How do you write an explicit formula for the sequence (8,11,16,23,32)

Notice that the sequence of differences is simply the sequence of odd numbers. So this is the sequence of squares shifted by a constant. Since 8-1=7, we have a(n) = n^2 +7

When given an arithmetic sequence with a negative term, how do you calculate what term the number will be when it reaches 0?

Some sequences do not even have 0 as one of the possibilities. So if you start with a negative and have a negative common difference, then there is no zero such as -2, -4, -6, -8, ... Others will have a common difference that skips over zero such as -4, 4, 12, 20, 28, ... If the sequence does include a zero, such as an initial value of - 9 and a common difference of 3, we get an equation f(n) = -9 + 3(n-1). set this equal to zero, 0 = -9 + 3(n-1) move the - 9 by adding and distribute to get 9 = 3n - 3, add 3 to both sides, 3n = 12., divide by 3 to get n = 4. Sequence would be -9, -6, -3, 0 ... which is what we were looking for. So if your equation skipped 0 as above f(n) = -4 +8(n-1) and did the same thing, then 0 = -4 + 8n - 8, add 12 to get 12 = 8n, n = 12/8 = 3/2, but a sequence does not have a 3/2 term, only 1, 2, 3 ... terms.

How would you find the explicit formula for a problem such as: "a" subscript 38 =-53.2, while the common difference is -1.1?

You need to know a common difference and one 'value' (sequence) in order to find the general form (formula). Since the common difference is -1.1, it would look like this: -1.1(n-1) + b (if you are starting from n=1). Then plug in "a" subscript 38 = -53.2 into the formula. Then it would be -1.1(38-1) + b = -53.2 All you have to do is solving for 'b' and it would be -12.5. Therefore, the general form (formula) would be -1.1(n-1) - 12.5 if you are starting from n=1.

how are the formulas for arithmetic sequences similar to functions?

Arithmetic sequences are functions. The difference between a linear function and a arithmetic sequence is that the first is continuous and the second is discrete, but the continuous line would go through all the points of its equivalent sequence.

which formula is more suitable to use recursive or explicit?

Most of the time, explicit has more power, but for a few occasions such as the Fibonacci sequence, the recursive is more applicable. The explicit equation is closer to the slope intercept form of the line that would go through all the poins of a particular arithmetic sequence.

Main content

Arithmetic sequences review

CCSS.Math:

Review arithmetic sequences and solve various problems involving them.

Parts and formulas of arithmetic sequences

In arithmetic sequences, the difference between consecutive terms is always the same. We call that difference the common difference.

For example, the common difference of the following sequence is plus, 2:

	start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma		5, comma		7, comma		9, comma, point, point, point

Arithmetic 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 arithmetic sequence whose first term is start color #11accd, k, end color #11accd and common difference is start color #ed5fa6, d, end color #ed5fa6:

a, left parenthesis, n, right parenthesis, equals, start color #11accd, k, end color #11accd, plus, left parenthesis, n, minus, 1, right parenthesis, start color #ed5fa6, d, end color #ed5fa6

This is the recursive formula of that sequence:

\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)+\maroonC d \end{cases}

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

Extending arithmetic sequences

Suppose we want to extend the sequence 3, comma, 8, comma, 13, comma, point, point, point We can see each term is start color #ed5fa6, plus, 5, end color #ed5fa6 from the previous term:

	start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma		8, comma		13, comma, point, point, point

So we simply add that difference to find that the next term is 18:

	start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6		start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma		8, comma		13, comma		18, comma, point, point, point

Problem 1

Current

What is the next term in the sequence minus, 5, comma, minus, 1, comma, 3, comma, 7, 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 3, comma, 8, comma, 13, comma, point, point, point We already know the common difference is start color #ed5fa6, plus, 5, end color #ed5fa6. We can also see that the first term is start color #11accd, 3, end color #11accd. Therefore, this is a recursive formula for the sequence:

\begin{cases}a(1) = \blueD 3 \\\\ a(n) = a(n-1)\maroonC{+5} \end{cases}

Problem 1

Current

Find k and d in this recursive formula of the sequence minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots.

\begin{cases}a(1) = k \\\\ a(n) = a(n-1)+d \end{cases}

k, equals

d, equals

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

Writing explicit formulas

Suppose we want to write an explicit formula for 3, comma, 8, comma, 13, comma, point, point, point We already know the common difference is start color #ed5fa6, plus, 5, end color #ed5fa6 and the first term is start color #11accd, 3, end color #11accd. Therefore, this is an explicit formula for the sequence:

a, left parenthesis, n, right parenthesis, equals, start color #11accd, 3, end color #11accd, start color #ed5fa6, plus, 5, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis

