If explicit formulas of the sequence are easier to find the 100th term of a sequence (based on the reflection question), why would we need recursive formulas? When are they useful?

You are right, We can do any sequence with explicit form just as easily as with recursive. They just wanted to make sure we knew that you can use both.

I have a question about your opinion on notation. So I am currently teaching this concept and after seeing this, I feel my students would have felt so much more comfortable using the notation you have as representing the arithmetic sequence as a function. The book we use uses asubn notation. Do you think there is a reason to use one notation over the other?

Certainly 𝑎(𝑛) helps tie in to functions. The only other practical consideration I can think of is that if a majority of mathematicians, books, websites, etc. use 𝑎ₙ then they should be familiar with it. But I don't know which is more common.

I feel like a robot using the explicit formula. Why on earth does it work? It seems phony, since we are always given the formulas that define the sequence in each exercise. How does one arrive to such formula?

Think y=mx+b. Recognize that the constant common difference is "m" and the term before the initial value is "b"

In problem seven it gives the problem Find c(8) in the sequence given by c(n)= 20-17(n-1) okay so how would i solve that problem i thought that you multiplied 17 by 17 and then subtracted it by 20 but when you click on the help button it says c(8)=20-17 (8-1) =20-17*7 =20-119 = -99 So my question is how they got the 17*7 and not 17*17

Because of the 8-1=7 and then you do 17*7 to get 119 and then subtract 119 from 20 to get -99

why do we have recursive formulas? I feel like explicit formulas is better and easier

First of all, it is useful in math to have multiply ways to think of/represent a concept. But there is also a good reason why we use recursive sequences. This is because some sequences are very difficult or impossible to represent explicitly but can be represented recursively. Take the Fibonacci sequence for example. You get each term by combining the two terms before it. The first few terms are 0, 1, 1, 2, 3, 5, 8, 13, 21,…. This can be expressed recursively as a(n)=a(n-2)+a(n-1), for a(0)=0 and a(1)=1. Not that bad. Now I encourage you to look up the explicit formula for the Fibonacci sequence. Which one do you think is simpler?

Is a(n) more confusing than "a" sub "n"? It makes me think "a multiplied to n" sometimes.

a_n (which is a sub n) typically means the nth term of a sequence. It can also be found in the definition of the sequence. a(n) (a of n) is definitely less used; if one wanted to define the sequence as a function, one would use f(n) or even f(x).

what is difference between explicit formula and recursiue formula because both equation looks similar jus we changed the side of our common difference d. i mean a(n)=a+(n-1)D or a(n)=a+D(n-1) we just changed the side of D . SO are they similar?

Actually the explicit formula for an arithmetic sequence is a(n)=a+(n-1)*D, and the recursive formula is a(n) = a(n-1) + D (instead of a(n)=a+D(n-1)). The difference is than an explicit formula gives the nth term of the sequence as a function of n alone, whereas a recursive formula gives the nth term of a sequence as a function of the previous term(s) of the sequence (and sometimes of n as well). In other words, an explicit formula tells how to calculate directly the nth term of a sequence, whereas a recursive formula tells how to get from one or more terms of the sequence to the next term. Have a blessed, wonderful day!

How are recursive formulas still functions? The second part of the recursive formula seems to me like using a word in that word's definition (because the letter used to show the function is used in the definition of the function, if that makes sense...?).

A function specifies a relationship between two variables. While a recursive formula does use itself to sequence back to its initial value, the result is still specifying the relationship between two variables. Also notice that one can change arithmetic sequences to recursive formulas and vice versa. Both are functions. Recursion itself, whether in math or in other places such as computer programming, can be a very different way of looking at a problem. Recursion is not limited to mathematics, but that's a whole different topic all together.

I still don't understand what you mean. I tried clicking 'I need help' but that didn't explain it well enough for me to understand

Don't understand what? Mabye I could help you.

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P.

assume that the common difference is y and the first term is x, so x + x + y + x + 2y + x + 3y + x + 4y + x + 5y + x + 6y = 63 which means 7x + 21y = 63 x + 7y + x + 8y + x + 9y + x + 10y + x + 11y + x + 12y + x + 13y = 161 so 7x + 70y = 161, subtract the first equation from the second and get 49y = 98, so y =2 and substitute this into either equation, 7x + 21(2) = 63, 7x = 21, x = 3. x is the first term, and two is the common difference, so f(n) = 2(n-1) + 3. f(28)= 2(27) + 3 = 57.

Main content

Course: Algebra 1 (Eureka Math/EngageNY) > Unit 3

Lesson 1: Topic A: Lessons 1-3: Arithmetic sequence intro

Intro to arithmetic sequence formulas

Google Classroom

Get comfortable with the basics of explicit and recursive formulas for arithmetic sequences.

