题目
题型:不详难度:来源:
都有f(x+1)=f(x)+2.数列{an}满足
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232552-39680.png)
(1)当x为正整数时,求f(n)的表达式;(2)设λ=3,求a1+a2+a3+…+a2n;
(3)若对任意n∈N*,总有anan+1<an+1an+2,求实数λ的取值范围.
答案
解析
试题分析:解:
(1)记bn=f(n),由f(x+1)=f(x)+2有bn+1﹣bn=2对任意n∈N*都成立,
又b1=f(1)=λ,所以数列bn为首项为λ公差为2的等差数列, 2分
故bn=2n+λ﹣2,即f(n)=2n+λ﹣2. 4分
(2)由题设λ=3
若n为偶数,则an=2n﹣1;若n为奇数且n≥3,则an=f(an﹣1)=2an﹣1+λ﹣2=2•2n﹣2+λ﹣2=2n﹣1+λ﹣2=2n﹣1+1
又a1=λ﹣2=1,
即
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232552-46563.png)
a1+a2+a3++a2n=(a1+a3++a2n﹣1)+(a2+a4++a2n)=(20+22++22n﹣2+n﹣1)+(21+23++22n﹣1)
=(1+21+22++22n﹣1)+n﹣1=22n+n﹣2. 8分
(3)当n为奇数且n≥3时,an+1an+2﹣anan+1=an+1(an+2﹣an)=2n[2n+1+λ﹣2﹣(2n﹣1+λ﹣2)]=3•22n﹣1>0; 10分
当n为偶数时,an+1an+2﹣anan+1=an+1(an+2﹣an)=(2n+λ﹣2)(2n+1﹣2n﹣1)]=3•2n﹣1(2n+λ﹣2),因为anan+1<an+1an+2,所以2n+λ﹣2>0,
∵n为偶数,∴n≥2,
∵2n+λ﹣2单增∴4+λ﹣2>0,即λ>﹣2
故λ的取值范围为(﹣2,+∞). 12分
点评:解决的关键是利用数列的通项公式以及数列的单调性来得到证明,属于中档题。
核心考点
试题【已知函数f(x)的图象经过点(1,λ),且对任意x∈R,都有f(x+1)=f(x)+2.数列{an}满足.(1)当x为正整数时,求f(n)的表达式;(2)设λ=】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232535-65365.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232535-85059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232535-74892.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232536-74804.png)
(1)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232536-72519.png)
(2)已知公比为
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232536-16887.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232536-83458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232537-60085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232537-67731.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232537-54582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232537-58432.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232536-83458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232437-33000.png)
(1)求:
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232437-21269.png)
(2)类比等差数列的前
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232437-26472.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232438-35340.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232343-12727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232343-84351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232343-85064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-57586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-11334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-26361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-77694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232345-43670.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232345-95477.png)
(Ⅰ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-57586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-26361.png)
(Ⅱ)将数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232344-26361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232345-76204.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232345-14077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232346-16343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232346-77564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232346-60883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232346-60883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232346-76090.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232129-67509.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232129-50553.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-57314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-54353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-39613.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232104-99520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232104-46218.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232104-99520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232104-92219.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-54353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232104-34232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-54353.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232105-67219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232105-49511.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232105-45849.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-57314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-39613.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232105-42914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232106-39344.png)
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232103-54353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232106-57925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232106-58860.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009232106-19691.png)
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