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! Chapter 1.8ThePrincipleofMathematicalInduction 1.8ThePrincipleofMathematicalInduction# The Theory1. For every natural number \(n\), let \(P_{n}\) be a proposition depending on \(n\), which can be true or false. If the conditions 1. \(P_{0}\) is true, 2. If \(P_{k}\) is true then \(P_{k+1}\) is true, are satisfied, then it follows that \(P_{n}\) is true for every natural number. 2. Note that if for a given integer \(m\) we change (a) for \(P_{m}\) is true, the respective conclusion is that \(P_{n}\) is true for all integers \(n\) greater than or equal to \(m\). 3. The Strong Version. Instead of (b), if we have that the truth of \(P_{m}\) for all \(m\frac{1}{1^{2}}+\frac{1}{2^{2}}+\cdots+\frac{1}{n^{2}}>\frac{3n}{ 2n+1}.\] 28. Show that all integers \(n\geq 8\) can be represented as a sum containing only the numbers \(3\) and \(5\). For example, \(8=3+5\), \(9=3+3+3\), \(10=5+5\). 29. Let \(n\) be a positive integer and let \(A\) be a set of \(2^{n+1}-1\) integers. Show that it is possible to choose \(2^{n}\) of them whose sum is a multiple of \(2^{n}\). 30. 1. Show that for all positive integers \(n\), \[\sum_{k=1}^{n-1}\frac{k}{(k+1)!}=1-\frac{1}{n!}\] 2. Show that for all integers \(n\geq 3\), the equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}=1\] is solvable in distinct positive integers. 31. Let \(F(n)\) denote the maximum number of regions that can be obtained drawing \(n\) straight lines in the plane. Determine a formula for \(F(n)\). 32. Show that all positive integers \(n\) can be represented as \[n=\pm 1^{2}\pm 2^{2}\pm\cdots\pm k^{2}\] for a certain positive integer \(k\) (depending on \(n\)) and some choice of signs \(+\) and \(-\). For example, \(4=1^{2}-2^{2}-3^{2}+4^{2}\). 33. Show that the number \[\frac{(2n)!(2m)!}{n!m!(n+m)!}\] is an integer for nonnegative integers \(m\) and \(n\)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