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Recognizing Geometric Sequences

Concept

Geometric Sequence

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

geometric sequence

Similar to other sequences, the first term is usually called a1,a_1, the second a2,a_2, and so on.

geometric progression
The number rr is called the common ratio of the sequence.
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Exercise

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

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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: 2and6. 2 \quad \text{and} \quad 6. If we let rr be the common ratio we get the equation 2r=6r=3. 2\cdot r = 6 \quad \Leftrightarrow \quad r=3. The common ratio, r,r, is 3.3. To find the next terms, we multiply by 3,3, three times. 1623=4864863=145814583=4374\begin{aligned} 162 \cdot 3 &= 486 \\ 486\cdot 3 &= 1458 \\ 1458\cdot 3 &= 4374 \end{aligned} In summary, the common ratio is 3,3, and the next three terms are 486,486, 1458,1458, and 4374.4374.

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,  1,\ 1.5,\ 2.25,\ 3.375,\ 5.0625, \ \ldots Here, the common ratio is 1.5,1.5, and the terms can be illustrated in a table.

This is very similar to the rate of change of an exponential function. The number 1.51.5 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.
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Exercise

A vet gives medicine to an axolotl for a week. The first dose is 3232 mg, and every day it's cut in half. List the doses in a sequence and graph it in a coordinate plane.

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Solution

The first dose is 3232 mg, so that is the first term. The next dose is half of that. 320.5=16. 32\cdot 0.5 = 16. The second term is 1616 mg. We can find the rest of the terms by continuing to multiply by 0.5.0.5. The sequence is 32, 16, 8, 4, 2, 1, 0.5. 32,\ 16,\ 8,\ 4,\ 2,\ 1,\ 0.5. If we let xx be the days and yy be the dose in mg we can graph the sequence.

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