In addition, a sequence can be thought of as an ordered list. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. A sequence is a function whose domain consists of a set of natural numbers beginning with (1). Use the information below to generate a citation. Lastly, find the next two terms of the sequence. If it is geometric, state the value of r. If it is arithmetic, state the value of d. Then you must include on every digital page view the following attribution: Algebra 2 Honors Name Sequences and Series Date Decide whether the sequence is arithmetic, geometric or neither. 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: And, yes, it is easier to just add them in this example, as there are only 4 terms. The values of a, r and n are: a 10 (the first term) r 3 (the 'common ratio') n 4 (we want to sum the first 4 terms) So: Becomes: You can check it yourself: 10 + 30 + 90 + 270 400. If you are redistributing all or part of this book in a print format, This sequence has a factor of 3 between each number. Sequences can have formulas that tell us how to find any term in the sequence. For example, 2,5,8 follows the pattern 'add 3,' and now we can continue the sequence. Some sequences follow a specific pattern that can be used to extend them indefinitely. Want to cite, share, or modify this book? This book uses the Sequences are ordered lists of numbers (called 'terms'), like 2,5,8. 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. 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 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. 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. Use an explicit formula for an arithmetic sequence.Ĭompanies often make large purchases, such as computers and vehicles, for business use.Use a recursive formula for an arithmetic sequence.If we look closely, we will see that we obtain the next term in the sequence by multiplying the previous term by the same number. This sequence is not arithmetic, since the difference between terms is not always the same. Find the common difference for an arithmetic sequence. Part 2: Geometric Sequences Consider the sequence 2, 4, 8, 16, 32, 64, ldots.Is 22 a number in the sequence with nth term = 4n+1 ?Īs 5.25 is not an integer this means that 22 is not a number in the sequence. If n (the term number) is an integer the number is in the sequence, if n is not an integer the number is not in the sequence. In order to work out whether a number appears in a sequence using the nth term we put the number equal to the nth term and solve it. In order to find any term in a sequence using the nth term we substitute a value for the term number into it. Mixing up working out a term in a sequence with whether a number appears in a sequence.Quadratic sequences have a common second difference d 2.Geometric sequences are generated by multiplying or dividing by the same amount each time – they have a common ratio r.Arithmetic sequences are generated by adding or subtracting the same amount each time – they have a common difference d.Mixing up arithmetic and geometric and quadratic sequences.
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