The d-value can be calculated by subtracting any two consecutive terms in an arithmetic sequence. This notation is necessary for calculating nth terms, or a n, of sequences. This means that if we refer to the fifth term of a certain sequence, we will label it a 5. Mathematicians also refer to generic sequences using the letter a along with subscripts that correspond to the term numbers as follows: Mathematicians use the letter d when referring to these difference for this type of sequence. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called the common differences. The fourth number plus -2 is the fifth number: 14 + (-2) = 12.īecause these sequences behave according to this simple rule of addiing a constant number to one term to get to another, they are called arithmetic sequences. This too works for any pair of consecutive numbers. Sequence C is a little different because we need to add -2 to the first number to get the second number. The third number plus 5 is the fourth number: 36 + 5 = 41, which will work throughout the entire sequence. This also works for any pair of consecutive numbers. The second number plus 3 is the third number: 8 + 3 = 11, and so on.įor sequence B, if we add 5 to the first number we will get the second number. This works for any pair of consecutive numbers. įor sequence A, if we add 3 to the first number we will get the second number. The following sequences are arithmetic sequences: Sequence A: 5, 8, 11, 14, 17. OpenStax CNX.Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. You can also download for free at For questions regarding this license, please contact If you use this textbook as a bibliographic reference, then you should cite it as follows: This work is licensed under a Creative Commons Attribution 4.0 International License. Glossary arithmetic sequence a sequence in which the difference between any two consecutive terms is a constant common difference the difference between any two consecutive terms in an arithmetic sequence In application problems, we sometimes alter the explicit formula slightly to.An explicit formula can be used to find the number of terms in a sequence.An explicit formula for an arithmetic sequence with common difference.As with any recursive formula, the initial term of the sequence must be given.A recursive formula for an arithmetic sequence with common difference.The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.The constant between two consecutive terms is called the common difference.An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant. Key Equations recursive formula for nth term of an arithmetic sequenceĮxplicit formula for nth term of an arithmetic sequence You can choose any term of the sequence, and add 3 to find the subsequent term.Īccess this online resource for additional instruction and practice with arithmetic sequences. 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. Find the common difference for an arithmetic sequence.
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