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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! (AHSME 1992) If \[\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}\] for three different positive numbers \(x\), \(y\) and \(z\), find \(\frac{x}{y}\). 10. Let \(a_{1}\), \(a_{2},\ldots\) be a sequence defined recursively by \(a_{1}=2\) and \[a_{n+1}=1-\frac{1}{a_{n}}\quad\mbox{for $n=1$, $2,\ldots$}.\] Find \(a_{2009}\). 11. Simplify \[\frac{3x^{2}+9x+7}{x+1}-\frac{2x^{2}+4x-3}{x-1}-\frac{x^{3}+x+1}{x^{2}-1}.\] 12. Compute \(\frac{22223^{3}+11112^{3}}{22223^{3}+11111^{3}}\). 13. Simplify \[\frac{a^{2}(\frac{1}{b}-\frac{1}{c})+b^{2}(\frac{1}{c}-\frac{1}{a})+c^{2}( \frac{1}{a}-\frac{1}{b})}{a(\frac{1}{b}-\frac{1}{c})+b(\frac{1}{c}-\frac{1}{a })+c(\frac{1}{a}-\frac{1}{b})}.\] 14. (Phillips Exeter Academy math materials, Richard Parris) Choose positive values for \(x_{0}\) and \(x_{1}\) that no one else would think of, then calculate seven more terms of the sequence defined recursively by \[x_{n}=\frac{1+x_{n-1}}{x_{n-2}}\quad\mbox{for $n=2$, $3,\ldots$}.\] What can you tell? **The problems!** 15. Simplify \[\frac{a^{4}-a^{2}b^{2}}{(a-b)^{2}}\div\frac{a(a+b)}{b^{2}}\cdot\frac{b^{3}}{a}.\]16. Which of the fractions \[A=\frac{7890123456}{8901234567}\quad\text{and}\quad B=\frac{7890123455}{8901234566}\] is greater? 17. Simplify \[\frac{x^{2}+2x-3}{x^{3}+7x^{2}+7x-15}.\] 18. Simplify \[\left(\frac{x^{2}-4x-4}{x+1}-\frac{x^{2}+8x+13}{x+2}\right)\div(11x^{2}+33x+21).\] 19. Let \(x_{0}\), \(x_{1}\), \(x_{2},\ldots\) be a sequence defined recursively by \(x_{0}=\pi\) and \[x_{n}=\frac{x_{n-1}-1}{x_{n-1}}\quad\text{for }n=1,\,2,\ldots.\] Compute \(x_{2009}\). 20. Simplify \[\frac{1}{x-1}-\frac{1}{x+1}-\frac{2}{x^{2}+1}-\frac{4}{x^{4}+1}-\frac{8}{x^{8 }+1}.\] 21. Let \(x_{0}\), \(x_{1}\), \(x_{2},\ldots\) be a sequence defined recursively by \(x_{0}=e\) and \[x_{n}=\frac{1-x_{n-1}}{1+x_{n-1}}\quad\text{for }n=1,\,2,\ldots.\] Compute \(x_{2009}\). 22. Simplify \[\frac{(1+ax)^{2}-(a+x)^{2}}{(1+bx)^{2}-(b+x)^{2}}\div\frac{(1+ay)^{2}-(a+y)^{2 }}{(1+by)^{2}-(b+y)^{2}}.\] 23. Simplify \[\frac{x-c}{(x-a)(x-b)}+\frac{b-c}{(a-b)(x-b)}+\frac{b-c}{(b-a)(x-a)}.\] 24. (HMMT 2002) Real numbers \(a\), \(b\) and \(c\) satisfy the equations \(a+b+c=26\) and \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=28\). Find the value of \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a}{c}+\frac{c}{b}+\frac{b}{a}.\] 25. Simplify \[\frac{1}{a}+\frac{1}{a^{2}-a}+\frac{1}{a^{2}-3a+2}+\frac{1}{a^{2}-5a+6}.