explicit formula

简明释义

显式公式

英英释义

例句

1.The explicit formula 显式公式 for the Fibonacci sequence allows us to find terms without recursion.

斐波那契数列的显式公式 explicit formula使我们能够不使用递归来找到项。

2.In physics, we often use an explicit formula 显式公式 to determine the trajectory of a projectile.

在物理学中，我们经常使用显式公式 explicit formula来确定抛射物的轨迹。

3.Using an explicit formula 显式公式 simplifies the process of calculating the area of a circle.

使用显式公式 explicit formula简化了计算圆面积的过程。

4.The explicit formula 显式公式 for the sum of an arithmetic series is essential for quick calculations.

算术级数和的显式公式 explicit formula对于快速计算至关重要。

5.In mathematics, an explicit formula 显式公式 can provide a direct way to calculate the nth term of a sequence.

在数学中，显式公式 explicit formula可以提供一种直接计算序列第n项的方法。

作文

In mathematics and various scientific fields, the term explicit formula refers to a mathematical expression that provides a direct way to calculate a specific value without needing to rely on iterative methods or recursion. An explicit formula allows one to determine the nth term of a sequence or the result of a function instantly, using a finite number of operations. This concept is particularly valuable in areas such as algebra, calculus, and number theory, where efficient computation is essential.For example, consider the arithmetic sequence defined by the first term a and a common difference d. The explicit formula for the nth term of this sequence can be expressed as: a_n = a + (n - 1)d. Here, you can directly compute the nth term without needing to calculate all the previous terms. This simplicity and efficiency make the explicit formula a powerful tool in mathematical problem-solving.On the other hand, some sequences are defined by recursive relations, where each term depends on the previous term. For instance, the Fibonacci sequence is defined recursively as F(n) = F(n-1) + F(n-2) with initial conditions F(0) = 0 and F(1) = 1. While it is possible to compute Fibonacci numbers using this recursive definition, it can be inefficient for large n due to repeated calculations. However, mathematicians have derived an explicit formula known as Binet's formula, which allows for direct computation of Fibonacci numbers without recursion: F(n) = (φ^n - (1 - φ)^n) / √5, where φ is the golden ratio (approximately 1.618). This explicit formula significantly simplifies the calculation process and demonstrates the power of having a direct method of obtaining results.The use of explicit formulas extends beyond sequences; they are also crucial in calculus. For example, when integrating functions, finding the area under a curve can often be computed using an explicit formula. The Fundamental Theorem of Calculus states that if F is an antiderivative of f on an interval [a, b], then the area A under the curve from a to b can be found using the explicit formula: A = F(b) - F(a). This allows for the quick calculation of definite integrals, showcasing the importance of explicit formulas in practical applications.In conclusion, the concept of an explicit formula is vital across many branches of mathematics and science. It provides a means to compute values directly and efficiently, saving time and effort compared to recursive methods. Understanding and utilizing explicit formulas can greatly enhance one's ability to solve complex problems and apply mathematical principles effectively. As we continue to explore the vast field of mathematics, the significance of explicit formulas will undoubtedly remain a key focus for students and professionals alike.

在数学和各种科学领域中，术语显式公式指的是一种数学表达式，它提供了一种直接计算特定值的方法，而无需依赖迭代方法或递归。显式公式允许人们立即确定序列的第n项或函数的结果，使用有限的运算次数。这个概念在代数、微积分和数论等领域尤其重要，因为高效的计算至关重要。例如，考虑一个由首项a和公差d定义的等差数列。该数列的第n项的显式公式可以表示为：a_n = a + (n - 1)d。在这里，您可以直接计算第n项，而无需计算所有前面的项。这种简单性和效率使得显式公式成为数学问题解决中的强大工具。另一方面，一些序列是通过递归关系定义的，其中每一项依赖于前一项。例如，斐波那契数列的递归定义为F(n) = F(n-1) + F(n-2)，初始条件为F(0) = 0和F(1) = 1。虽然可以使用这种递归定义计算斐波那契数，但对于较大的n来说，由于重复计算，这可能效率不高。然而，数学家们推导出了一个名为比内公式的显式公式，它允许直接计算斐波那契数，而无需递归：F(n) = (φ^n - (1 - φ)^n) / √5，其中φ是黄金比例（约为1.618）。这个显式公式显著简化了计算过程，并展示了拥有直接获得结果的方法的力量。显式公式的使用不仅限于序列；它们在微积分中也至关重要。例如，在对函数进行积分时，计算曲线下方的面积通常可以使用显式公式。微积分基本定理指出，如果F是f在区间[a, b]上的一个原函数，那么从a到b的曲线下的面积A可以使用显式公式计算：A = F(b) - F(a)。这使得快速计算定积分成为可能，展示了显式公式在实际应用中的重要性。总之，显式公式的概念在数学和科学的许多分支中至关重要。它提供了一种直接和高效地计算值的方法，相比递归方法节省了时间和精力。理解和利用显式公式可以极大地增强一个人解决复杂问题和有效应用数学原理的能力。随着我们继续探索广阔的数学领域，显式公式的重要性无疑将继续成为学生和专业人士的重点关注。