assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 英国补考|组合学代写Combinatorics代考|Relation between Recursive and Direct Formulas

Is it possible to define the same sequence by formulas of two types: by a recursive formula and a direct one? There is no exact answer to this question in a general setting. It depends on the range of methods allowed for the construction of the formulas of both types. Having no intention to give a comprehensive answer, we provide some sensible recommendations to find the answer to this reasonable question in important practical cases.

First, we consider the transition from a direct formula to a recursive one. This transition is always possible, though there is not much sense in it as it can be performed in infinitely many ways. There is a simple example that illustrates this. Let us have the sequence
$$a_n=2^{n-1}$$
This is a geometric progression with the common ratio of 2 and the initial term 1.
Below, there are several transformations of this direct formula into a recursive one:

1. $a_n-a_{n-1}=2^{n-1}-2^{n-2}=2^{n-2}$, hence
$$a_n=a_{n-1}+2^{n-1} ;$$ $a_n \cdot a_{n-1}=2^{n-1} \cdot 2^{n-2}=2^{2 n-3}$, which yields
2. $$3. a_n=\frac{2^{2 n-3}}{a_{n-1}} 4.$$
5. $\frac{a_n}{a_{n-1}}=\frac{2^{n-1}}{2^{n-2}}=2$, hence $a_n=2 a_{n-1}$
6. Thus, the sequence defined by the direct formula $a_n=2^{n-1}$ can also be defined by recurrence relations:
7. $$8. a_1=1, a_n=a_{n-1}+2^{n-1}, 9.$$
10. or
11. $$12. a_1=1, a_n=\frac{2^{2 n-3}}{a_{n-1}}, 13.$$
14. or
15. $$16. a_1=1, a_n=2 \cdot a_{n-1}, 17.$$
18. or by infinite number of others.
19. Serious problems can be encountered while attempting the reverse transition from a recursive formula to a direct one. In fact, such a transition is not always possible. And when it actually is, performing it requires more than just technical exercise. In most cases, the success of transition is down to the combination of erudition, creativity, and luck.

## 英国补考|组合学代写Combinatorics代考|Recurrence Relations in Combinatorial Problems

Is there any relation between sequences and combinatorial problems? Yes, there is. Moreover, sequences appear in combinatorial problems mostly in the context of recurrence relations.

Example 1.35. There is a path leading to a rabbit hole. The path is a line of squares. Walking on this path a rabbit jumps into the nearest square or one square further, randomly choosing from these two options. How many ways are there for the rabbit to reach the n-th square?

In order to solve this problem, we need to define a formula (direct or recursive) of a certain sequence. Which sequence is that? And how is this sequence related to the problem?
Denote the sought amount in any appropriate way, say, by $\gamma_n$. The index $n$ is not only appropriate here but even necessary as the answer should depend on $n$. Having answered the question of the problem, that is, having determined the amount of ways for the rabbit to reach the $n$-th square, we will find the answer to an infinite amount of questions concerning the exact values of $n: 1,2,3,4, \ldots$. Having the formula for arbitrary $n$, we will know $\gamma_1$, and $\gamma_2$, and $\gamma_3$, and so on. In other words, we will know the law of expansion of the sequence $\left(\gamma_n\right)$, and thus will be able to calculate every element (at least potentially). Therefore, although the question seems to be posed in respect of one number, it actually requires us to find the law of expansion of a certain numeric sequence. The sequence, the $n$-th element of which denotes the number of ways, in which the rabbit can reach the $n$-th square.

How can this problem be addressed? What could we start with? First, we must clearly understand the situation: what is known and what is to be found. Our aim is clear: we need to guess the law of a certain numerical sequence. What do we know about this sequence? What does the statement of the problem tell us about it? Obviously, the statement of the problem describes the law of the sequence. It appears to be nonsense: we need to find a rule, which is known from the very beginning. However, at the beginning of the problem and in the question we encounter essentially different laws of expansion of the sequence $\left(\gamma_n\right)$. In the statement of the problem, there is a purely descriptive characterization of the sequence. Relying solely on this characterization it is very hard to determine, say, $\gamma_{20}$. And the task is to discover the quantitative law of the sequence building upon the qualitative description. We have nothing to begin with, except for the aggregation of actual data about the sequence. We directly calculate (thoroughly considering different options) several initial members.

# 组合学代考

## 英国补考|组合学代写Combinatorics代考|Relation between Recursive and Direct Formulas

$$a_n=2^{n-1}$$

1. $a_n-a_{n-1}=2^{n-1}-2^{n-2}=2^{n-2}$ ，因此
$$a_n=a_{n-1}+2^{n-1} ;$$
$a_n \cdot a_{n-1}=2^{n-1} \cdot 2^{n-2}=2^{2 n-3}$, , 产生
2. $\$ \$$3. a_n =\backslash f frac{2^{2 n-3}}\left{a_{-}{n-1}\right} 4. \\ 5. \frac{a_n}{a_{n-1}}=\frac{2^{n-1}}{2^{n-2}}=2 ， \quad 因此 a_n=2 a_{n-1} 6. 因此，由直接公式定义的序列 a_n=2^{n-1} 也可以由递归关系定义: 7. \\ 8. a_1=1, a_n=a_ {n-1}+2^{\wedge}{n-1} 9. \ \$$
10. 或者
11. $\$ \$$12. a_1=1, a_n= \left{\right. frac \left{2^{\wedge}{2 \mathrm{n}-3}\right}\left{a_{-}{n-1}\right}, 13. \ \$$
14. 或者
15. $\$ \
16. a_1=1, a_n=2 $\backslash c d o t a_{-}{n-1}$,
17. \$\$
18. 或无数其他人。
19. 在尝试从递归公式到直接公式的反向转换时，可能会遇到严重的问题。事实上，这种转变并不总是可能的。实际上，执行它需要的不仅仅是技术练习。在 大多数情况下，转型的成功取决于博学、创造力和运气的结合。

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师