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Chapter 5: Problem 138

What is a geometric sequence? Give an example with your description.

Short Answer

Expert verified
A geometric sequence is a numerical sequence where the ratio between every two successive numbers is constant. For instance, \(2, 6, 18, 54, 162,\) where the ratio is 3.

Step by step solution

01

Definition

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This can be written mathematically as: \(a, ar, ar^2, ar^3, ar^4, ...\), where \(a\) is the first term and \(r\) is the common ratio.
02

Properties of Geometric Sequences

The important property of any geometric sequence is that the ratio of any two consecutive terms is constant throughout the sequence.
03

Example of a Geometric Sequence

An example of a geometric sequence could be: \(2, 6, 18, 54, 162, ...\). In this sequence, the first term (a) is 2 and the common ratio (r) is 3. So each term is 3 times the previous term.

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(18,6,2, \frac{2}{3}, \ldots\)

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(1.5,-3,6,-12, \ldots\)

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. \(1,5,9,13, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{200}\), when \(a_{1}=60, r=1\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(5,15,45,135, \ldots\)
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