\] 26. Simplify \[\left(\frac{x+2}{x+3}-\frac{x+1}{x+2}+\frac{x+4}{x+5}-\frac{x+3}{x+4}\right) \div\frac{x^{2}+7x+13}{x^{2}+8x+15}.\]27. * Simplify \[\left(\frac{a_{1}^{4}}{a_{1}^{3}+a_{1}^{2}a_{2}+a_{1}a_{2}^{2}+a_{2}^{ 3}}+\frac{a_{2}^{4}}{a_{2}^{3}+a_{2}^{2}a_{3}+a_{2}a_{3}^{2}+a_{3}^{3}}+\cdots+ \frac{a_{n}^{4}}{a_{n}^{3}+a_{n}^{2}a_{1}+a_{n}a_{1}^{2}+a_{1}^{3}}\right)\] \[-\left(\frac{a_{2}^{4}}{a_{1}^{3}+a_{1}^{2}a_{2}+a_{1}a_{2}^{2}+a_{ 2}^{3}}+\frac{a_{3}^{4}}{a_{2}^{3}+a_{2}^{2}a_{3}+a_{2}a_{3}^{2}+a_{3}^{3}}+ \cdots+\frac{a_{1}^{4}}{a_{n}^{3}+a_{n}^{2}a_{1}+a_{n}a_{1}^{2}+a_{1}^{3}} \right).\] 28. * (AMC12 2001, Richard Parris) Consider sequences of positive real numbers of the form \(x,2000,y,\dots\), in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of \(x\) does the term 2001 appear somewhere in the sequence? 29. Given real numbers \(x\), \(y\) and \(z\) with \(xyz=1\), compute \[\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}.\] 30. * Let \(a,\ b,\ c\) and \(d\) be distinct (not necessarily real) numbers such that \(a+b+c+d=9\) and \(a^{2}+b^{2}+c^{2}+d^{2}=10\). Evaluate \[\frac{a^{5}}{(a-b)(a-c)(a-d)}+\frac{b^{5}}{(b-a)(b-c)(b-d)}+\frac{c^{5}}{(c-a) (c-b)(c-d)}+\frac{d^{5}}{(d-a)(d-b)(d-c)}.\] 1.4Partial Fractions and Telescoping Sums**Definition**1. Find constants \(a\) and \(c\) such that the expressions \[\frac{x+2}{x^{2}-3x}\quad\text{and}\quad\frac{a}{x}+\frac{c}{x-3}\] are equivalent. We say that the fraction \(\frac{x+2}{x^{2}-3x}\) is _decomposed_ as the sum of the _partial-fractions_\(\frac{a}{x}\) and \(\frac{c}{x-3}\). This algebraic process is called _partial fraction decomposition_. **2. Decompose \(\frac{4x^{2}-3x-4}{x^{3}+x^{2}-2x}\). 3. Decompose \(\frac{13x+14}{2x^{3}-13x^{2}-7x}\). 4. So far, we have dealt with examples with their denominators being the products of distinct linear expressions. (Note that \(x^{2}-3x=x(x-3)\), \(x^{3}+x^{2}-2x=x(x-1)(x-2)\), etc.) How is a fraction decomposed when a polynomial divides its denominator multiple times? We illustrate our method with the following example. Find \(a\), \(b\), and \(c\) such that \[\frac{x^{2}+2x+3}{(x+2)^{3}}=\frac{a}{x+2}+\frac{b}{(x+2)^{2}}+\frac{c}{(x+2) ^{3}}.\] 5. Decompose \(\frac{x^{3}-4x-1}{x(x-1)^{3}}\). 6. Decompose \(\frac{x^{4}}{x^{2}+4x+4}\). 7. A polynomial is _reducible_ if it can be written as the product of two polynomials with degree at least 1. Otherwise, it is _irreducible_. For example, \(x^{3}+1\) is reducible (why?) and \(x^{2}-x+1\) is irreducible. Here is an example dealing with a fraction where an irreducible polynomial divides its denominator. Find \(a\), \(b\), and \(c\) such that \[\frac{1}{x^{3}+1}=\frac{a}{x+1}+\frac{bx+c}{x^{2}-x+1}.