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.

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.

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.

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.

are there any lessons on solving arithmetic sequences

https://www.khanacademy.org/math/algebra/sequences/introduction-to-arithmetic-squences/v/explicit-and-recursive-definitions-of-sequences This starts a string of lessons

Main content

Algebra 1

Course: Algebra 1 > Unit 9

Lesson 4: Constructing geometric sequences

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:

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}

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		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		2, 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.

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Bianca-Karina Kumnig
Posted 4 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)?
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(4 votes)
Answer
- Bsparkle Glitter
  Posted 4 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
  (16 votes)
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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(3 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 ...”
  (17 votes)
RUBY.R.
Posted 4 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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(5 votes)
Answer
- Aiena
  Posted 4 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
  (5 votes)
kat
Posted 4 years ago. Direct link to kat's post “What does the n-1 mean ag...”
What does the n-1 mean again?
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(4 votes)
Answer
- JF
  Posted 4 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)
Jimin Park
Posted 4 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 4 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
  (7 votes)
Arbaaz Ibrahim
Posted 4 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
John Murphy
Posted 4 months 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 4 months 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
  (4 votes)
Lee Isaiah
Posted 5 years ago. Direct link to Lee Isaiah's post “Why do the explicit and r...”
Why do the explicit and recursive formulas work?
Button navigates to signup pageComment on Lee Isaiah's post “Why do the explicit and r...”
(5 votes)
Answer
- sade coore
  Posted 5 years ago. Direct link to sade coore's post “Because there all number ...”
  Because there all number that are put together in the right place to make something work
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  (0 votes)
kushal Tirupathi
Posted 6 years ago. Direct link to kushal Tirupathi's post “are there any lessons on ...”
are there any lessons on solving arithmetic sequences
Button navigates to signup pageComment on kushal Tirupathi's post “are there any lessons on ...”
(0 votes)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “https://www.khanacademy.o...”
  https://www.khanacademy.org/math/algebra/sequences/introduction-to-arithmetic-squences/v/explicit-and-recursive-definitions-of-sequences
  This starts a string of lessons
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  (5 votes)
smiller3
Posted 3 years ago. Direct link to smiller3's post “at the end of last exampl...”
at the end of last example the equivalentproblem removed the n-1 and n as the exponent . Why
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(2 votes)
Answer