2023八年级上·江苏·专题练习
1 . 我们知道:任意一个有理数与无理数的和为无理数,任意一个不为零的有理数与一个无理数的积为无理数,而零与无理数的积为零.由此可得:如果
,其中a、b为有理数,x为无理数,那么
且
.运用上述知识,解决下列问题:
(1)如果
,其中a、b为有理数,那么
,
;
(2)如果
,其中a、b为有理数,求
的平方根.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21ddf68448b62a8da683ac4da36e6279.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
(1)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bff08df51b9b9b02c5744065479695b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380bbacf854e30e2e747fc286d2b9997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ccd4162c7d09f970cb77cadacdbe521.png)
(2)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a2c0b8f59209a31da6ab784b0bb8ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d993cf0a090de3b01f1dda52c6fdc9f.png)
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23-24八年级上·吉林长春·期末
2 . 观察下列各式:
,
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f2efb7c8a74a4a88d128eaf8b42b30a.png)
(1)由此可推测:
______;
(2)依照上述规律,写出
的推测过程;
(3)请你猜想出能表示以上式子的一般规律,用含
(
表示整数)的等式表示出来,并说明理由;
(4)请直接用(3)中的规律计算
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2f620b40e8bc8300e028e5adf218c3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d42c28ab7a1d275129d7e176a26a846.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f80510e9009407eb0fd09c359058b8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f2efb7c8a74a4a88d128eaf8b42b30a.png)
(1)由此可推测:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/634b5928adfdd795a5ad8da7b014693f.png)
(2)依照上述规律,写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/787a01b05b6355c1e54a5f74aaae1ac0.png)
(3)请你猜想出能表示以上式子的一般规律,用含
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(4)请直接用(3)中的规律计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c44b3a46af57cf46e76445bd4813061f.png)
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3 . 观察下列各式:
,
,
.
(1)猜想:
_______;
(2)你发现的规律是:
_______;(
为正整数)
(3)用规律计算:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f725bf58e1df5a21eed840533a328ece.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0bbea24f1e12e309624d0b63978c7ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d5b50ab6e085c722f1d5c1791dd42f6.png)
(1)猜想:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4169918cc613b7b9c517cf2c893987a.png)
(2)你发现的规律是:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf68f3d35db125770c770cc7be6faf9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)用规律计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4bc6f29c43e5ff7bcdc3cd56bfcfba2.png)
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4 . 观察下列一组式子的变形过程,然后回答问题:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/805ac4378d81908a9ec62b5da7a50028.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f52e24fbfbe194c8bd6c1845a4370a7.png)
.
,
.
(1)用含n(n为正整数)的关系式表示上述各式子的变形规律为_______.
(2)利用上面的结论,求下列式子的值:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/805ac4378d81908a9ec62b5da7a50028.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f52e24fbfbe194c8bd6c1845a4370a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/805ac4378d81908a9ec62b5da7a50028.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/760f1b4bd19f34c76f5299882065aaa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/902b9dc06e6b4a1b36e7d6ca10ae18d1.png)
(1)用含n(n为正整数)的关系式表示上述各式子的变形规律为_______.
(2)利用上面的结论,求下列式子的值:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b6c5f3d793c4e200992ae1abfc4d0c3.png)
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5 . 观察下列各式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e21c771935207f871f494361a9732879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/336269aed514ecd40a270f847c93d7eb.png)
请你根据上面三个等式提供的信息,猜想:
(1)
.
(2)请你按照上面每个等式反映的规律,写出用n(n为正整数)表示的等式 .
(3)利用上述规律计算:
(仿照上式写出过程).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf95b3a5cec45d465ac90ee1fd0c1917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e21c771935207f871f494361a9732879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/336269aed514ecd40a270f847c93d7eb.png)
请你根据上面三个等式提供的信息,猜想:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab99bc402442dce79f3ba1a99d626fc5.png)
(2)请你按照上面每个等式反映的规律,写出用n(n为正整数)表示的等式 .
(3)利用上述规律计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dd0dfcff70f366dafd8cb2fd1c4bb8e.png)
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2023-12-27更新
|
126次组卷
|
2卷引用:山东省济南市莱芜区莲河学校2023-2024学年七年级上学期12月月考数学试题
22-23七年级下·全国·课后作业
6 . 已知
,
,
,
,…,
.