\] 8. Decompose \(\frac{1}{(x^{2}+1)(x^{2}+4)}\). 9. Decompose \(\frac{5x^{3}-3x^{2}+2x-1}{x^{4}+x^{2}}\). 10. Decompose \(\frac{x^{2}}{x^{4}-1}\).11. Decompose \(\dfrac{x+4}{x^{3}+2x-3}\). 12. Decompose \(\dfrac{x^{4}+3x^{2}-4x+5}{(x-1)^{2}(x^{2}+1)}\). **Telescoping sums** 13. Evaluate \[\dfrac{1}{2!}+\dfrac{2}{3!}+\cdots+\dfrac{2008}{2009!}.\] 14. Given that \[\dfrac{1}{1\times 2}+\dfrac{1}{3\times 4}+\cdots+\dfrac{1}{2007\times 2008}= \dfrac{1}{n+1}+\dfrac{1}{n+2}+\cdots+\dfrac{1}{2n},\] compute \(n\). 15. Simplify \[\dfrac{1}{x-1}+\dfrac{1}{(x-1)(x-2)}+\dfrac{1}{(x-2)(x-3)}+\cdots+\dfrac{1}{(x- 2008)(x-2009)}.\] 16. [AHSME 1991] If \(T_{n}=1+2+\cdots+n\) and \[P_{n}=\dfrac{T_{2}}{T_{2}-1}\cdot\dfrac{T_{3}}{T_{3}-1}\cdots\dfrac{T_{n}}{T_{n }-1},\] for \(n\geq 2\), compute \(P_{1993}\). 17. Evaluate \[\dfrac{3}{1!+2!+3!}+\dfrac{4}{2!+3!+4!}+\cdots+\dfrac{2001}{1999!+2000!+2001!}.\] 18. Find the value of \[\dfrac{1}{3^{2}+1}+\dfrac{1}{4^{2}+2}+\dfrac{1}{5^{2}+3}+\cdots.\] 19. Determine the value of the sum \[\dfrac{3}{1^{2}\cdot 2^{2}}+\dfrac{5}{2^{2}\cdot 3^{2}}+\dfrac{7}{3^{2}\cdot 4^{2} }+\cdots+\dfrac{29}{14^{2}\cdot 15^{2}}.\] 20. Compute \[\sum_{n=0}^{\infty}\dfrac{n}{n^{4}+n^{2}+1}.\] **The Problems!** 21. Decompose \(\dfrac{27-17x}{7x^{2}+41x-6}\). 22. Decompose \(\dfrac{3x-4}{x^{2}-3x+2}\).23. Decompose \(\frac{x^{2}+1}{(x+1)^{2}(x+2)}\). 24. Decompose \(\frac{4x^{3}-7x}{x^{4}-5x^{2}+4}\). 25. Decompose \(\frac{5x^{2}+12x+29}{2x^{3}+10x^{2}-3x-15}\). 26. Simplify \[\frac{2}{(x+1)(x+3)}+\frac{2}{(x+3)(x+5)}+\cdots+\frac{2}{(x+2007)(x+2009)}.\] 27. [AIME1 2002, Florin Pop] Consider the sequence defined by \(a_{k}=\frac{1}{k^{2}+k}\) for \(k\) greater than or equal to 1. Given that \[a_{m}+a_{m+1}+\cdots+a_{n-1}=\frac{1}{29},\] for positive integers \(m\) and \(n\) with \(m\) less than \(n\), find \(m+n\). 28. [AIME2 2008] The sequence \(\{a_{n}\}\) is defined by \[a_{0}=1,a_{1}=1,\quad\text{and}\quad a_{n}=a_{n-1}+\frac{a_{n-1}^{2}}{a_{n-2}} \quad\text{for}\quad n\geq 2.\] The sequence \(\{b_{n}\}\) is defined by \[b_{0}=1,b_{1}=3,\quad\text{and}\quad b_{n}=b_{n-1}+\frac{b_{n-1}^{2}}{b_{n-2}} \quad\text{for}\quad n\geq 2.\] Find \(\frac{b_{32}}{a_{32}}\). 29. [AIME2 2002, Leo Schneider] Find the integer closest to \(1000\sum_{n=3}^{1000}\frac{1}{n^{2}-4}\). 30图片描述

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