arithmetical series

简明释义

1. 算术级数; 2. 等差级数;

英英释义

例句

1.The concept of an arithmetical series (算术级数) is fundamental in algebra and number theory.

算术级数 的概念在代数和数论中是基础的。

2.You can find the nth term of an arithmetical series (算术级数) using the formula a_n = a + (n-1)d.

你可以使用公式 a_n = a + (n-1)d 来找到 算术级数 的第 n 项。

3.The sum of the first ten terms in an arithmetical series (算术级数) can be calculated using the formula S = n/2 * (a + l).

前十项的和可以使用公式 S = n/2 * (a + l) 来计算，n 是项数，a 是首项，l 是末项。

4.In an arithmetical series (算术级数), each term increases by a constant difference.

在一个 算术级数 中，每一项都以一个恒定的差值增加。

5.To calculate the total amount saved over a year with monthly deposits, you can model it as an arithmetical series (算术级数).

要计算一年内每月存款的总金额，你可以将其建模为一个 算术级数 。

作文

In mathematics, an arithmetical series is a sequence of numbers in which the difference between consecutive terms is constant. This concept is fundamental in understanding various mathematical principles and has numerous applications in real life. The basic form of an arithmetical series can be represented as follows: if we have a first term 'a' and a common difference 'd', the series can be expressed as a, a+d, a+2d, a+3d, and so on. The nth term of this series can be calculated using the formula: Tn = a + (n-1)d.The significance of the arithmetical series extends beyond simple number sequences. It is widely used in fields such as finance, physics, and computer science. For instance, when calculating the total cost of a series of items purchased at a constant price, one can use the arithmetical series to find the total amount spent. Suppose you buy five books, each costing $10. The total cost can be calculated as an arithmetical series: 10 + 10 + 10 + 10 + 10 = 50. Here, the first term is 10, and the common difference is 0 since all terms are identical.Moreover, the arithmetical series is essential for solving problems that involve summation. The sum of the first n terms of an arithmetical series can be calculated using the formula: S_n = n/2 * (2a + (n-1)d). This formula allows us to quickly compute the sum without having to add each term individually, which is particularly useful when dealing with large sets of data.In addition to practical applications, the arithmetical series has historical significance. Ancient mathematicians, such as those from the Greek and Indian civilizations, explored the properties of these series long before modern mathematics took shape. They recognized patterns within numbers and developed methods to calculate sums efficiently. This foundational work laid the groundwork for later advancements in algebra and calculus.Understanding the arithmetical series also enhances critical thinking skills. When students learn about this concept, they develop the ability to identify patterns and relationships among numbers. This skill is not only applicable in mathematics but also in everyday problem-solving scenarios. For example, when planning a budget, one might recognize that expenses can be categorized into fixed and variable costs, drawing parallels to the fixed nature of the arithmetical series.In conclusion, the arithmetical series is a vital mathematical concept that plays a crucial role in various disciplines. Its simplicity and applicability make it a valuable tool for both theoretical exploration and practical decision-making. Whether in academic settings or daily life, mastering the arithmetical series equips individuals with the skills necessary to navigate numerical challenges effectively. As we continue to encounter mathematical concepts throughout our lives, a firm grasp of the arithmetical series will undoubtedly serve us well in a multitude of scenarios.

在数学中，算术级数是一个数字序列，其中连续项之间的差是恒定的。这个概念在理解各种数学原理方面是基础，并且在现实生活中有许多应用。算术级数的基本形式可以表示如下：如果我们有一个首项 'a' 和一个公差 'd'，则该系列可以表示为 a, a+d, a+2d, a+3d，依此类推。这个系列的第 n 项可以使用公式计算：Tn = a + (n-1)d。算术级数的重要性超越了简单的数字序列。它在金融、物理和计算机科学等领域被广泛使用。例如，当计算一系列以固定价格购买的商品的总成本时，可以使用算术级数来找出总花费。假设你买了五本书，每本书的价格是10美元。总成本可以计算为一个算术级数：10 + 10 + 10 + 10 + 10 = 50。在这里，首项是10，而公差是0，因为所有项都是相同的。此外，算术级数对于解决涉及求和的问题至关重要。前 n 项的和可以使用公式计算：S_n = n/2 * (2a + (n-1)d)。这个公式使我们能够快速计算和，而不必逐项相加，这在处理大量数据时尤其有用。除了实际应用之外，算术级数还具有历史意义。古代数学家，如希腊和印度文明的人，早在现代数学形成之前就探索了这些系列的性质。他们认识到数字之间的模式，并开发了有效计算和的方法。这些基础工作为后来的代数和微积分的发展奠定了基础。理解算术级数也提高了批判性思维能力。当学生学习这个概念时，他们培养了识别数字之间的模式和关系的能力。这种技能不仅适用于数学，而且适用于日常问题解决场景。例如，在制定预算时，人们可能会认识到费用可以分为固定成本和可变成本，从而与算术级数的固定性质进行比较。总之，算术级数是一个重要的数学概念，在各个学科中发挥着关键作用。它的简单性和适用性使其成为理论探索和实际决策的重要工具。无论是在学术环境还是日常生活中，掌握算术级数都能为个人提供有效应对数字挑战所需的技能。随着我们继续在生活中遇到数学概念，对算术级数的牢固掌握无疑将在多种场景中为我们服务。