Problem 1

Current

Write an explicit formula for minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots

a, left parenthesis, n, right parenthesis, equals

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

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Robert Bass
Posted 5 years ago. Direct link to Robert Bass's post “How do you write an expli...”
How do you write an explicit formula for the sequence (8,11,16,23,32)
Button navigates to signup pageComment on Robert Bass's post “How do you write an expli...”
(8 votes)
Answer
- Tahlequah
  Posted 4 years ago. Direct link to Tahlequah 's post “Notice that the sequence ...”
  Notice that the sequence of differences is simply the sequence of odd numbers. So this is the sequence of squares shifted by a constant. Since 8-1=7, we have
  a(n) = n^2 +7
  Comment on Tahlequah 's post “Notice that the sequence ...”
  (6 votes)
lindah17
Posted 5 years ago. Direct link to lindah17's post “Why was the Arithmetic Se...”
Why was the Arithmetic Sequence review so hard on my mind? It took me to look at the explanation to understand the answers to every darn question. And, I am yet so confused on this section.
Button navigates to signup pageComment on lindah17's post “Why was the Arithmetic Se...”
(8 votes)
Answer
- Kshiviraj2177
  Posted 2 years ago. Direct link to Kshiviraj2177's post “I watched all the videos ...”
  I watched all the videos and I'm ready for my semester exams. The concept is really easy once you get to know it.
  Comment on Kshiviraj2177's post “I watched all the videos ...”
  (0 votes)
Levi Dawit
Posted 5 years ago. Direct link to Levi Dawit's post “When given an arithmetic ...”
When given an arithmetic sequence with a negative term, how do you calculate what term the number will be when it reaches 0?
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(3 votes)
Answer
- David Severin
  Posted 5 years ago. Direct link to David Severin's post “Some sequences do not eve...”
  Some sequences do not even have 0 as one of the possibilities. So if you start with a negative and have a negative common difference, then there is no zero such as -2, -4, -6, -8, ... Others will have a common difference that skips over zero such as -4, 4, 12, 20, 28, ... If the sequence does include a zero, such as an initial value of - 9 and a common difference of 3, we get an equation f(n) = -9 + 3(n-1). set this equal to zero, 0 = -9 + 3(n-1) move the - 9 by adding and distribute to get 9 = 3n - 3, add 3 to both sides, 3n = 12., divide by 3 to get n = 4. Sequence would be -9, -6, -3, 0 ... which is what we were looking for. So if your equation skipped 0 as above f(n) = -4 +8(n-1) and did the same thing, then 0 = -4 + 8n - 8, add 12 to get 12 = 8n, n = 12/8 = 3/2, but a sequence does not have a 3/2 term, only 1, 2, 3 ... terms.
  Button navigates to signup page
  (4 votes)
busygirl022705
Posted 4 years ago. Direct link to busygirl022705's post “How would you find the ex...”
How would you find the explicit formula for a problem such as:
"a" subscript 38 =-53.2, while the common difference is -1.1?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- David Lee
  Posted 4 years ago. Direct link to David Lee's post “You need to know a common...”
  You need to know a common difference and one 'value' (sequence) in order to find the general form (formula). Since the common difference is -1.1, it would look like this: -1.1(n-1) + b (if you are starting from n=1).
  Then plug in "a" subscript 38 = -53.2 into the formula. Then it would be -1.1(38-1) + b = -53.2
  All you have to do is solving for 'b' and it would be -12.5.
  Therefore, the general form (formula) would be -1.1(n-1) - 12.5 if you are starting from n=1.
  Comment on David Lee's post “You need to know a common...”
  (5 votes)
isabella.knight
Posted 5 years ago. Direct link to isabella.knight's post “how are the formulas for ...”
how are the formulas for arithmetic sequences similar to functions?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- David Severin
  Posted 5 years ago. Direct link to David Severin's post “Arithmetic sequences are ...”
  Arithmetic sequences are functions. The difference between a linear function and a arithmetic sequence is that the first is continuous and the second is discrete, but the continuous line would go through all the points of its equivalent sequence.
  Button navigates to signup page
  (4 votes)
rmikush
Posted 3 years ago. Direct link to rmikush's post “how do you write a recurs...”
how do you write a recursive formula for 1, 1, 0, -1, -1, 0, 1, 1
Button navigates to signup pageComment on rmikush's post “how do you write a recurs...”
(3 votes)
Answer
Valarie Torres
Posted 3 years ago. Direct link to Valarie Torres's post “Consider the arithmetic s...”
Consider the arithmetic sequence 27, 13, -1, ...
The explicit rule for the sequence in terms of n is a(n)= 27-14(n-1). If the nth term is -841, find the value of n.
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(2 votes)
Answer
- Kim Seidel
  Posted 3 years ago. Direct link to Kim Seidel's post “If the nth term is -841, ...”
  If the nth term is -841, then you were just given the result. Basically a(n)=-841
  So, change the equation into -841=27-14(n-1).
  You can now solve for n.
  Hope this helps.
  Button navigates to signup page
  (1 vote)
468234
Posted a year ago. Direct link to 468234's post “How would you write a exp...”
How would you write a explicit formula when f(0)=5, f(x)=f(x-1)+(2x-1)?
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(2 votes)
Answer
Kanyon Christman
Posted 2 months ago. Direct link to Kanyon Christman's post “Hi, sup bruuuuhh”
Hi, sup bruuuuhh
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(2 votes)
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Benjamin Araujo
Posted 2 months ago. Direct link to Benjamin Araujo's post “which formula is more sui...”
which formula is more suitable to use recursive or explicit?
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(1 vote)
Answer
- David Severin
  Posted 2 months ago. Direct link to David Severin's post “Most of the time, explici...”
  Most of the time, explicit has more power, but for a few occasions such as the Fibonacci sequence, the recursive is more applicable. The explicit equation is closer to the slope intercept form of the line that would go through all the poins of a particular arithmetic sequence.
  Comment on David Severin's post “Most of the time, explici...”
  (2 votes)

Algebra 1

Course: Algebra 1 > Unit 9

Arithmetic sequences review

Parts and formulas of arithmetic sequences

Extending arithmetic sequences

Writing recursive formulas

Writing explicit formulas

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