Complete the Sequence

4.15.3 Complete the Sequence

4.15.3 • Complete the Sequence

Activity

Write the following terms and definitions in your math notebook:

A geometric sequence is a sequence in which each term is found by multiplying the previous term by the same constant.

The number that is multiplied by each term is called the common ratio.

1. Complete each geometric sequence.

1.5, 3, 6, _a__, 24, _b__

Blank “a”:

Compare your answer:

Blank “b”:

Compare your answer:

b. 40, 120, 360, __a__, __b__

Blank “a”:

Compare your answer:

1,080

Blank “b”:

Compare your answer:

3,240

c. 200, 20, 2, __a__, 0.02, __b__

Blank “a”:

Compare your answer:

0.2

Blank “b”:

Compare your answer:

0.002

d. $\frac{1}{7}$ , __a__, $\frac{9}{7}$ , $\frac{27}{7}$ , __b__

Blank “a”:

Compare your answer:

$\frac{3}{7}$

Blank “b”:

Compare your answer:

$\frac{81}{7}$

e. 24, 12, 6, __a__, __b__

Blank “a”:

Compare your answer:

Blank “b”:

Compare your answer:

1.5

2. For each sequence, find its common ratio.

a. 1.5, 3, 6, ___, 24, ___

Compare your answer:

b. 40, 120, 360, ___, ___

Compare your answer:

c. 200, 20, 2, ___, 0.02, ___

Compare your answer:

$\frac{1}{10}$

d. $\frac{1}{7}$ , ___, $\frac{9}{7}$

$\frac{27}{7}$

Compare your answer:

e. 24, 12, 6, ___, ___

Compare your answer:

$\frac{1}{2}$

Video: Completing Geometric Sequences

Watch the following video to learn more about geometric sequences.

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Self Check

Which of the following sequences is geometric?

10, 7, 4, 1, . . .
3, 5, 8, 10, 13, . . .
48, 24, 12, 6, . . .
5, 10, 15, 20 . . .

Additional Resources

Finding a Common Ratio

Definition of a Geometric Sequence

A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If $a_{1}$ is the initial term of a geometric sequence and $r$ is the common ratio, the sequence will be

{ $a_{1}, a_{1} \cdot r$ , $a_{1} \cdot r^{2}$ , $a_{1} \cdot r^{3}$ . . .}

How to:

Given a set of numbers, determine if they represent a geometric sequence.

Divide each term by the previous term.
Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.

Example

Is the sequence geometric? If so, find the common ratio.

1, 2, 4, 8, 16, . . .
48, 12, 4, 2, . . .

Is a geometric sequence. Each term is multiplied by 2, so 2 is the common ratio.
This is not geometric because each term is not multiplied by the same constant. There is not a common ratio.

Try it

Finding a Common Ratio

Is the sequence geometric? If so, find the common ratio.

100, 20, 4, $\frac{4}{5}$ , . . .

Here is how to determine if the sequence is geometric:

Each term is $\frac{1}{5}$ of the previous term. So, the common ratio is $\frac{1}{5}$ , and this is a geometric sequence.