Recognizing Geometric Sequences

Concept

Geometric Sequence

A sequence where each term is multiplied by the same number, $r,$ to get the next term is a geometric sequence. The number $r$ can be any non-zero real number. In the following example, $r$ is $2,$ making each term twice as large as the previous.

Similar to other sequences, the first term is usually called $a_1,$ the second $a_2,$ and so on.

The number

r

is called the common ratio of the sequence.

Find the common ratio

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Exercise

Consider the following geometric sequence. $2,\ 6,\ 18,\ 54,\ 162, \ \ldots$ Determine the common ratio and find the next three terms.

Show Solution

Solution

In geometric sequences, the terms increase or decrease by a common ratio. Since we know that this sequence is geometric, it's enough to find the ratio between two consecutive terms. The ratio for the others must then be the same. Let's take the first two: $2 \quad \text{and} \quad 6.$ If we let $r$ be the common ratio we get the equation $2\cdot r = 6 \quad \Leftrightarrow \quad r=3.$ The common ratio, $r,$ is $3.$ To find the next terms, we multiply by $3,$ three times. $\begin{aligned} 162 \cdot 3 &= 486 \\ 486\cdot 3 &= 1458 \\ 1458\cdot 3 &= 4374 \end{aligned}$ In summary, the common ratio is $3,$ and the next three terms are $486,$ $1458,$ and $4374.$

Concept

Geometric Sequences and Exponential Functions

The terms in a geometric sequence increase or decrease by the same factor for each term. Consider the geometric sequence $1,\ 1.5,\ 2.25,\ 3.375,\ 5.0625, \ \ldots$ Here, the common ratio is $1.5,$ and the terms can be illustrated in a table.

This is very similar to the rate of change of an exponential function. The number $1.5$ would, in that case, be the constant multiplier. In fact, when graphing a geometric sequence in a coordinate plane, it resembles the graph of an exponential function.

By making this comparison, geometric sequences can be considered functions. They have the same characteristics as exponential functions, but where exponential functions are continuous, both the domain and range for geometric sequences are discrete.

Graph the geometric sequence