![]() ![]() ![]() 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: The Fibonacci numbers form a famous sequence in mathematics that was investigated by Leonardo of Pisa (1170 1250), who is better known as Fibonacci. 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. The variable, a n represents the n t h term and a n 1. In other words, n takes on the values 1 (first term), 2 (second term), 3 (third term), etc. Generally, the variable n is used to represent the term number. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term. A recursive rule for a sequence is a formula which tells us how to progress from one term to the next in a sequence. ![]() 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. a ( n) 3 + 2 ( n 1) In the formula, n is any term number and a ( n) is the n th term. Specifically, you might find the formulas a n a + ( n 1) d (arithmetic) and a n a r n 1 (geometric). Here is an explicit formula of the sequence 3, 5, 7. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. If you look at other textbooks or online, you might find that their closed formulas for arithmetic and geometric sequences differ from ours. Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Recursive sequences do not have one common formula. Lower case a 2 is the second number in the sequence and so on. Usually, we learn about this function based on the arithmetic-geometric sequence, which has terms with a common difference between them. Use an explicit formula for a geometric sequence. Lower case a 1 is the first number in the sequence. Recursive Function is a function that repeats or uses its own previous term to calculate subsequent terms and thus forms a sequence of terms.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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