1 . 将正方形
分割成
个全等的小正方形(图1、图2分别给出了
的情形),在每个正方形的顶点各放置一个数,使位于正方形
的四边及平行于某边的任一直线上的数都分别依次成等差数列.若顶点
处的四个数互不相同且和为1,记所有顶点上的数和为
,则有
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022205c82846dfa3420c67402a278fea.png)
______ ,…,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d64205bb7d470602fb525b02e72a81b.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/188ba12941ea122a7775a4999e129161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d15d4afcc5736bb747eeffcffd3f4464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf8c7f5f6a35f7ed88271a077619aef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022205c82846dfa3420c67402a278fea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d64205bb7d470602fb525b02e72a81b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/29/8a0dc8d3-051d-4666-9a45-8e1c699ed7bd.png?resizew=254)
您最近一年使用:0次
2 . 已知有序数对
,有序数对
,定义“
变换”:
,
,
,可以将有序数对
转化为有序数对
.
(1)对于有序数对
,不断进行“
变换”,能得到有序数对
吗?请说明理由.
(2)设有序数对
经过一次“
变换”得到有序数对
,且有序数对
的三项之和为2024,求
的值.
(3)在(2)的条件下,若有序数对
经过
次“
变换”得到的有序数对的三项之和最小,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ba3b8caed50a276997b18ffe786db9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ade970602ccabf3cc41cbe91ddd9e354.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf084ae8f3e0f87e8511fc9016639d8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff9e9969b1ce7b665ece26b23d5e5595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1090785e572862906ea30f8f2a739e34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
(1)对于有序数对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd690d529356ccb1c8680255476b9f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e996353e1191b4381d47f8973fb04c6.png)
(2)设有序数对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ba3b8caed50a276997b18ffe786db9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9ebcb129e9534dfa22667162ccab477.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/916bb2cc1b29574ff95b47567c59ee0c.png)
(3)在(2)的条件下,若有序数对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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3 . 若
表示自然数
的最大奇因数,例如
,
,
,记
(
为自然数),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
______ .,
的通项公式为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91eb1a74ed4eb789a5cf6bf0d08900a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ac85888a9b802aea6eb688edac8f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18443df3018a6806c7ded2bf11ed70a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9bf2837ae4419c83bb69ad10ab7ef14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/640faa4d58105334dafd1cf218f30ae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
您最近一年使用:0次
2024·山东·模拟预测
4 . 将数列
从首项开始从左到右依次排列,得到数组
,
,
,…,
,然后执行以下操作:将
移到
右侧,然后剔除
,再将
移到
右侧,然后剔除
,继续以上操作,即将最左边的数移到最右边,然后剔除新数组最左边的数,直到剩下最后一个数
.若令此操作为
,则
,且确定
的值可确定
的值,如
,
,
.
(1)证明:
;
(2)证明:
;
(3)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.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)
![](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/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354feaea520a5a8be1a23491d8d5a338.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11d9b67e005c4a598b63cf77d2f8444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68af8d457cb86ace285b58beed4481ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60581b1690e001201b5db7c17d2d9a6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/940b6a01805abaf331c42fa30fee999e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7bcf93832ee27c7bca8832ae48eba6c.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee74648f3750bd91f96cc165eb5ea95b.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf41559b6cce92cc5997d3179b2a7483.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e6583dc871d2fd9ee2bb4816b6b3198.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f3d5dc3c54741677409d2dc5ba8093e.png)
您最近一年使用:0次
5 . 已知数列
:
,
,
,
,
,
,
,
,
,
,
,其中第
项为
,接下来的
项为
,
,接下来的
项为
,
,
,再接下来的
项为
,
,
,
,依此类推,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b06e95b57b7a81cd81d05557a11fa92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b06e95b57b7a81cd81d05557a11fa92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
A.![]() |
B.![]() |
C.存在正整数![]() ![]() ![]() ![]() |
D.有且仅有![]() ![]() ![]() |
您最近一年使用:0次
6 . 求正整数
,使得
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63022eef355eb0776a1ee956c8fdf8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab797c5e8378285cff446f8bca5e9012.png)
您最近一年使用:0次
7 . 设数列
,
为
的满足下列性质
的排列
的个数,性质T:排列中仅存在一个
,使得
.
(1)求
的值,并写出
时其中一种排列的情形.
(2)若
,求满足性质
的所有排列的情形.
(3)求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b3c6f8a0d8adcd7bb9cb3d0214c8f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e260b088f071983f254ce8f5163fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7b19b82b9c52bccc2d326c151962da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/535577e9213fa6b2a2bed70460fc4077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de6bc6e1861d56ea5eb320e6c569338e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b8800620cdf97b3db972d64ab8e4886.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf2f1c1409a06278e847e6b573cef254.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368691ff6590140094865d9a7a702d09.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c03a70f7cba93600ed0a3164efadbcc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(3)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
您最近一年使用:0次
8 . 如图,将正整数按下表的规律排列,把行与列交叉处的那个数称为某行某列的元素,记作
,如第2行第4列的数是15,记作
,则有序数对
是____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb574762e9dc1df3fc79fa957b1fd49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0485cc492f0f7fd84375fd26d4d12585.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49bd66006400c3413106caa2fc94fa2d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/14/8d8501a9-764d-4edf-86bc-031e42d95659.png?resizew=185)
您最近一年使用:0次
9 . 如图中的三角形称为谢尔宾斯基三角形,每个图都是取前一个图中的每个黑色三角形三边的中点将其分成四个小三角形,并将中间三角形变为白色,白色三角形不变.若第一个三角形的面积为1,第n个图中白色部分的面积记为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
______ .著名的洛卡斯数列
满足
,
,
,
中所有既是偶数,又是3的倍数的项从小到大排列构成一个新的数列,该数列的第n项为
,则数列
的前n项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ab233fde57c65ad8591abac0f6a370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1a25c00e9653b88ec05ac86bd86ac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8903a7918cdf1a954fb699243f150242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa98ea8cb9b47e3c698c606b65689ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ab233fde57c65ad8591abac0f6a370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/155e9d6bd5789408a2fd8b04bf08f1d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c2864e2ec3416cc4c081ac1f71a0af.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/25/8adf23b7-b4bb-46bb-8ea9-2715d47be9da.png?resizew=484)
您最近一年使用:0次
10 . 对20进行“乘以2”或“减去3”的一种运算,得到的结果再进行“乘以2”或“减去3”的一种运算,…,一直进行这样运算,每进行一种运算记作一次运算,已知运算n次后,得到结果为49,则n的最小值为____________ .
您最近一年使用:0次