2017上海中学高一周练、

上海中学高一数学周练卷

2017.6.1

一. 填空题 1.设234212121212

333333

n n n S -=

-+-+???+-，则lim n n S →∞=

2.n →∞

-=

3.用数学归纳法证明123(21)(1)(21)n n n +++???++=++时，从“n k =到1n k =+”时，左边需增加的代数式是

4.数列2211,(12),(122),,(1222),n -+++???+++???+???的前n 项和是

5.

计算：lim(log n →∞

=

6.数列{}n a 按以下条件给出：111

2

2,2,n n n n a a a n a a n ++=??

=+??=?当为奇数时当为偶数时

，则2015a =

7.若通项公式1

(1)(32n n a n

λ=-+

+的数列{}n a 的各项都是正数，则λ的取值范围是 8.设各项皆为非零实数实数的数列{}n a 的前n 项和是n S ，且11a =，对任意*n N ∈都有

11

2

n n n S a a +=，则n a =

9.设数列{}n a 的前n 项和1lg n n S a b =+，则使得lim 1n n S →∞

=成立的实数b 的取值范围是

10.如图，一个粒子的起始位置为原点，在第一象限内于两正半轴上运动，第一秒运动到(0,1)，而后它接着 按图示在x 轴、y 轴的垂直方向来回运动，且每秒移动 一个单位长度，那么经过2015秒时，这个粒子所处位 置的坐标是

11.若在由正整数构成的无穷数列{}n a 中，对任意的正整数n ，都有1n n a a +≤，且对任意的正整数k ，该数列中恰有21k -个k ，则2015a =

12.有五只猴子摘了一些桃子，打算隔天早上起来分了吃，晚上的时候，第一只猴子偷偷起来把桃子分成五堆，还多了一个，就把多了的那个吃掉，并拿走了一堆，第二只猴子也偷偷起来将桃子分成了五堆，还是又多了一个，同样吃掉了这一颗桃子，并拿走了其中一堆，第三只、第四只、第五只猴子都做了同样的事情，则这堆桃子最少有

个

二. 选择题

13.某个命题与自然数n 有关，如果n k =时命题成立，可以推得1n k =+时该命题也成立，现知道15n =时该命题不成立，则（ ）

A. 16n =时该命题不成立

B. 16n =时该命题成立

C. 14n =时该命题不成立

D. 14n =时该命题成立

14.已知1lim 1n

n

n a b a →∞-=+（b 为常数），则a 的取值范围是（

）

A. a R ∈且1a ≠

B. ||1a >

C. 11a -<≤

D. 0a =或1a =

15.等差数列{}n a 中，n n S m =，m m S n

=（m n ≠，*,,m n N ∈），则m n S +的值（ ）

A. 大于4

B. 等于4

C. 小于4

D. 大于2或小于4

16.已知A 、B 为ABC ?的内角，tan A 是以4-为第三项，4为第七项的等差数列的公差，

tan B 是以1

3

为第三项，9为第六项的等比数列的公比，则此三角形为（

）

A. 钝角三角形

B. 锐角三角形

C. 等腰直角三角形

D. 非等腰的直角三角形

三. 简答题

17.求11sin cos lim sin cos n n n n n θθθθ++→∞-+，其中

3(0,4

π

θ∈；18.已知数列{}n a 是公差为d 的等差数列，证明：数列{}n a 的通项公式是1(1)n a a n d =+-；

19.用哪个数学归纳法证明：223521n n n +?++（*n N ∈）能被25整除；

20.我们将分母为正整数，分子为1的分数叫做单位分数，如果两个单位分数的分母是两个连续正整数，就称这两个单位分数为连续单位分数，证明：对任意不小于2的正整数n ，

111111234212n n

-

+-+???+--可以写成n 个连续单位分数的和；

参考答案

一. 填空题 1.

18

2.1

3.(22)(23)k k +++

4.122n n +--

5.1

6.1007624?-

7.5[3,)2

-8.n

9.b <10.(9,44)

11.4512.3121

二. 选择题 13.C

14.C

15.A 16.B

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