AEF - Geometric Sequences (Lesson)

Geometric Sequences

Recall that a sequence is a special type of function. Each element in a sequence is called a term; these values would be considered the range. Each term is paired with a position number, and these values would be considered the domain. The domains of sequences are consecutive integers and usually start at 0 or 1.

Position Number (domain)	$n$	1	2	3	4
Term of Sequence (range)	$f(n)$ or $a_n$	3	9	27	81

From the table, you can see that the "fourth term in the sequence is 81" or f(4) = 81 or a₄ = 81.

The explicit rule of a sequence is a rule that will allow you to determine any term in the sequence by using n, the position number. The explicit rule for the sequence above is f(n) = 3ⁿ or a_n = 3ⁿ

A geometric sequence is a special type of sequence in which the ratio of consecutive terms is constant. This ratio is referred to as the common ratio.

Determine if each of the following sequences is geometric. If so, give the common ratio:

1, 4, 9, 16
64, 16, 4, 1
7, 12, 17, 22
-5, -10, -20, -40

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

There is a rule for finding the explicit rule for any geometric sequence:

Geometric Sequences Explicit Rule

a_n = nth term
a₁ = first term
r = common ratio
n = term position

Let's try one:

Makers of Japanese swords in the 1400s repeatedly folded and hammered the metal to form layers. The folding process increased the strength of the sword.

The table shows how the number of layers depends on the number of folds. Write an explicit rule for the geometric sequence described by the table:

Number of Folds	n	1	2	3	4
Number of Layers	a_n	2	4	8	16

First, let's find the common ratio by dividing consecutive terms.

16/8 = 2

8/4 = 2

4/2 = 2

Now that we know r = 2, we can write the explicit rule by using the first term, or a₁.

$LaTeX: a_n=a_1(r)^{n-1}$ $a_n=a_1(r)^{n-1}$

$LaTeX: a_n=2(2)^{n-1}$ $a_n=2(2)^{n-1}$

A recursive rule for a sequence defines the terms of the sequence by relating it to one or more previous terms.

Recursive Rule Image

So the recursive rule for the table above would be: a₁ = 2 and $LaTeX: a_n=a_{n-1}\cdot2\:$ $a_n=a_{n-1}\cdot2\:$ for n > 2

Watch this video to try a few problems:

Geometry Sequences Practice

The first term of a geometric sequence is 6 and the common ratio is -8. Find the 7^th term.
Write an explicit rule for the geometric sequence: -4, -12, -36, ... ?
Using the rule you wrote in #2, find the 12^th term.
Given that a₂ = 12 and a₄ = 192 find the explicit rule given that r is positive.

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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