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### Course: Get ready for Precalculus > Unit 6

Lesson 3: Constructing geometric sequences- Explicit & recursive formulas for geometric sequences
- Recursive formulas for geometric sequences
- Explicit formulas for geometric sequences
- Converting recursive & explicit forms of geometric sequences
- Converting recursive & explicit forms of geometric sequences
- Geometric sequences review

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# 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 $2$ :

Geometric sequence formulas give $a(n)$ , the ${n}^{\text{th}}$ term of the sequence.

This is the ${k}$ and common ratio is ${r}$ :

**explicit**formula for the geometric sequence whose first term isThis is the

**recursive**formula of that sequence:*Want to learn more about geometric sequences? Check out this video.*

## Extending geometric sequences

Suppose we want to extend the sequence $54,18,6,\text{\u2026}$ We can see each term is ${\times {\displaystyle \frac{1}{3}}}$ from the previous term:

So we simply multiply that ratio to find that the next term is $2$ :

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

## Writing recursive formulas

Suppose we want to write a recursive formula for $54,18,6,\text{\u2026}$ We already know the common ratio is ${{\displaystyle \frac{1}{3}}}$ . We can also see that the first term is ${54}$ . Therefore, this is a recursive formula for the sequence:

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

## Writing explicit formulas

Suppose we want to write an explicit formula for $54,18,6,\text{\u2026}$ We already know the common ratio is ${{\displaystyle \frac{1}{3}}}$ and the first term is ${54}$ . Therefore, this is an explicit formula for the sequence:

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

## Want to join the conversation?

- yo is sequence rare problem in real life?(7 votes)
- 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.(41 votes)

- 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)?(5 votes)
- 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.(22 votes)

- how do you get the equivalent formula?(6 votes)
- 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.(10 votes)

- How do you type n-1 as a power in the answer box? It keeps changing the -1 to a "regular" -1(4 votes)
- You need to put parentheses around the exponent. For example:

2^(n-1)

Hope this helps.(10 votes)

- What does the n-1 mean again?(5 votes)
- The n-1 refers to the number of times you multiply the common ratio.(3 votes)

- A geometric sequence involves a common difference that is added to each term, yes or no.(2 votes)
- 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.(11 votes)

- 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(4 votes)
- 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.(4 votes)

- do you think khan academy's exercises prepare you for exams?(3 votes)
- 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.(2 votes)
- 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.(5 votes)

- When writing the explicit formula, is it necessary to put the common ratio in parentheses?(4 votes)