Before taking this lesson, make sure you know the basics of arithmetic sequences and have some experience with evaluating functions and function domain.

What is a formula?

We are used to describing arithmetic sequences like this:

3, 5, 7, …

But there are other ways. In this lesson, we'll be learning two new ways to represent arithmetic sequences: recursive formulas and explicit formulas. Formulas give us instructions on how to find any term of a sequence.

To remain general, formulas use

n

to represent any term number and

a (n)

to represent the

n^{th}

term of the sequence. For example, here are the first few terms of the arithmetic sequence 3, 5, 7, ...

$n$ ‍	$a (n)$ ‍
(The term number)	(The $n^{th}$ ‍ term)
$1$ ‍	$3$ ‍
$2$ ‍	$5$ ‍
$3$ ‍	$7$ ‍

We mentioned above that formulas give us instructions on how to find any term of a sequence. Now we can rephrase this as follows: formulas tell us how to find $a (n)$ ‍ for any possible $n$ ‍.

Check your understanding

1) Find $a (4)$ ‍ in the sequence 3, 5, 7, ...

a (4) =

	$+ 2 ↷$ ‍		$+ 2 ↷$ ‍		$+ 2 ↷$ ‍
$3$ ‍		$5$ ‍		$7$ ‍		$9$ ‍
$a (1)$ ‍		$a (2)$ ‍		$a (3)$ ‍		$a (4)$ ‍

2) For any term number $n$ ‍, what does $a (n - 1)$ ‍ represent?

Let's look at a few specific values for

n

$n$ ‍	$a (n - 1)$ ‍	Interpretation of $a (n - 1)$ ‍
$2$ ‍	$a (2 - 1) = a (1)$ ‍	The first term, which is the term before $a (2)$ ‍
$3$ ‍	$a (3 - 1) = a (2)$ ‍	The second term, which is the term before $a (3)$ ‍
$4$ ‍	$a (4 - 1) = a (3)$ ‍	The third term, which is the term before $a (4)$ ‍

In general, we see that $a (n - 1)$ ‍ is the term that comes before $a (n)$ ‍.

Recursive formulas of arithmetic sequences

Recursive formulas give us two pieces of information:

The first term of a sequence
The pattern rule to get any term in a sequence from the term that comes before it

Here is the recursive formula of our sequence 3, 5, 7, ... along with the interpretation for each part.

{\begin{cases} a (1) = 3 & \leftarrow The first term is three. \\ a (n) = a (n - 1) + 2 & \leftarrow Add two to the previous term. \end{cases}

In order to find the fifth term, for example, we need to extend the sequence term by term:

$a (n)$ ‍	$= a (n - 1) + 2$ ‍
$a (1)$ ‍			$= 3$ ‍
$a (2)$ ‍	$= a (1) + 2$ ‍	$= 3 + 2$ ‍	$= 5$ ‍
$a (3)$ ‍	$= a (2) + 2$ ‍	$= 5 + 2$ ‍	$= 7$ ‍
$a (4)$ ‍	$= a (3) + 2$ ‍	$= 7 + 2$ ‍	$= 9$ ‍
$a (5)$ ‍	$= a (4) + 2$ ‍	$= 9 + 2$ ‍	$= 11$ ‍

Cool! This formula gives us the same sequence as described by 3, 5, 7, ...

Check your understanding

Now it's your turn to find terms of sequences using their recursive formulas.

Just as we used

a (n)

to represent the

n^{th}

term of the sequence 3, 5, 7, ..., we can use other letters to represent other sequences. For example, we can use

b (n)

c (n)

, or

d (n)

3) Find $b (4)$ ‍ in the sequence given by

{\begin{cases} b (1) = - 5 \\ b (n) = b (n - 1) + 9 \end{cases}

b (4) =

The meaning of this formula can be verbalized as follows:

The first term is negative five, and any other term is the term before it plus nine.

In order to find

b (4)

, we need to extend the sequence term by term:

$b (n)$ ‍	$= b (n - 1) + 9$ ‍
$b (1)$ ‍			$= - 5$ ‍
$b (2)$ ‍	$= b (1) + 9$ ‍	$= - 5 + 9$ ‍	$= 4$ ‍
$b (3)$ ‍	$= b (2) + 9$ ‍	$= 4 + 9$ ‍	$= 13$ ‍
$b (4)$ ‍	$= b (3) + 9$ ‍	$= 13 + 9$ ‍	$= 22$ ‍

Therefore, $b (4) = 22$ ‍.

