Clearly a line of length \(n\) units takes the same time to articulate regardless of how it is composed. A line of length \(n\) contains \(n\) units where each short syllable is one unit and each long syllable is two units. Observe that each term is r times the previous term. in which first term a1a and other terms are obtained by multiplying by r. A geometric series is of the form a,ar,ar2,ar3,ar4,ar5. Suppose also that each long syllable takes twice as long to articulate as a short syllable. Recursive formula for a geometric sequence is ana (n-1)xxr, where r is the common ratio. Suppose we assume that lines are composed of syllables which are either short or long. In particular, about fifty years before Fibonacci introduced his sequence, Acharya Hemachandra (1089 – 1173) considered the following problem, which is from the biography of Hemachandra in the MacTutor History of Mathematics Archive: Historically, it is interesting to note that Indian mathematicians were studying these types of numerical sequences well before Fibonacci. 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. Since the question was originally only asking for the value of the third term we know our solution only needs to be the value of the third term which is 9. Step 5: We found the recursive sequence we were looking for: 1,3,9. You can choose any term of the sequence, and add 3 to find the subsequent term. This recursive formula is a geometric sequence. 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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