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Geometric Sequences

Recognizing Geometric Sequences


Geometric Sequence

A geometric sequence is a sequence where the ratio r of any term to its preceding term is a constant other than 0. This ratio is called the common ratio. In the following geometric sequence, the first term is 3, and the common ratio is 2.

Each term of a geometric sequence is multiplied by the common ratio r to get the next term. Like any other sequence, the first term of a geometric sequence is denoted by a1, the second a2, and so on.

Therefore, geometric sequences have the following form.

Consider the following geometric sequence.
Determine the common ratio and find the next three terms.
Show 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:
If we let r be the common ratio we get the equation
The common ratio, r, is 3. To find the next terms, we multiply by 3, three times.
In summary, the common ratio is 3, and the next three terms are 486, 1458, and 4374.


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
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.

A vet gives medicine to an axolotl for a week. The first dose is 32 mg, and every day it's cut in half. List the doses in a sequence and graph it in a coordinate plane.

Show Solution
The first dose is 32 mg, so that is the first term. The next dose is half of that.
The second term is 16 mg. We can find the rest of the terms by continuing to multiply by 0.5.
The sequence is
If we let x be the days and y be the dose in mg we can graph the sequence.
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