Posted on Categories:基础编程, 计算机代写

# 计算机代写|基础编程代写Fundamental of Programming代考|Mathematical Induction

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 计算机代写|基础编程代写Fundamental of Programming代考|Mathematical Induction

Let $P(n)$ be some statement about the integer $n$; for example, $P(n)$ might be ” $n$ times $(n+3)$ is an even number,” or “if $n \geq 10$, then $2^n>n^3$.” Suppose we want to prove that $P(n)$ is true for all positive integers $n$. An important way to do this is:
a) Give a proof that $P(1)$ is true.
b) Give a proof that “if all of $P(1), P(2), \ldots, P(n)$ are true, then $P(n+1)$ is also true”; this proof should be valid for any positive integer $n$.
As an example, consider the following series of equations, which many people have discovered independently since ancient times:
$$\begin{gathered} 1=1^2, \ 1+3=2^2, \ 1+3+5=3^2, \ 1+3+5+7=4^2, \ 1+3+5+7+9=5^2 . \end{gathered}$$
We can formulate the general property as follows:
$$1+3+\cdots+(2 n-1)=n^2$$

## 计算机代写|基础编程代写Fundamental of Programming代考|Numbers, Powers, and Logarithms

Let us now begin our study of numerical mathematics by taking a good look at the numbers we are dealing with. The integers are the whole numbers
$$\ldots,-3,-2,-1,0,1,2,3, \ldots$$
(negative, zero, or positive). A rational number is the ratio (quotient) of two integers, $p / q$, where $q$ is positive. A real number is a quantity $x$ that has a decimal expansion
$$x=n+0 . d_1 d_2 d_3 \ldots$$
where $n$ is an integer, each $d_i$ is a digit between 0 and 9 , and the sequence of digits doesn’t end with infinitely many $9 \mathrm{~s}$. The representation (1) means that
$$n+\frac{d_1}{10}+\frac{d_2}{100}+\cdots+\frac{d_k}{10^k} \leq x<n+\frac{d_1}{10}+\frac{d_2}{100}+\cdots+\frac{d_k}{10^k}+\frac{1}{10^k},$$
for all positive integers $k$. Examples of real numbers that are not rational are $\pi=3.14159265358979 \ldots$, the ratio of circumference to diameter in a circle; $\phi=1.61803398874989 \ldots$, the golden ratio $(1+\sqrt{5}) / 2$ (see Section 1.2.8).
A table of important constants, to forty decimal places of accuracy, appears in Appendix A. We need not discuss the familiar properties of addition, subtraction, multiplication, division, and comparison of real numbers.

Difficult problems about integers are often solved by working with real numbers, and difficult problems about real numbers are often solved by working with a still more general class of values called complex numbers. A complex number is a quantity $z$ of the form $z=x+i y$, where $x$ and $y$ are real and $i$ is a special quantity that satisfies the equation $i^2=-1$. We call $x$ and $y$ the real part and imaginary part of $z$, and we define the absolute value of $z$ to be
$$|z|=\sqrt{x^2+y^2}$$

## 计算机代写|基础编程代写Fundamental of Programming代考|Mathematical Induction

a)给出$P(1)$为真的证明。
b)给出“如果$P(1), P(2), \ldots, P(n)$全部为真，那么$P(n+1)$也为真”的证明;这个证明对任何正整数$n$都是有效的。

$$\begin{gathered} 1=1^2, \ 1+3=2^2, \ 1+3+5=3^2, \ 1+3+5+7=4^2, \ 1+3+5+7+9=5^2 . \end{gathered}$$

$$1+3+\cdots+(2 n-1)=n^2$$

## 计算机代写|基础编程代写Fundamental of Programming代考|Numbers, Powers, and Logarithms

$$\ldots,-3,-2,-1,0,1,2,3, \ldots$$
(负、零或正)。有理数是两个整数$p / q$之比(商)，其中$q$为正数。实数是一个有十进制展开的量$x$
$$x=n+0 . d_1 d_2 d_3 \ldots$$

$$n+\frac{d_1}{10}+\frac{d_2}{100}+\cdots+\frac{d_k}{10^k} \leq x<n+\frac{d_1}{10}+\frac{d_2}{100}+\cdots+\frac{d_k}{10^k}+\frac{1}{10^k},$$

$$|z|=\sqrt{x^2+y^2}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。