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.

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.

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.

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.

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.

Main content

Algebra 1

Course: Algebra 1 > Unit 9

Lesson 4: Constructing geometric sequences

Geometric sequences review

Google Classroom

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

	$\times 2 ↷$ ‍		$\times 2 ↷$ ‍		$\times 2 ↷$ ‍
$1,$ ‍		$2,$ ‍		$4,$ ‍		$8, …$ ‍

Geometric sequence formulas give

a (n)

, the

n^{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 (n) = k \cdot r^{n - 1}

This is the recursive formula of that sequence:

{\begin{cases} a (1) = k \\ a (n) = a (n - 1) \cdot r \end{cases}

Want to learn more about geometric sequences? Check out this video.

Extending geometric sequences

Suppose we want to extend the sequence

54, 18, 6, …

We can see each term is

\times \frac{1}{3}

from the previous term:

	$\times \frac{1}{3} ↷$ ‍		$\times \frac{1}{3} ↷$ ‍
$54,$ ‍		$18,$ ‍		$6, …$ ‍

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

2

	$\times \frac{1}{3} ↷$ ‍		$\times \frac{1}{3} ↷$ ‍		$\times \frac{1}{3} ↷$ ‍
$54,$ ‍		$18,$ ‍		$6,$ ‍		$2, …$ ‍

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, …

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:

{\begin{cases} a (1) = 54 \\ a (n) = a (n - 1) \cdot \frac{1}{3} \end{cases}

Problem 1

Find $k$ ‍ and $r$ ‍ in this recursive formula of the sequence $\frac{1}{2}, 2, 8, \dots$ ‍.

{\begin{cases} a (1) = k \\ a (n) = a (n - 1) \cdot r \end{cases}

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, …

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 (n) = 54 \cdot {(\frac{1}{3})}^{n - 1}

Problem 1

Write an explicit formula for $\frac{1}{2}, 2, 8, \dots$ ‍

a (n) =

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

Want to join the conversation?

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dohanlingan
Posted 4 years ago. Direct link to dohanlingan's post “yo is sequence rare probl...”
yo is sequence rare problem in real life?
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(5 votes)
Answer
- David Severin
  Posted 4 years ago. Direct link to David Severin's post “It is all over the place ...”
  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.
  Comment on David Severin's post “It is all over the place ...”
  (30 votes)
Bianca-Karina Kumnig
Posted 5 years ago. Direct link to Bianca-Karina Kumnig's post “if you are given t6= 160 ...”
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)?
Button navigates to signup pageComment on Bianca-Karina Kumnig's post “if you are given t6= 160 ...”
(5 votes)
Answer
- Bsparkle Glitter
  Posted 5 years ago. Direct link to Bsparkle Glitter's post “If you use recursive: ...”
  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.
  Button navigates to signup page
  (21 votes)
RUBY.R.
Posted 5 years ago. Direct link to RUBY.R.'s post “how do you get the equiva...”
how do you get the equivalent formula?
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(6 votes)
Answer
- Aiena
  Posted 5 years ago. Direct link to Aiena's post “Hi Ruby, To get an equiv...”
  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.
  Button navigates to signup page
  (7 votes)
Jimin Park
Posted 5 years ago. Direct link to Jimin Park's post “A geometric sequence invo...”
A geometric sequence involves a common difference that is added to each term, yes or no.
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(2 votes)
Answer
- Aiena
  Posted 5 years ago. Direct link to Aiena's post “Hi Jimin, A geometric se...”
  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.
  Button navigates to signup page
  (10 votes)
kat
Posted 5 years ago. Direct link to kat's post “What does the n-1 mean ag...”
What does the n-1 mean again?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- JF
  Posted 5 years ago. Direct link to JF's post “The n-1 refers to the num...”
  The n-1 refers to the number of times you multiply the common ratio.
  Button navigates to signup page
  (3 votes)
Mateo
Posted 7 months ago. Direct link to Mateo's post “Bro the "Converting recur...”
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
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Timo
  Posted 6 months ago. Direct link to Timo's post “Same here. I figured out ...”
  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.
  Button navigates to signup page
  (4 votes)
Marion Backbier
Posted 5 months ago. Direct link to Marion Backbier's post “How do you type n-1 as a ...”
How do you type n-1 as a power in the answer box? It keeps changing the -1 to a "regular" -1
Button navigates to signup pageComment on Marion Backbier's post “How do you type n-1 as a ...”
(3 votes)
Answer
- Kim Seidel
  Posted 5 months ago. Direct link to Kim Seidel's post “You need to put parenthes...”
  You need to put parentheses around the exponent. For example:
  2^(n-1)
  Hope this helps.
  Button navigates to signup page
  (5 votes)
sarra
Posted 5 months ago. Direct link to sarra's post “do you think khan academy...”
do you think khan academy's exercises prepare you for exams?
Button navigates to signup pageComment on sarra's post “do you think khan academy...”
(3 votes)
Answer
John Murphy
Posted a year ago. Direct link to John Murphy's post “Finally I get to ask a qu...”
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.
Button navigates to signup pageComment on John Murphy's post “Finally I get to ask a qu...”
(2 votes)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “To enter n-1 as an expone...”
  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.
  Button navigates to signup page
  (5 votes)
Arbaaz Ibrahim
Posted 5 years ago. Direct link to Arbaaz Ibrahim's post “When writing the explicit...”
When writing the explicit formula, is it necessary to put the common ratio in parentheses?
Button navigates to signup pageComment on Arbaaz Ibrahim's post “When writing the explicit...”
(4 votes)
Answer