线性代数网课代修|ENGO361代写|ENGO361英文辅导|ENGO361

如果你也在线性代数linearalgebra这个学科遇到相关的难题,请随时添加vx号联系我们的代写客服。我们会为你提供专业的服务。 linearalgebra™长期致力于留学生网课服务,涵盖各个网络学科课程:金融学Finance,经济学Economics,数学Mathematics,会计Accounting,文学Literature,艺术Arts等等。除了网课全程托管外,linearalgebra™也可接受单独网课任务。无论遇到了什么网课困难,都能帮你完美解决! [AIME 1997] The function \(f\), defined by \[f(x)=\frac{ax+b}{cx+d},\] where \(a\), \(b\), \(c\), and \(d\) are nonzero real numbers, has the properties \(f(19)=19\), \(f(97)=97\), and \(f(f(x))=x\), for all values of \(x\), except \(-\frac{d}{c}\). Find the unique number that is not in the range of \(f\). 31. Let \(\{r_{n}\}_{n\geq 1}\) be a sequence of real numbers such that \(r_{1}=2\), and \(r_{n}=r_{1}r_{2}\cdots r_{n-1}+1\) for \(n=2,3,\ldots\). Find the first 10 terms of this sequence. Show that \[1-\frac{1}{r_{1}}-\frac{1}{r_{2}}-\cdots-\frac{1}{r_{n}}=\frac{1}{r_{1}r_{2} \cdots r_{n}}\] for all \(n\geq 1\). (Unit fractions \(\frac{1}{n}\), where \(n\) is a positive integer, are also called _Egyptian fractions_.)32. Consider the sequence \(a_{0},a_{1},\ldots\) with \(a_{0}=\frac{1}{2}\) and \[a_{k}=a_{k-1}+\frac{a_{k}a_{k-1}}{2008}\quad\text{for}\quad k=1,2,\ldots.\] Compute \(a_{2008}\). 33. [AIME2 2005, Zuming Feng] Let \(m\) be a positive integer, and let \(a_{0},\ldots,a_{m}\) be a sequence of real numbers such that \(a_{0}=37\), \(a_{1}=72\), \(a_{m}=0\) and \[a_{k+1}=a_{k-1}-\frac{3}{a_{k}}\] for \(k=1,2,\ldots,m-1\). Find \(m\). 34. [AIME 1987] Compute \[\frac{(10^{4}+324)(22^{4}+324)(34^{4}+324)(46^{4}+324)(58^{4}+324)}{(4^{4}+32 4)(16^{4}+324)(28^{4}+324)(40^{4}+324)(52^{4}+324)}.\] 35. [Putnam 1979] Let \(x_{n}\) be a sequence of nonzero real numbers such that \[x_{n}=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}\] for \(n=3,4,\ldots\). Establish necessary and sufficient conditions on \(x_{1}\) and \(x_{2}\) for \(x_{n}\) to be an integer for infinitely many values of \(n\). 1.5Simplifying Square Root Expressions**Definition**1. If \(x^{2}=N\), then \(x\) is a square root of \(N\). The principal square root of a nonnegative number is its nonnegative square root. The symbol \(\sqrt{N}\) represents the principal square root of \(N\). The negative square root of \(N\) is written \(-\sqrt{N}\). The following properties hold: 1. \(\left(\sqrt{a}\right)^{2}=a\quad(a\geq 0)\) 2. \(\sqrt{a^{2}}=|a|=\begin{cases}a&\text{if }a>0 0&\text{if }a=0 -a&\text{if }a0)\) 5. \(\left(\sqrt{a}\right)^{n}=\sqrt{a^{n}}\quad(a\geq 0)\)** 2. Evaluate \(\left(\sqrt{3}+\sqrt{5}+\sqrt{7}\right)\left(\sqrt{3}+\sqrt{5}-\sqrt{7}\right) \left(\sqrt{3}-\sqrt{5}+\sqrt{7}\right)\left(-\sqrt{3}+\sqrt{5}+\sqrt{7}\right)\). 3. Evaluate \(\frac{\sqrt{15}+\sqrt{35}+\sqrt{21}+5}{\sqrt{3}+2\sqrt{5}+\sqrt{7}}\). 4. Evaluate \[\frac{1}{2\sqrt{1}+\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\cdots+\frac{1}{10 0\sqrt{99}+99\sqrt{100}}.\] 5. Evaluate \[\sqrt{1+\frac{1}{1^{2}}+\frac{1}{2^{2}}}+\sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{ 2}}}+\cdots+\sqrt{1+\frac{1}{2008^{2}}+\frac{1}{2009^{2}}}.\] 6. Simplify \(\sqrt{x^{2}-2x+1}-\sqrt{x^{2}-4x+4}+\sqrt{x^{2}+6x+9}\). 7. Given that \(P_{1}=\sqrt{(a+c)^{2}+b^{2}},P_{2}=\sqrt{a^{2}+(b+c)^{2}}\), and \(P_{3}=\sqrt{(a+b)^{2}+c^{2}}\), where \(a>b>c>0\), find the maximum and minimum of \[\{P_{1}P_{2},P_{1}P_{3},P_{2}P_{3},P_{1}^{2},P_{2}^{2},P_{3}^{2}\}.\] 8. Prove that \(\sqrt{a^{2}+\frac{1}{b^{2}}+\frac{a^{2}}{(ab+1)^{2}}}=\bigg{|}a+\frac{1}{b}- \frac{a}{ab+1}\bigg{|}\). 9. Compute \(\sqrt{1+1999^{2}}+\frac{1999^{2}}{2000^{2}}-\frac{1}{2000}\).10. Simplify \(\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-\sqrt{2}\cdot\sqrt{6-2\sqrt{5}}\). 11. Evaluate \(\sqrt{8+\sqrt{40+8\sqrt{5}}}+\sqrt{8-\sqrt{40+8\sqrt{5}}}\). 12. If \(\sqrt{19-8\sqrt{3}}=x+y\), where \(x\) is an integer and \(0\leq y0\). 3. two complex, non-real solutions that are conjugates of each other if \(b^{2}-4acn\), be the solutions to \(x-\dfrac{1}{x}=1990\). Find the value of \[n\left(\dfrac{1-m^{3}}{1-m}\right)+1.\] 13. What is the least integral value of \(t\) for which the roots of the equation \(x^{2}+2(t+1)x+9t-5=0\) are unequal negative numbers? 14. What are all integers \(k\) for which \(x^{2}+kx+k+17=0\) has integral roots? **The Problems!** 15. Determine the nature of the solutions of each equation: 1. \(4x^{2}-12x+9=0\). 2. \(y^{2}=\dfrac{y}{2}+\dfrac{3}{5}\). 16图片描述

发表回复

您的电子邮箱地址不会被公开。 必填项已用 * 标注