定义:
,
,
,…,按此规律类推,
Sn=a1+a2+a3+…+an=________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/778dc8fa2140878bf8670bf79c01a39a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/658934b379e419c9a95b72a62c4c83ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6e8bc7ff8b9430b53fb94fcd9f5fc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f87b344d6788ce20022d74ff13e5e78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a051247d27d94ca427954f974e7f5b84.png)
定义:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17ef4e6e5b989b593c86f8abc615fb95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90ebd53812cfb6634b6880751b7e2467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aadd602845bd18fa9adfc0c8cee354f3.png)
Sn=a1+a2+a3+…+an=
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名校
7 . 下列说法正确的有( )
①如果
则
②若
是四次三项式,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e05bd7d9bb486e2b3660af90dff8d609.png)
③若
,则
④
,
,
,
,
,
,
,则
的末位数字是9
①如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25bd336e2b190038609fbc5b41023f12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78c3b0a0ebf055745e368a9ccd6a66fe.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0c4596045d0012c89035ad66af6dee2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e05bd7d9bb486e2b3660af90dff8d609.png)
③若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da0219208c068423dd31f170b6fc5edd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955087ea9bcecd6fed53c3bb9a0eef7d.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97a8d2e5a71a80815b3454e2fe4140e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/738317df2665c72d92812b7714675549.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a52685b77856d28af3beed027d725506.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14f48b3a441172a0073be2877988167b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/914832f3edf43812c7ffa2c7fea02620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4d58a9daa5736ec597957257887a5d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c96ebeb9ebb81fd6452d11746ec0634.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa3cdec22172e28e9ce74eb1ae5154be.png)
A.1 | B.2 | C.3 | D.4 |
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8 . 阅读材料并解决问题:
求
的值.
令
,等式两边同时乘2,则
,
两式相减得
,所以
.
依据以上计算方法,计算![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f8f8300dd7b6c3a98018d1f9817a7c.png)
________ .
求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b61b0fcf02d4435d74196901285bc72.png)
令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02cb84a482cd21587c12f56624d17cf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4825b6762e4eb03f548278b0b6439e8f.png)
两式相减得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aa19d7f3930366e301cf8f7b61ec8fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2147e73b499dc252c37b3d419b8cf918.png)
依据以上计算方法,计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f8f8300dd7b6c3a98018d1f9817a7c.png)
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9 .
是一个求和符号,英文名称:sigma,汉语名称西格玛。例如:
,当
时
;
,当
时
;
,当
时
;
下列说法正确的个数是( )
①当
时,
;
②当
时,若
,则
;
③如果
,那么
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e5550d8659980c02488a57afd5964ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1348cad215d68433d28173c6a1af74c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8068be2ea6278a60e8be970ff8a7a2b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a6cc40f07ce15f88a717cb1d4b1ea51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e2e58e3a6b01b8b71a29a3af33cb43a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16a0f198ea05831307199b8815b93da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526cd8fb3b3c181ee2a79a7fcc94f96d.png)
下列说法正确的个数是( )
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b61f98a8ac38df3ec07c7e81758a8b.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41ca1b8398fbd1dcbdc5e859d674c65a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0471deae2956862a92b230e493f26419.png)
③如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b38c1ab251160bb0bd2d88e4aeca550f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f93c2aa7cfeffad5350333c05c4a22cc.png)
A.0个 | B.1个 | C.2个 | D.3个 |
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名校
10 . 已知有依次排列的两个数
,
,将第一个数乘2的积加上第二个数得到第三个数记为
,即
;将第二个数乘2的积加上第三个数得到第四个数记为
,即
;将第三个数乘2的积加上第四个数得到第五个数记为
,即
,……,以此类推,下列说法:①
:②
;③
;④
.其中正确的个数是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2693734765399876e9e93cdb110231c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48bf2138c305e7d3ab3c2999eec10c60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/522f5ab8b58a9b2495adcc2e662b9fa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2df0a21c492ffc6c149a235048423b07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1534d4139200e3c3db707a2c70e0f1b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8987581e5ee8498f02c1b4aca5a5f017.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1371d06bd3a1f53adb56bfff897722b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fb77921859746a8fc2ebc7c7b57ef9.png)
A.1 | B.2 | C.3 | D.4 |
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