Arithmetic sequences and series12/9/2023 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, An arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference. ![]() 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 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. 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. You need to know the nth term formula for an arithmetic sequence. Formulas used with arithmetic sequences and. + ( a1 + ( n - 1) d) An arithmetic series is the adding together of the terms of an arithmetic sequence. Arithmetic Series: Sn a1 + ( a1 + d) + ( a1 + 2 d) + ( a1 + 3 d) +. That constant difference is known as the common difference of the sequence. We will be working with finite sums (the sum of a specific number of terms). The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. In an arithmetic sequence, the difference between consecutive terms in the sequence is constant. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value. 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. 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. What is an arithmetic sequence For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. 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. Arithmetic sequences follow a pattern of adding a fixed amount from one term to the next.The number being added to each term is constant (always the same). This decrease in value is called depreciation. ![]() The book-value of these supplies decreases each year for tax purposes.
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