If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra 1

### Course: Algebra 1>Unit 9

Lesson 4: Constructing geometric sequences

# Geometric sequences review

Review geometric sequences and solve various problems involving them.

## Parts and formulas of geometric sequence

In geometric sequences, the ratio between consecutive terms is always the same. We call that ratio the common ratio.
For example, the common ratio of the following sequence is 2:
start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
1, comma2, comma4, comma8, comma, point, point, point
Geometric sequence formulas give a, left parenthesis, n, right parenthesis, the n, start superscript, start text, t, h, end text, end superscript term of the sequence.
This is the explicit formula for the geometric sequence whose first term is start color #11accd, k, end color #11accd and common ratio is start color #ed5fa6, r, end color #ed5fa6:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, k, end color #11accd, dot, start color #ed5fa6, r, end color #ed5fa6, start superscript, n, minus, 1, end superscript
This is the recursive formula of that sequence:
$\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)\cdot\maroonC r \end{cases}$

## Extending geometric sequences

Suppose we want to extend the sequence 54, comma, 18, comma, 6, comma, point, point, point We can see each term is start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 from the previous term:
start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6
54, comma18, comma6, comma, point, point, point
So we simply multiply that ratio to find that the next term is 2:
start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6
54, comma18, comma6, comma2, comma, point, point, point
Problem 1
• Current
What is the next term in the sequence start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots?

Want to try more problems like this? Check out this exercise.

## Writing recursive formulas

Suppose we want to write a recursive formula for 54, comma, 18, comma, 6, comma, point, point, point We already know the common ratio is start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6. We can also see that the first term is start color #11accd, 54, end color #11accd. Therefore, this is a recursive formula for the sequence:
$\begin{cases}a(1) = \blueD{54} \\\\ a(n) = a(n-1)\cdot\maroonC{\dfrac{1}{3}} \end{cases}$
Problem 1
• Current
Find k and r in this recursive formula of the sequence start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots.
$\begin{cases}a(1) = k \\\\ a(n) = a(n-1)\cdot r \end{cases}$
k, equals
r, equals

Want to try more problems like this? Check out this exercise.

## Writing explicit formulas

Suppose we want to write an explicit formula for 54, comma, 18, comma, 6, comma, point, point, point We already know the common ratio is start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6 and the first term is start color #11accd, 54, end color #11accd. Therefore, this is an explicit formula for the sequence:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, 54, end color #11accd, dot, left parenthesis, start color #ed5fa6, start fraction, 1, divided by, 3, end fraction, end color #ed5fa6, right parenthesis, start superscript, n, minus, 1, end superscript
Problem 1
• Current
Write an explicit formula for start fraction, 1, divided by, 2, end fraction, comma, 2, comma, 8, comma, dots
a, left parenthesis, n, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• if you are given t6= 160 and t10=2560 how do you find the common ratio of r using the formula a.r(to the power of n-1)?
• If you use recursive:

t(6) = 160
t(7) = 160 x r
t(8) = 160 x r x r
t(9) = 160 x r x r x r
t(10) = 160 x r x r x r x r = 160 x r^4

since t(10)= 2560;
160 x r^4 = 2560
r^4 = 2560/160
r^4 = 16
r^2 = 4
r = 2

the common ratio is 2.

Feel free to correct me when I am wrong. I am new af as well.
• yo is sequence rare problem in real life?
• It is all over the place in real life. Every time you buy something, it is a sequence. If you buy 1 bag of chips, it costs .50, 2 bags cost 1.00, 3 bags cost 1.50, etc.
If you are paid 8 dollars an hour, this sequence is $8,$16, \$24, etc.
Touchdowns in football are 6, 12, 18, etc., but extra points can be 0, 1, or 2 additional points.
• how do you get the equivalent formula?
• Hi Ruby,

To get an equivalent formula of an explicit geometrical formula, you just need to manipulate the standard formula.

Let's have a look at it with an example. The explicit formula that we are given is-
a(n) = 54⋅(1/3)ˆn−1
a(n) = 54⋅ (1/3)ˆn ⋅ (1/3)ˆ-1
a(n) = 54⋅ (1/3)ˆn ⋅ 3 (as 1/3 ˆ-1 = 3)
a(n) = 162⋅ (1/3)ˆn

So, an equivalent formula of our example is
a(n) = 162⋅ (1/3)ˆn

I hope this helped.

Aiena.
• What does the n-1 mean again?
• The n-1 refers to the number of times you multiply the common ratio.
• A geometric sequence involves a common difference that is added to each term, yes or no.
• Hi Jimin,

A geometric sequence has a COMMON RATIO that is MULTIPLIED between any two terms.

What you are talking about is an "arithmetic sequence". In arithmetic sequences, you have a COMMON DIFFERENCE that is added to the next term.

I hope this clarified your doubt.

Aiena.
• When writing the explicit formula, is it necessary to put the common ratio in parentheses?
• Finally I get to ask a question. I was unable to find an acceptable answer to any of these questions, but after looking at the hints it shows the solution has an n-1 exponent. I can't get it to accept exponents that have anything more than a simple letter. How do you make a superscript like the hint shows. I don't understand the how to format acceptable inputs. I know that I have probably missed something real basic. I have not searched help yet for this issue, which I will do now.
• To enter n-1 as an exponent, you need to put parentheses around the n-1. Otherwise, most systems / calculators assume the exponent is one number or variable. For example:
2^(n-1)

Hope this helps.