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## 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

CCSS.Math: , , ,

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 #ed5fa6 | start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6 | ||||
---|---|---|---|---|---|---|

1, comma | 2, comma | 4, comma | 8, 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:This 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, 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 #ed5fa6 | start color #ed5fa6, times, start fraction, 1, divided by, 3, end fraction, \curvearrowright, end color #ed5fa6 | |||
---|---|---|---|---|

54, comma | 18, comma | 6, 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 #ed5fa6 | ||||||
---|---|---|---|---|---|---|

54, comma | 18, comma | 6, comma | 2, comma, point, point, point |

*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:

*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:

*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)?(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.(13 votes)

- What does the n-1 mean again?(4 votes)
- The n-1 refers to the number of times you multiply the common ratio.(4 votes)

- how do you get the equivalent formula?(4 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.(5 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.(6 votes)

- When writing the explicit formula, is it necessary to put the common ratio in parentheses?(4 votes)
- yo is sequence rare problem in real life?(3 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.(2 votes)

- Why do the explicit and recursive formulas work?(5 votes)
- Because there all number that are put together in the right place to make something work(0 votes)

- at the end of last example the equivalentproblem removed the n-1 and n as the exponent . Why(2 votes)
- are there any lessons on solving arithmetic sequences(0 votes)
- More of a technical question... How do I write the power of n-1? More explicitly how to write "-" or other signs? Whenever I want to press "^" it brings me down...(1 vote)
- You need to use parentheses. For example: 2^(n-1)(1 vote)