![]() There is another formula used to find the n th term of a geometric sequence given its previous term and the common ratio which is called the recursive formula of the geometric sequence. r = common ratio of the geometric sequence.a = first term of the geometric sequence.So in general, the n th term of a geometric sequence is, , where 'a' is the first term and 'r' is the common ratio. We have already seen that a geometric sequence is of the form a, ar, ar 2, ar 3. is an infinite sequence where the last term is not defined. Infinite geometric sequenceĪn infinite geometric sequence is a geometric sequence that contains an infinite number of terms. 13122 is a finite geometric sequence where the last term is 13122. They areĪ finite geometric sequence is a geometric sequence that contains a finite number of terms. There are two types of geometric sequences based on the number of terms in them. is a geometric sequence where a = √2 and r = -1 is a geometric sequence where a = π and r = 2 is a geometric sequence where a = -4 and r = -1/2 is a geometric sequence where a = 1/4 and r = 1/2 The common ratio can be either a positive or a negative number. where 'a' is the first term and 'r' is the common ratio of the sequence. So a geometric sequence is in form a, ar, ar 2. In other words, in a geometric sequence, every term is multiplied by a constant which results in its next term. This ratio is known as a common ratio of the geometric sequence. Geometric Sequence vs Arithmetic SequenceĪ geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant. Sum of Infinite Geometric Sequence Formula Here we shall learn more about each of the above-mentioned geometric sequence formulas along with their proofs and examples. The geometric sequences can be finite or infinite. The sum of an infinite geometric sequence.The recursive formula of a geometric sequence.Here, we learn the following geometric sequence formulas: The common ratio of a geometric sequence can be either negative or positive but it cannot be 0. Here is an example of a geometric sequence is 3, 6, 12, 24, 48. i.e., To get the next term in the geometric sequence, we have to multiply with a fixed term (known as the common ratio), and to find the preceding term in the sequence, we just have to divide the term by the same common ratio. It is a sequence in which every term (except the first term) is multiplied by a constant number to get its next term. A geometric sequence is a special type of sequence. 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: 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. 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. ![]()
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