![]() ![]() +(a + l) 2S(n) = n(a + l)ġ4.6 Geometric Series Geometric Sequence : 3, 9, 27, 81, … Geometric Series : 3 + 9 + 27 + 81ġ4.6 Geometric Series Formula of Geometric Series S(n) = a + aR + aR2 +aR3+ …. + a + d+ aġ4.5 Arithmetic Series S(n) = a + a + d + a + 2d + a + 3d + ………. + a + (n - 1)d lġ4.5 Arithmetic Series Formula of Arithmetic Series S(n) = l + l - d + l - 2d + l - 3d + …. Let’s consider a sequence : T(1), T(2), T(3), T(4), …., T(n)ġ4.5 Arithmetic Series Arithmetic Sequence : 2, 5, 8, 11, … Arithmetic Series : 2 + 5 + 8 + 11 + ….ġ4.5 Arithmetic Series Formula of Arithmetic Series S(n) = a + a + d + a + 2d + a + 3d + …. We usually denote the sum of the first n term of a series by the notation S(n). G.P.) is a sequence having a common ratio.ġ4.3 Geometric Sequence Illustrative Examplesġ4.3 Geometric Sequence Geometric Means When x, y and z are three consecutive terms of geometric sequence, the middle term y is called the geometric mean of x and z.ġ4.3 Geometric Sequence Geometric Means Insert two geometric means between 16 and -54.ġ4.3 Geometric Sequence Insert two geometric means between 16 and -54.ġ4.4 Series The expression T(1) + T(2) + T(3) +….+ T(n) is called a series. A.P.) is a sequence having a common difference.ġ4.2 Arithmetic Sequence Illustrative Examplesġ4.2 Arithmetic Sequence Arithmetic Means When a, b and c are three consecutive terms of arithmetic sequence, the middle term b is called the arithmetic mean of a and c.ġ4.2 Arithmetic Sequence Arithmetic Means Insert two arithmetic means between 11 and 35.ġ4.2 Arithmetic Sequence Insert two arithmetic means between 11 and 35.ġ4.3 Geometric Sequence A geometric sequence(G.S. So, the sequence can be represented by the general term T(n) = 2n or Tn = 2n The sequence is formed from timing 2 to the previous term.ġ4.2 Arithmetic Sequence An arithmetic sequence(A.S. 0, sin20o, 2sin30o, 3sin40o arithmetic sequenceġ4.1 Sequences Consider the following sequence:1, 3, 5, 7, 9, …., 111 3 is the second term of the sequence, mathematically, T(2) = 3 or T2 = 3 1 is the first term of the sequence,mathematically, T(1) = 1 or T1 = 1 5 is the third term of the sequence, mathematically, T(3) = 5 or T3 = 5 111 is the nth term of the sequence, mathematically, T(n) = 111 or Tn = 111ġ4.1 Sequences Consider the sequence 2, 4, 8, 16, …. Arithmetic and Geometric Sequences and their Summationġ4.1 Sequences arithmetic sequence geometric sequence geometric sequence geometric sequence Find the next two terms of the following sequences : 2, 5, 8, 11,…… 2, 6, 18, 54, …. 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 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.Find the common difference for an arithmetic sequence. ![]()
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