![]() ![]() Note that this is actually an arithmetic sequence with a first term of We deduce that the first five terms of this sequence are Similarly, by substituting □ = 2, 3, and 4 into the given ![]() □ = 1 into the recursive formula and use the fact that The recursive formula □ = □ + 5 . Here, we are given the first term □ = − 1 3 together with To generate a sequence from its recursive formula, we need to know the first Recall that a recursive formula of the form □ = □ ( □ ) defines each term of a sequence as a function Let us now try an example to practice this skill.Įxample 1: Finding the First Five Terms of a Sequence Using Its Recursive Formulaįind the first five terms of the sequence with general term That we will meet later on in this explainer. Types of sequence, including arithmetic sequences, geometric sequences, and others The strength of recursive formulas is that they enable us to describe many different Note that this is actually anĪrithmetic sequence with a first term of 8 and a common difference of − 4. Therefore, the first five terms of this sequence are Into the formula and use the fact that □ = − 4 to get To find □ , we substitute □ = 3 into theįormula and use the fact that □ = 0 to getįinally, to find □ , we substitute □ = 4 Similarly, to find □ , we substitute □ = 2 into the formula and use the fact that □ = 1 into the recursive formula □ = □ − 4 To find the second term, □ , we substitute Sequence defined by the recursive formula Note that this is actually anĪrithmetic sequence with a first term of 10 and a common difference of 1.Īs another example, suppose we are asked to find the first five terms of the Therefore, the first four terms of this sequence are Into the formula and use the fact that □ = 1 2 to get Into the formula and use the fact that □ = 1 1 to getįinally, to find □ , we substitute □ = 3 Similarly, to find □ , we substitute □ = 2 To find the second term, □ , we substitute □ = 1 into the recursive formula We already know the first term, which is □ = 1 0 . Suppose we areĪsked to find the first four terms of the sequence defined by the recursive This process is best illustrated through some specific examples. In this way, we canīuild up the sequence until it has as many terms as we wish. ![]() The formula with □ = 2 to derive the value of Once we know the value of □ , we can use ![]() □ = □ ( □ ) , we can use the formula with If we know the first term, □ , and the recursive formula Of a sequence using a preceding term or terms.Ī recursive formula of the form □ = □ ( □ ) ĭefines each term of a sequence as a function of the previous term. Definition: Recursive Formula of a SequenceĪ recursive formula (sometimes called a recurrence relation) is a formula that 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, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The yearly salary values described form a geometric sequence because they change by a constant factor each year. In this section, we will review sequences that grow in this way. When a salary increases by a constant rate each year, the salary grows by a constant factor. His salary will be $26,520 after one year $27,050.40 after two years $27,591.41 after three years and so on. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. He is promised a 2% cost of living increase each year. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Use an explicit formula for a geometric sequence.Use a recursive formula for a geometric sequence.List the terms of a geometric sequence.Find the common ratio for a geometric sequence. ![]()
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