recurrence formula

简明释义

递推公式

英英释义

例句

1.Using the recurrence formula 递推公式, we can compute the factorial of a number recursively.

使用 递推公式 recurrence formula，我们可以递归地计算一个数字的阶乘。

2.We derived the recurrence formula 递推公式 for the geometric series in our last math class.

我们在上次数学课上推导了几何级数的 递推公式 recurrence formula。

3.In solving the Fibonacci sequence, we often use the recurrence formula 递推公式 F(n) = F(n-1) + F(n-2).

在解决斐波那契数列时，我们常常使用 递推公式 recurrence formula F(n) = F(n-1) + F(n-2)。

4.The recurrence formula 递推公式 is essential for calculating the number of ways to climb stairs.

这个 递推公式 recurrence formula 对于计算爬楼梯的方式数量是至关重要的。

5.The recurrence formula 递推公式 helps us find the terms of a sequence efficiently.

这个 递推公式 recurrence formula 帮助我们有效地找到数列的项。

作文

In the realm of mathematics, particularly in the study of sequences and series, the concept of a recurrence formula plays a pivotal role. A recurrence formula is essentially an equation that recursively defines a sequence. Each term of the sequence is defined as a function of its preceding terms. This is a powerful tool that allows mathematicians to express complex relationships in a manageable form.To illustrate, consider the Fibonacci sequence, one of the most famous examples in mathematics. The Fibonacci sequence is defined by the recurrence relation: F(n) = F(n-1) + F(n-2), with seed values F(0) = 0 and F(1) = 1. Here, each term is the sum of the two preceding terms. The beauty of this recurrence formula lies in its simplicity and the way it generates an infinite sequence of numbers that have fascinating properties and applications in various fields such as computer science, biology, and even art.Understanding how to derive and utilize a recurrence formula can significantly enhance problem-solving abilities. For instance, when faced with a complex problem that involves a sequence, rather than trying to compute each term individually, one can apply a recurrence formula to find the nth term efficiently. This method not only saves time but also provides deeper insights into the structure of the problem.Moreover, recurrence formulas are not limited to simple sequences. They can be used to describe more complicated scenarios, such as those found in dynamic programming. In computer science, many algorithms rely on recurrence formulas to optimize solutions to problems by breaking them down into smaller subproblems. For example, the well-known algorithm for calculating the minimum path in a grid can be expressed using a recurrence formula that considers the best path to each cell based on previously calculated paths.The application of recurrence formulas extends beyond academia into real-world scenarios. In finance, for example, the calculation of compound interest can be modeled using a recurrence formula. If we let A(n) represent the amount of money after n periods, the recurrence formula can be expressed as A(n) = A(n-1) * (1 + r), where r is the interest rate. This allows investors to predict future earnings based on past performance, demonstrating the practical utility of these mathematical constructs.In conclusion, the recurrence formula is a fundamental concept that bridges theoretical mathematics and practical application. Its ability to define sequences and solve complex problems efficiently makes it an invaluable tool in various fields. By mastering recurrence formulas, students and professionals alike can enhance their analytical skills and better understand the intricate patterns that govern both mathematical theories and real-world phenomena. As we continue to explore the vast landscape of mathematics, the significance of recurrence formulas will undoubtedly remain a cornerstone of our understanding and innovation.

在数学领域，特别是在序列和级数的研究中，递推公式的概念发挥着关键作用。递推公式本质上是一个递归定义序列的方程。序列中的每一项都被定义为其前几项的函数。这是一个强大的工具，使数学家能够以可管理的形式表达复杂的关系。为了说明这一点，考虑斐波那契数列，这是数学中最著名的例子之一。斐波那契数列由递推关系定义：F(n) = F(n-1) + F(n-2)，种子值为F(0) = 0和F(1) = 1。在这里，每一项都是前两项的和。这个递推公式的美在于它的简单性，以及它生成无限数字序列的方式，这些数字具有迷人的属性和在计算机科学、生物学甚至艺术等各个领域的应用。理解如何推导和利用递推公式可以显著增强解决问题的能力。例如，当面临涉及序列的复杂问题时，与其试图逐项计算，不如应用递推公式来有效地找到第n项。这种方法不仅节省时间，还提供了对问题结构的更深刻见解。此外，递推公式并不限于简单的序列。它们可以用于描述更复杂的场景，例如动态规划中发现的场景。在计算机科学中，许多算法依赖于递推公式通过将问题分解为更小的子问题来优化解决方案。例如，计算网格中最小路径的著名算法可以用递推公式表示，该公式考虑基于先前计算的路径到达每个单元格的最佳路径。递推公式的应用超越了学术界，进入现实世界的场景。例如，在金融领域，复利的计算可以用递推公式建模。如果我们让A(n)表示n个周期后的金额，递推公式可以表示为A(n) = A(n-1) * (1 + r)，其中r是利率。这使得投资者能够根据过去的表现预测未来的收益，展示了这些数学构造的实际效用。总之，递推公式是一个基本概念，它连接了理论数学和实际应用。它定义序列和有效解决复杂问题的能力使其成为各个领域中不可或缺的工具。通过掌握递推公式，学生和专业人士都可以增强他们的分析能力，更好地理解支配数学理论和现实现象的复杂模式。随着我们继续探索广阔的数学领域，递推公式的重要性无疑将继续成为我们理解和创新的基石。