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Chapter 8: Problem 77

What is the common ratio in a geometric sequence?

Short Answer

Expert verified
The common ratio in a geometric sequence is found by dividing any term by its preceding term, given by the formula \(r = \frac{a_{n+1}}{a_n}\) .

Step by step solution

01

Understand Geometric Sequence

In order to understand the common ratio, it is important first to recognize the structure of a geometric sequence. A geometric sequence typically looks like this: \(a, ar, ar^2, ar^3, ar^4, ..., ar^n\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
02

Formula for Common Ratio

To calculate the common ratio of a geometric sequence, divide any term by its preceding term. This can be represented by the formula: \(r = \frac{a_{n+1}}{a_n}\), where \(a_{n+1}\) is any term in the sequence and \(a_n\) is the term that immediately precedes it.
03

Example

For example, suppose we have this geometric sequence: 5, 15, 45, 135, 405. To find the common ratio, choose any term and divide it by the previous term. For example, \(r = \frac{45}{15} = 3\). Therefore, the common ratio is 3.

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