782075.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034e4d52bd5ae47074a93c0647f67399.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9661053f3ef4cfa926e5d5fd5c6555f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f1ada25dccde00dfff2525360188a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c975849cb845a1eb30e25d6bb0782075.png)
定义2:记无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da889327e4b9a31336a88e6da53334d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7968de1fe9004760db9a41c24df809b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7968de1fe9004760db9a41c24df809b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c975849cb845a1eb30e25d6bb0782075.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/859cf5bf57a50d2da19c0bb926ce9c18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e61e145e5a49ebbe72c3b9ba1f7cdde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a6f5c5649285cbabda20a452db04f38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7968de1fe9004760db9a41c24df809b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c975849cb845a1eb30e25d6bb0782075.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8557a6d85a35cd171e43087afd1b0576.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38c95a62ca5cb2f440792632ec36595b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd6ebf0a370d321e89a8f9921041a5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae4c872929a492d8bcd9e649f190a66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8671bcd155dd76d76d83573c6f20e930.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cfb19f0c37a72b33083ae9319f11a74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6961967d7e48061a9cbb14f597e73d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7968de1fe9004760db9a41c24df809b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae4c872929a492d8bcd9e649f190a66.png)
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解题方法
8 . 通过以下操作得到一系列数列:第1次,在2,3之间插入2与3的积6,得到数列2,6,3;第2次,在2,6,3每两个相邻数之间插入它们的积,得到数列2,12,6,18,3;类似地,第3次操作后,得到数列:2,24,12,72,6,108,18,54,3.按上述这样操作11次后,得到的数列记为
,则
的值是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/044f83b1fbe7f1153e6282c929c92cc4.png)
A.6 | B.12 | C.18 | D.108 |
您最近一年使用:0次
2022-01-19更新
|
1720次组卷
|
5卷引用:专题10 数列通项公式的求法 微点5 构造法
(已下线)专题10 数列通项公式的求法 微点5 构造法专题06数列(已下线)专题5-1 等差等比性质综合-22022年1月浙江省普通高中学业水平考试数学试题2023年浙江省普通高中学业水平考试押题预测数学试题
9 . 数列
:1,1,2,3,5,8,…,称为斐波那契数列,该数列是由意大利数学家菜昂纳多·斐波那契(Leonardo Fibonacci)从观察兔子繁殖而引入,故又称为“兔子数列”.数学上,该数列可表述为
,
.对此数列有很多研究成果,如:该数列项的个位数是以60为周期变化的,通项公式
等.借助数学家对人类的此项贡献,我们不难得到
,从而易得
+
+
+…+
值的个位数为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8323901a49cac29afd7d62864f088077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c3ded683593d177e9dc890938422cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/334b002c0edfc3f32f9b65b129c03dcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc41fa7aef562e9dd92ba267be7813a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23cabc65fcefdd191930fbfa89cf125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a8553a873958eb9d14c89c59ab5c317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a2fc7063bcdce2bee4cb0f4d748d72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da690136762870d3082276b147353aaf.png)
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10 . 设
是定义域为
的函数,如果对任意的
、
均成立, 则称
是“平缓函数”.
(1)若
, 试判断
和
是否为“平缓函数” ? 并说明理由; (参考公式:
时,
恒成立)
(2)若函数
是“平缓函数”, 且
是以 1为周期的周期函数, 证明:对任意的
、
, 均有
;
(3)设
为定义在
上函数, 且存在正常数
使得函数
为“平缓函数”. 现定义数列
满足:
, 试证明:对任意的正整数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1076aeb555debc8bf15c586375d64c83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2fe919afe3583b1ef0dadbd23102a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70458945dae62f8f6a553dfaa8eb723a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b40b099989abb2d15ddf60413c8a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/640461197eaf70c0ebe1eb89daf383c1.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33890c6b0bf167514d44139d9dca0154.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49a2b43fdce5aaae58c0907de23cbc6c.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee6bf90a1bbeea09e1b7206975a99f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b66efdaa9fe939b2b1a7326aaf5e35d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f422484ad3a17eeebc4d15a3fb6289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/848ea90ecbd6c506a78058f983ff5614.png)
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