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## Algebra (all content)

### Course: Algebra (all content)>Unit 4

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 $2$:
$×2\phantom{\rule{0.167em}{0ex}}↷$$×2\phantom{\rule{0.167em}{0ex}}↷$$×2\phantom{\rule{0.167em}{0ex}}↷$
$1,$$2,$$4,$$8,\text{…}$
Geometric sequence formulas give $a\left(n\right)$, the ${n}^{\text{th}}$ term of the sequence.
This is the explicit formula for the geometric sequence whose first term is $k$ and common ratio is $r$:
$a\left(n\right)=k\cdot {r}^{n-1}$
This is the recursive formula of that sequence:
$\left\{\begin{array}{l}a\left(1\right)=k\\ \\ a\left(n\right)=a\left(n-1\right)\cdot r\end{array}$

## Extending geometric sequences

Suppose we want to extend the sequence $54,18,6,\text{…}$ We can see each term is $×\frac{1}{3}$ from the previous term:
$×\frac{1}{3}\phantom{\rule{0.167em}{0ex}}↷$$×\frac{1}{3}\phantom{\rule{0.167em}{0ex}}↷$
$54,$$18,$$6,\text{…}$
So we simply multiply that ratio to find that the next term is $2$:
$×\frac{1}{3}\phantom{\rule{0.167em}{0ex}}↷$$×\frac{1}{3}\phantom{\rule{0.167em}{0ex}}↷$$×\frac{1}{3}\phantom{\rule{0.167em}{0ex}}↷$
$54,$$18,$$6,$$2,\text{…}$
Problem 1
What is the next term in the sequence $\frac{1}{2},2,8,\dots$?

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

## Writing recursive formulas

Suppose we want to write a recursive formula for $54,18,6,\text{…}$ We already know the common ratio is $\frac{1}{3}$. We can also see that the first term is $54$. Therefore, this is a recursive formula for the sequence:
$\left\{\begin{array}{l}a\left(1\right)=54\\ \\ a\left(n\right)=a\left(n-1\right)\cdot \frac{1}{3}\end{array}$
Problem 1
Find $k$ and $r$ in this recursive formula of the sequence $\frac{1}{2},2,8,\dots$.
$\left\{\begin{array}{l}a\left(1\right)=k\\ \\ a\left(n\right)=a\left(n-1\right)\cdot r\end{array}$
$k=$
$r=$

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

## Writing explicit formulas

Suppose we want to write an explicit formula for $54,18,6,\text{…}$ We already know the common ratio is $\frac{1}{3}$ and the first term is $54$. Therefore, this is an explicit formula for the sequence:
$a\left(n\right)=54\cdot {\left(\frac{1}{3}\right)}^{n-1}$
Problem 1
Write an explicit formula for $\frac{1}{2},2,8,\dots$
$a\left(n\right)=$

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

## Want to join the conversation?

• 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.
• 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.
• 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.
• 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.
• What does the n-1 mean again?
• The n-1 refers to the number of times you multiply the common ratio.
• Bro the "Converting recursive and explicit for..." wasn't working for me! so when I went here and got everything right I knew that that thing was just glitched up lol
• Same here. I figured out you'd have to write the power of n (without the -1) first, then put it in parenthesis, and only then can you write the needed -1 in there too so it is recognised correctly as part of the exponent. Then it works.
• How do you type n-1 as a power in the answer box? It keeps changing the -1 to a "regular" -1
• You need to put parentheses around the exponent. For example:
2^(n-1)
Hope this helps.