![]() ![]() We begin by thinking of a function as a map between variables. Substitute the common ratio into the recursive formula for a geometric sequence. And because, the constant factor is called the common ratio20. Find the common ratio by dividing any term by the preceding term. A geometric sequence18, or geometric progression19, is a sequence of numbers where each successive number is the product of the previous number and some constant. Be prepared to think about functions from many different points of view. How To: Given the first several terms of a geometric sequence, write its recursive formula. We want to use our emphasis on sequences to come to terms with these ideas. Unfortunately, many students have subtle misconceptions about how to think about functions. Functions are at the heart of everything we do in calculus. Subsection 13.2.2 Functions as Maps ¶īefore we discuss more about projection functions, we take a short diversion to review some core concepts about functions in general. When the recurrence relation for a sequence \(x\) is solved for the next value as a dependent variable in terms of an expression involving of the previous term, we call this map or function the projection function because it allows us to project future values based on current values. For example, consider the sequence introduced above,įor recursively defined sequences, the equation that describes the relationship between consecutive terms of the sequence is called the recurrence relation. The simplest pattern-based sequences follow simple recursive patterns.Īn arithmetic sequence is a sequence whose terms change by a fixed increment or difference. When a sequence can be defined so that the next value can found knowing only the previous value, we say the sequence has a recursive definition. We often think of sequences in terms of a pattern for how to find the values. It was introduced in 1202 by Leonardo Fibonacci. The sequence shown in this example is a famous sequence called the Fibonacci sequence. Subsection 13.2.1 Arithmetic and Geometric Sequences Seeing the pattern for an explicit formula for an arithmetic sequence or a geometric sequence will be easy as compared to finding explicit formulas for sequences that do not fall into these categories. We visualize the role of projection functions as maps between sequence values and through cobweb diagrams. We will learn about projection functions used in such recursive definitions. We review some basic ideas about functions. Arithmetic and geometric sequences are two familiar examples of sequences with recursive definitions. In this section, we consider recursively defined sequences. For example, the sequence \((7,10,13,16,\ldots)\) is easy to recognize that each term is found by adding \(3\) to the previous term. Another approach is to look for a pattern in how terms are generated from earlier terms. There were several examples of this in the previous section. One approach is to look at the values of individual terms and see if there is an explicit formula relating the index with the formula. When looking for patterns in sequences, we usually explore two possibilities. This is the explicit formula for the geometric sequence whose first term is k and common ratio is r : a ( n) k r n 1. Section 13.2 Recursive Sequences and Projection Functions ¶ Overview. Geometric sequence formulas give a ( n), the n th term of the sequence. ![]()
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