1 . 相传古希腊毕达哥拉斯学派的数学家常用小石子在沙滩上摆成各种形状来研究数,并根据小石子所排列的形状把数分成许多类.现有三角形数表按如图的方式构成,其中项数
:第一行是以1为首项,2为公差的等差数列.从第二行起,每一个数是其肩上两个数的和,例如:
;
为数表中第
行的第
个数.
和
;
(2)一般地,证明一个与正整数
有关的命题,可按下列步骤进行:①证明当
时命题成立;②以“当
时命题成立”为条件,推出“当
时命题也成立.”完成这两个步骤就可以断定命题对
开始的所有正整数
都成立,这种方法即数学归纳法.请证明:数表中除最后2行外每一行的数都依次成等差数列,并求
关于
的表达式;
(3)若
,
,试求一个等比数列
,使得
,且对于任意的
,均存在实数
,当
时,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/008c3c308a9a18f5a3bad6c67cacf113.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdab9718ae9ad2732585fa25b760a956.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7242d5f694c3c7c9530f5ef0cd1447b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1192d0aa416fc19f7f4b842cf6717808.png)
(2)一般地,证明一个与正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2636b1b9ad69adc8b268d3513a59b7ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/469410cf8d7cd28620a58363cb5cbb6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d85b80a9c97bd7106dcbfb34199b1e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8f1ae8e6654806b02cd359fb484ea4e.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e789526ee5eab677295edf78fefb00f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4a92d4463e0a56109a13d60b640e0a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2995a87642de38c4a7c79c133fb2d1bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c83f7e578f082cbba0e39cff3c2c5da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdc83e348654e938962f3fd0c04e023f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efe9dbc75f393b682c8a90fe7277ab4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9afaaa196735c0c02f05f97fda5534a4.png)
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2024-03-06更新
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2卷引用:湖北省云学名校联盟2023-2024学年高二下学期3月联考数学试卷