题目
题型:不详难度:来源:
设正整数数列{an}满足:a2=4,且对于任何
n∈N*,有
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015729-23300.gif)
(1)求a1,a3;
(2)求数列{ an }的通项an.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-49005.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72550.gif)
(2)对任意
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-62379.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-47339.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-70699.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-50689.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-56664.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-96957.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015732-45184.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-49005.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015732-70619.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015732-85568.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015732-82516.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72550.gif)
(2)方法一:由
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-49005.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-79929.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
下面用数学归纳法证明.
1
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-70699.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-52792.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
2
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-21479.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-59932.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-84856.gif)
由①得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-48275.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-30802.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-20526.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015735-16526.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015735-37590.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015735-23792.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015735-41044.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015736-45953.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015736-93596.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015736-21848.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015736-65046.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-84856.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
由1
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-62379.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
(2)方法二:
由
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-49005.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-79929.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
下面用数学归纳法证明.
1
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015731-70699.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-52792.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
2
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-21479.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-59932.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-84856.gif)
由①得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015736-16897.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015737-60854.gif)
由②左式,得
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015737-37597.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015737-69276.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015737-32939.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015738-74936.gif)
又由②右式,
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015738-39976.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015738-73628.gif)
因为两端为正整数,则
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015738-31601.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015738-36933.gif)
又因
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015735-16526.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015739-18813.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015739-92221.gif)
据③④
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015736-65046.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015734-84856.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
由1
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015733-18968.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-62379.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015730-72084.gif)
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-86193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-73327.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-38458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-64663.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-55506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-24894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-26517.png)
(I)若
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015709-88940.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-32351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-64883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-56276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-64246.png)
(II)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-17455.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-28496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015711-20070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015711-72267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015711-84114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015711-54455.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015710-56276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015711-99604.png)
已知各项全不为零的数列{ak}的前k项和为Sk,且Sk=
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015655-19653.gif)
(Ⅰ)求数列{ak}的通项公式;
(Ⅱ)对任意给定的正整数n(n≥2),数列{bk}满足
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015655-53282.gif)
求b1+b2+…+bn.
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015647-34673.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015648-12420.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015648-90862.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015648-45944.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015648-56716.gif)
A.2009 | B.2010 | C.4018 | D.4020 |
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015641-39348.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191013/20191013015642-22644.gif)
A.1 | B.2 | C.3 | D.4 |
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