Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, formulas are formulas of the first order arithmetic (with constants for the primitive recursive functions of natural numbers). Then you must include on every physical page the following attribution: Definition: Arithmetic sequence is a list of numbers where each number is equal to the previous number, plus a constant. If you are redistributing all or part of this book in a print format, Recursive Sequence A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. a1, an a (n-1) +d Consider an example arithmetic sequence. Want to cite, share, or modify this book? This book uses the The recursive rule of an arithmetic sequence gives the first term of the sequence and a recursive equation. You can choose any term of the sequence, and add 3 to find the subsequent term. In this case, the constant difference is 3. The sequence below is another example of an arithmetic sequence. In this article, we will discuss the definition of a recursive function, its formula, and the procedure of creating the recursive formula for the given sequence with solved examples. For this sequence, the common difference is –3,400. Each term increases or decreases by the same constant value called the common difference of the sequence. sequence, defined by recurrence and its first term, until the indicated index. The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Arithmetic Sequence Formula: an a1 +d(n 1) a n a 1 + d ( n - 1). In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value. There are few recursive formulas to find the n th term based on the pattern of the given data. The first term of the sequence The pattern rule to get any term from its previous terms. The truck will be worth $21,600 after the first year $18,200 after two years $14,800 after three years $11,400 after four years and $8,000 at the end of five years. Recursive Formula is a formula that defines the each term of sequence using the previous/preceding terms. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. After five years, she estimates that she will be able to sell the truck for $8,000. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.Īs an example, consider a woman who starts a small contracting business. This decrease in value is called depreciation. The book-value of these supplies decreases each year for tax purposes. If you know the n th term of an arithmetic sequence and you know the common difference, d, you can find the (n+1) th term using the recursive formula a n+1 a n +d.
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