![]() In this equation, F(n) is equal to the sum of the previous two terms in the sequence. For example, the explicit formula for the Fibonacci sequence is: An explicit formula defines each term of the sequence as a function of n, the position of that term in the sequence. ![]() There are two main ways to define a recursive sequence: through an explicit formula or through a recurrence relation. In other words, each term in the sequence is determined by the terms that came before it. Recursive Sequence FormulaĪ recursive sequence is a sequence in which each term is defined as a function of the previous term. For example: 1!, 2!, 3!, 4! Is a factorial sequence because each term is multiplied by the one preceding it. For example: 1, 1, 2, 3, 5 is the Fibonacci sequence because each subsequent number is the sum of the previous two numbers.įactorial sequences multiply each term by the one before it. For example: 1/2, 1/4, 1/8 is a geometric sequence because each term is halved from the one before it.įibonacci sequences have a special relationship between each term where the next term is the sum of the previous two terms. Geometric sequences have a common ratio between each term. For example: 3, 5, 7, 9, 11 is an arithmetic sequence because the common difference is 2. There are four types of recursive sequences: arithmetic, geometric, Fibonacci, and factorial.Īrithmetic sequences have a common difference between each term. For example, the Fibonacci numbers can also be defined by the difference equation:į(n) – F(n-1) = F(n-1) – F(n- Types of Recursive Sequences For example, the Fibonacci numbers can be defined by the explicit formula:Ī difference equation for a recursive sequence defines each term in terms of the difference between two consecutive terms. The recurrence relation can be defined in terms of an explicit formula or a difference equation.Īn explicit formula for a recursive sequence defines each term in terms of one or more previous terms. The most important thing to understand about recursive sequences is the recurrence relation, which defines how each term in the sequence is related to the previous terms. Linear recursive sequences have a constant recurrence relation, while nonlinear recursive sequences have a variable recurrence relation. There are two types of recursive sequences: linear and nonlinear. They can be used to solve problems in computer science and engineering, such as computing the Fibonacci numbers. Recursive sequences are used to model many real-world phenomena, such as population growth and compound interest. In other words, each subsequent term in the sequence is the result of applying some mathematical function to the previous terms. In mathematics, a recursive sequence is a sequence defined recursively by two initial values and a recurrence relation. A simple example of this is the Fibonacci sequence, where each term is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13… In this post, we’ll explore what recursive sequences are, some famous examples, and how to calculate them. ![]() That is, the nth term of th e sequence is defined in terms of the previous terms. ![]() Recursive Sequence Definitions & ExamplesĪ recursive sequence is a sequence whose terms are defined recursively.
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