4) Find $c (3)$ ‍ in the sequence given by

{\begin{cases} c (1) = 20 \\ c (n) = c (n - 1) - 17 \end{cases}

c (3) =

The meaning of this formula can be verbalized as follows:

The first term is 20, and any other term is the term before it minus 17.

In order to find

c (3)

, we need to extend the sequence term by term:

$c (n)$ ‍	$= c (n - 1) - 17$ ‍
$c (1)$ ‍			$= 20$ ‍
$c (2)$ ‍	$= c (1) - 17$ ‍	$= 20 - 17$ ‍	$= 3$ ‍
$c (3)$ ‍	$= c (2) - 17$ ‍	$= 3 - 17$ ‍	$= - 14$ ‍

Therefore, $c (3) = - 14$ ‍.

5) Find $d (5)$ ‍ in the sequence given by

{\begin{cases} d (1) = 2 \\ d (n) = d (n - 1) + 0.4 \end{cases}

d (5) =

The meaning of this formula can be verbalized as follows:

The first term is two, and any other term is the term before it plus 0.4.

In order to find

d (5)

, we need to extend the sequence term by term:

$d (n)$ ‍	$= d (n - 1) + 0.4$ ‍
$d (1)$ ‍			$= 2$ ‍
$d (2)$ ‍	$= d (1) + 0.4$ ‍	$= 2 + 0.4$ ‍	$= 2.4$ ‍
$d (3)$ ‍	$= d (2) + 0.4$ ‍	$= 2.4 + 0.4$ ‍	$= 2.8$ ‍
$d (4)$ ‍	$= d (3) + 0.4$ ‍	$= 2.8 + 0.4$ ‍	$= 3.2$ ‍
$d (5)$ ‍	$= d (4) + 0.4$ ‍	$= 3.2 + 0.4$ ‍	$= 3.6$ ‍

Therefore, $d (5) = 3.6$ ‍.

Explicit formulas of arithmetic sequences

Here is an explicit formula of 3, 5, 7, ...

a (n) = 3 + 2 (n - 1)

This formula allows us to simply plug in the number of the term we are interested in to get the value of that term.

In order to find the fifth term, for example, we need to plug

n = 5

into the explicit formula.

\begin{aligned} a (5) & = 3 + 2 (5 - 1) \\ = 3 + 2 \cdot 4 \\ = 3 + 8 \\ = 11 \end{aligned}

Lo and behold, we get the same result as before!

Check your understanding

6) Find $b (10)$ ‍ in the sequence given by $b (n) = - 5 + 9 (n - 1)$ ‍.

b (10) =

7) Find $c (8)$ ‍ in the sequence given by $c (n) = 20 - 17 (n - 1)$ ‍.

c (8) =

8) Find $d (21)$ ‍ in the sequence given by $d (n) = 2 + 0.4 (n - 1)$ ‍.

d (21) =

Sequences are functions

Notice that the formulas we used in this lesson work like functions: We input a term number

n

, and the formula outputs the value of that term

a (n)

Sequences are in fact defined as functions. However,

n

cannot be any real number value. There's no such thing as the negative fifth term or the 0.4th term of a sequence.

This means that the domain of sequences—which is the set of all possible inputs of the function—is the positive integers.

A note about notation

We've been writing

a (4)

, for example, to represent the fourth term, but other sources sometimes write

a_{4}

Both notations are fine to use. We prefer

a (4)

because it emphasizes that sequences are functions.

Reflection question

9) Which type of formula is more useful for quickly finding the 100th term of an arithmetic sequence?

To find the 100th term using the recursive formula, we have to extend the sequence term-by-term until we get to 100. That will take a very long time!

To find the 100th term using the explicit formula, all we need to do is to plug

n = 100

into the formula and evaluate. This doesn't take more time than finding the second term, or the 1000th term, for that matter.

In conclusion, the explicit formula is more useful for finding the 100th term of an arithmetic sequence.

Challenge problem

10) The explicit formula of an arithmetic sequence is

f (n) = 3 - 4 (n - 1)

Which term of the sequence is equal to -65?

