Geometric sequence, properties, examples, limit, sense of variation, graphical representation
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This document provides an in-depth analysis of geometric sequences, including their definition, properties, and examples. It covers the calculation of terms, the limit of the sequence, and the graphical representation of geometric sequences. The document also explores the sense of variation of a geometric sequence and provides solutions to various problems.
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[...] If so the next part arithmetic sequence with common difference . Yes so the next part is a geometric progression of ratio . Example: The sequence defined by is an arithmetic-geometric sequence. Let's calculate the first terms of this sequence: The function associated with this arithmetic-geometric sequence is an affine function defined on ? par . Example: An individual deposits 3,000 ? on an account with an interest rate of per year. Each year, he deposits an additional 300 It is noted that the amount saved during the year . [...]
[...] This real constant is usually denoted by is called the ratio of the geometric sequence thus defined. A geometric sequence is always defined by its first term and its ratio. Notation: Consider a sequence geometric. We can then use the following notations: Therefore, to demonstrate that a sequence is geometric, we need to calculate the ratios for different values of and to deduce that these reports are not only constant, that is, independent of but especially equal to which is nothing other than the reason for the sequence. [...]
[...] To show that then calculate . What is the nature of the sequence? ? Justify your answer. Express in accordance with . Calculate the number of tons of iron extracted in rounded to the unit. Express the total amount of iron extracted between and). Calculate the amount of iron that this company will be able to extract if the operation continues indefinitely under the same conditions. Solution : (independent of The sequence is therefore geometric with ratio and the first term . [...]
[...] In order to dynamize it, he injects an additional amount into his capital each month, which decreases by 30% each month. Calculate the total amount of capital invested at the end of the eighth year. Let's assume the capital invested at the end of -year. Initially, the capital is: . After a year, the new capital is: After two years, the new capital is: In total, the invested capital is: VII. Arithmetic-geometric sequences A. Definition Let and two real numbers given. We define an arithmetic-geometric sequence. by the data of its first term ? [...]
[...] To do this, we construct the symmetric of with respect to the first bisector . Then we start again with , to place , then etc. By graphical reading, the conjectures that can be made are: - conjecture n°1: The sequence is strictly decreasing and bounded. All terms are included between 4 and 10. - conjecture n°2: It seems that the sequence converge and admits for limit the abscissa of the intersection point of the line with the first bisector. [...]
