College Entrance Examination Questions

I. known quantity column $\{a_{n}\}$ satisfies $a_{1}=1/2$ and $a_{n+1}=a_{n}-a_{n}^{2}$

(1) Proof: $1\leq \frac{a_{n}}{a_{n+1}}\leq 2$

(2) The first $n$ of the sequence $\{a_{n}^{2}\}$ and the $s_{n}$, prove $\frac{1}{2 (n+2)}\leq \frac{s_{n}}{n}\leq\frac{1}{2 (n+1)}$

two. in the sequence $\{a_{n}\}$, $a _{1}=3$, $a _{n+1}a_{n}+\lambda a_{n+1}+u a_{n}^2=0$

(1) If $\lambda=0,u=-2$, find the formula of the $\{a_{n}\}$ of series

(2) If $\lambda = \frac{1}{k_{0}} (K_{0}\in N_{+},k_{0}\geq 2), u=-1$, prove $2+\frac{1}{3k_{0}+1}<a_{k_{0}+1}<2+\frac {1} {2k_{0}+1}$

three. Set function $f (x) =e^{mx}+x^{2}-mx$

(1) Proof: $f (x) $ in $ (-\infty,0) $ monotonically decreasing in $ (0,+\infty) $ monotonically incrementing

(2) if to any $x_{1},x_{2}\in [ -1,1]$, there are $|f (X_{1})-F (x_{2}) |\leq e-1$, the value of the $m$ range.

four. known function $f (x) =\ln (x+1), g (x) =k x (k \in R) $

(1) Proof: When $x>0$, $f (x) <x$

(2) Proof: When the $k<1$ exists $x_{0}>0$, so that arbitrary $x\in (0,x_{0}) $ constant has $f (x) >g (x) $

(3) Determine that all $k$ may only make the presence of $t>0$, to any $x\in (0,T) $ constant has $|f (x)-G (x) |<x^{2}$

Five. known function $f (x) =-2 (x+a) \ln x +x^{2}-2ax-2a^{2}+a (a>0) $

(1). Set $g (x) $ is the $f (x) $ of the derivative, discusses the monotonicity of $g (x) $

(2) Proof of existence $a\in (0,1) $ makes $f (x) \geq 0$ in the interval $ (1,+\infty) $ constant and $f (x) =0$ in the interval $ (1,+\infty) $ within the unique solution

Six. known $a>0$ function $f (x) =e^{ax}\sin x (X\geq 0) $ record $x_{n}$ for $f (x) $ from small to large $n$ extreme points, proving

(1) Sequence $f (X_{n}) $ is geometric series

(2) If the $a\geq \frac{1}{\sqrt{e^{2}-1}}$, then all $x_{n}<|f (X_{n}) |$ Heng set up

Seven. Set function $f (x) =\ln (x+1) +a (x^{2}-x), a \in R $

(1) Discussion of the number of function $f (x) $ Extreme points

(2) any $x,f (x) \geq 0$ is established to seek the value range of $a$

eight. set $x_{n}$ is the horizontal axis of the tangent of the curve $y=x^{2n+2}+1$ at point $ ($) $ at the intersection of the $x$ axes

(1) The formula for solving the $\{x_{n}\}$ of sequence numbers

(2) Remember $t_{n}=x_{1}^{2}x_{3}^{2}\cdot x_{2n-1}^{2}$, prove $t_{n}\geq \frac{1}{4n}$

nine. sequence $\{a_{n}\}$ Meet $a_{1}+2a_{2}+\cdot+na_{n}=4-\frac{n+2}{2^{n-1}}$

(1) The first $n$ and $t_{n}$ of the sequence $\{a_{n}\}$

(2) Make $b_{1}=a_{1}$, $b _{n}=\frac{t_{n-1}}{n}+ (1+\frac{1}{2}+\frac{1}{3}+\cdot+\frac{1}{n}) A_{n} (N\geq 2) $, proving $\{b_ {n}\}$ $n$ and $s_{n}$ meet $s_{n}<2+2\ln n$

10. Known quantity column $\{a_{n}\}$ satisfies $a_{1}\in n_{+},a_{1}\leq 36$ and $a_{n}\geq when 18$ $a_{n+1}=2a_{n}$, other $a_{n+1}=2a_{n}-36$, Kee Collection $m=\{a_{n}|n\in n_{+}\}$

(1) If $a_{1}=6$ writes out all elements of $m$

(2) If the set $m$ exists an element is a multiple of 3 then all elements in $m$ are multiples of 3

(3) The maximum value of the number of elements in the set $m$

11. known set $x=\{1,2,3\},y_{n}=\{1,2,3,\cdot,n\}$, set $s_{n}=\{(A, B) | a\in x,b\in y_{n}\}$ and $s_{n}$ element satisfies $a$ divisible by $b$ or $b$ divisible by $ A $, $f (n) $ represents the number of elements contained in the $s_{n}$, and writes out an expression of $f (n) $.

College Entrance Examination Questions