![]() The common ratio is also the base of an exponential function as shown in Figure 9.4.2. The sequence of data points follows an exponential pattern. Substitute the common ratio into the recursive formula for geometric sequences and define a1. A recursive formula for an arithmetic sequence with common difference d is given by +dn. The common ratio can be found by dividing the second term by the first term.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.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.The recursive formula for a sequence allows you to find the value of the n th term in the sequence if you know the value of the (n-1) th term in the sequence.Ī sequence is an ordered list of numbers or objects. However, we know that (a) is geometric and so this interpretation holds, but (b) is not. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. and are often referred to as positive integers. The graph of each sequence is shown in Figure 6.4.1. The natural numbers are the numbers in the list 1, 2, 3. The natural numbers are the counting numbers and consist of all positive, whole numbers. The index of a term in a sequence is the term’s “place” in the sequence. ![]() Geometric sequences are also known as geometric progressions. For example in the sequence 2, 6, 18, 54., the common ratio is 3.Įxplicit formulas define each term in a sequence directly, allowing one to calculate any term in the sequence without knowing the value of the previous terms.Ī geometric sequence is a sequence with a constant ratio between successive terms. For example: In the sequence 5, 8, 11, 14., the common difference is "3".Įvery geometric sequence has a common ratio, or a constant ratio between consecutive terms. ![]() Arithmetic sequences are also known are arithmetic progressions.Įvery arithmetic sequence has a common or constant difference between consecutive terms. \)Īn arithmetic sequence has a common difference between each two consecutive terms. Given two consecutive terms of the arithmetic sequence, say a n and a n + 1, we have the relation a n + 1 a n + r.
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