1 . 阅读以下案例,并参考此案例化简
.
案例:观察恒等式
左右两边
的系数.
因为等式右边
,
所以等式右边
的系数为
,
又等式左边
的系数为
,
所以
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69e90e7e76ef809b523c0a04990bcad2.png)
案例:观察恒等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1e1cbe85ec76ba8d640ce431cd45b46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
因为等式右边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b89a3c1118e65cdaf568dd3898aba88.png)
所以等式右边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/857f298cc4509de873d41b4671c8ae4c.png)
又等式左边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f56b40ed700b253ce7099a15c452446.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75e51a525c34b847b66ef5c5d83734a1.png)
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2 . 设
,
,
.
(1)当
时,试比较
与1的大小;
(2)根据(1)的结果猜测一个一般性结论,并加以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45561b8f752d03ca63558057641d5dd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f7e8aaabe94cf71bcaaf5694a4dcdfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4967a0f83ec59ad5a74ce1c3653a2451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67e905a3a1f7d003229c907db8d2c843.png)
(2)根据(1)的结果猜测一个一般性结论,并加以证明.
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解题方法
3 . 观察下列不等式:
,
,
,
,…….
(1)根据这些不等式,归纳出一个关于正整数n的命题;
(2)用数学归纳法证明(1)中得到的命题.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a11174c92d9ed0ebc6acb7be0ff2369.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b092a0ab1a0a35e25ab5dc4ce7b08f48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c3d3ac03284da9b56bd2e6f0cbea94a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de48b2eaa886f4c94e1c9da4cabe5300.png)
(1)根据这些不等式,归纳出一个关于正整数n的命题;
(2)用数学归纳法证明(1)中得到的命题.
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4 . 如图所示的三角形数阵叫“莱布尼兹调和三角形”,它们是由整数的倒数组成的,第n行有n个数且两端的数均为
,每个数是它下一行左右相邻两数之和,如
,
,
,…,则第11行第8个数(从左往右数)为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78efce0b9458e7d0775730af10785496.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48e793a209cbb7698b63ce86071061bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/660f41a92328772f61ade4e991d5ac0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ebcc3f6b9a6e9b4dcd68382ba32a00b.png)
![](https://img.xkw.com/dksih/QBM/2022/7/4/3015415672111104/3017353059336192/STEM/6cda17fcf69145378e12715d52514a02.png?resizew=214)
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5 . 徽州的刺绣有着悠久的历史,如图①②③④为徽州刺绣最简单的四个图案,这些图案都是由小正方形构成,小正方形的个数越多刺绣越漂亮,现按同样的规律刺绣(小正方形的摆放规律相同),设第n个图案包含
个小正方形,则
( )
![](https://img.xkw.com/dksih/QBM/2022/7/4/3015346201075712/3015925110013952/STEM/9e2bfc2c85fb45e38eff9caf03d7f211.png?resizew=586)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d4fc8faefb26b233d4aa9dbef043aae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/636023a11bb3a8e98729fbfcbb308b01.png)
![](https://img.xkw.com/dksih/QBM/2022/7/4/3015346201075712/3015925110013952/STEM/9e2bfc2c85fb45e38eff9caf03d7f211.png?resizew=586)
A.61 | B.64 | C.65 | D.66 |
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3卷引用:云南省红河哈尼族彝族自治州弥勒市第一中学2021-2022学年高二下学期第四次月考数学试题
6 . 已知数列1,
,
,
,…,
(
)的前
项和为
.
(1)求
,
,
;
(2)猜想前
项和
,并证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60f466b4dda536642c8707527f614bd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724e7575ee2ce15e8b934729709ad515.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5ef448eca63582453acc7f8f6baaa51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dd496172242f8939bc56ccd64fe7ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97769855336d73371930df1f187875e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f30f56664446f32dbbc2c5f12a99374.png)
(2)猜想前
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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7 . 已知
,
,
通过观察等式的规律,写出一般性规律的命题,并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d456109abcdc10c219e4c48d4024b71f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6892a9ada2c38b28bd6f2254489c87f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bda41669c06f42c92b99f4ee888054d6.png)
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8 . 1202年意大利数学家斐波那契出版了他的《算盘全书》,在书中收录了一个有关兔子繁殖的问题.他从兔子繁殖规律中发现了“斐波那契数列”,具体数列为:1,1,2,3,5,8,13,…,即从该数列的第三项开始,每个数字都等于前两个相邻数字之和.已知数列
为斐波那契数列,其前n项和为
,且满足
,则当
时,
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f7ba29ada127d208b75cc47677ed706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4abd335037bb7895b9eb592d16934933.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f01d3d53cd7492197b1d1f5c3ea936e.png)
A.1 | B.2 | C.![]() | D.![]() |
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9 . 已知数列
中,
,
.
(1)求
,
,
,
的值;
(2)根据(1)的计算结果,猜想
的通项公式,并用数学归纳法证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f7652005ab389cc57a4a58de33070f3.png)
(1)求
![](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/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
(2)根据(1)的计算结果,猜想
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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辽宁省县级重点高中协作体2021-2022学年高二下学期期中考试数学试题河南省南阳市第一中学2021-2022学年高二下学期第五次月考理科数学试题辽宁省大连市第一〇三中学2021-2022学年高二下学期期中考试数学试题(已下线)4.4 数学归纳法-2022-2023学年高二数学《基础·重点·难点 》全面题型高分突破(苏教版2019选择性必修第一册)(已下线)4.4 数学归纳法(1)(已下线)4.4 数学归纳法(精练)-【题型分类归纳】2022-2023学年高二数学同步讲与练(人教A版2019选择性必修第二册)
10 .
(
,
),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca8c81b918419fea27b76d8292da429c.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27f56b2449d86748d535e4e73e1f525e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca8c81b918419fea27b76d8292da429c.png)
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