Term number

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hcomet2062
Posted 7 years ago. Direct link to hcomet2062's post “If explicit formulas of t...”
If explicit formulas of the sequence are easier to find the 100th term of a sequence (based on the reflection question), why would we need recursive formulas? When are they useful?
Button navigates to signup pageComment on hcomet2062's post “If explicit formulas of t...”
(26 votes)
Answer
- MrLogic642
  Posted 7 years ago. Direct link to MrLogic642's post “You are right, We can do ...”
  You are right, We can do any sequence with explicit form just as easily as with recursive. They just wanted to make sure we knew that you can use both.
  Comment on MrLogic642's post “You are right, We can do ...”
  (24 votes)
Mrs. Beck
Posted 8 years ago. Direct link to Mrs. Beck's post “I have a question about y...”
I have a question about your opinion on notation. So I am currently teaching this concept and after seeing this, I feel my students would have felt so much more comfortable using the notation you have as representing the arithmetic sequence as a function. The book we use uses asubn notation. Do you think there is a reason to use one notation over the other?
Button navigates to signup pageButton navigates to signup page
(19 votes)
Answer
- Jordan Cooper
  Posted 6 years ago. Direct link to Jordan Cooper's post “Certainly 𝑎(𝑛) helps ti...”
  Certainly 𝑎(𝑛) helps tie in to functions.
  
  The only other practical consideration I can think of is that if a majority of mathematicians, books, websites, etc. use 𝑎ₙ then they should be familiar with it. But I don't know which is more common.
  Button navigates to signup page
  (14 votes)
farchettiensis
Posted 6 years ago. Direct link to farchettiensis's post “I feel like a robot using...”
I feel like a robot using the explicit formula. Why on earth does it work? It seems phony, since we are always given the formulas that define the sequence in each exercise. How does one arrive to such formula?
Button navigates to signup pageButton navigates to signup page
(10 votes)
Answer
- La Brant, Geoff
  Posted 6 years ago. Direct link to La Brant, Geoff's post “Think y=mx+b. Recognize ...”
  Think y=mx+b. Recognize that the constant common difference is "m" and the term before the initial value is "b"
  Comment on La Brant, Geoff's post “Think y=mx+b. Recognize ...”
  (18 votes)
Tiya Sharma
Posted 6 years ago. Direct link to Tiya Sharma's post “what is difference betwee...”
what is difference between explicit formula and recursiue formula because both equation looks similar jus we changed the side of our common difference d.
i mean a(n)=a+(n-1)D or a(n)=a+D(n-1)
we just changed the side of D .
SO are they similar?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Ian Pulizzotto
  Posted 6 years ago. Direct link to Ian Pulizzotto's post “Actually the explicit for...”
  Actually the explicit formula for an arithmetic sequence is a(n)=a+(n-1)*D, and the recursive formula is a(n) = a(n-1) + D (instead of a(n)=a+D(n-1)).
  
  The difference is than an explicit formula gives the nth term of the sequence as a function of n alone, whereas a recursive formula gives the nth term of a sequence as a function of the previous term(s) of the sequence (and sometimes of n as well). In other words, an explicit formula tells how to calculate directly the nth term of a sequence, whereas a recursive formula tells how to get from one or more terms of the sequence to the next term.
  
  Have a blessed, wonderful day!
  Comment on Ian Pulizzotto's post “Actually the explicit for...”
  (21 votes)
Eva
Posted 8 months ago. Direct link to Eva's post “why do we have recursive ...”
why do we have recursive formulas? I feel like explicit formulas is better and easier
Button navigates to signup pageComment on Eva's post “why do we have recursive ...”
(9 votes)
Answer
- Tanner P
  Posted 8 months ago. Direct link to Tanner P's post “First of all, it is usefu...”
  First of all, it is useful in math to have multiply ways to think of/represent a concept.
  
  But there is also a good reason why we use recursive sequences. This is because some sequences are very difficult or impossible to represent explicitly but can be represented recursively.
  
  Take the Fibonacci sequence for example. You get each term by combining the two terms before it. The first few terms are 0, 1, 1, 2, 3, 5, 8, 13, 21,…. This can be expressed recursively as a(n)=a(n-2)+a(n-1), for a(0)=0 and a(1)=1. Not that bad. Now I encourage you to look up the explicit formula for the Fibonacci sequence. Which one do you think is simpler?
  Comment on Tanner P's post “First of all, it is usefu...”
  (12 votes)
Frenchy Starfire (I'm offiline so don't try anything funny XD)
Posted 7 years ago. Direct link to Frenchy Starfire (I'm offiline so don't try anything funny XD)'s post “In problem seven it gives...”
In problem seven it gives the problem Find c(8) in the sequence given by c(n)= 20-17(n-1) okay so how would i solve that problem i thought that you multiplied 17 by 17 and then subtracted it by 20 but when you click on the help button it says
c(8)=20-17 (8-1)
=20-17*7
=20-119
= -99
So my question is how they got the 17*7 and not 17*17
Button navigates to signup pageComment on Frenchy Starfire (I'm offiline so don't try anything funny XD)'s post “In problem seven it gives...”
(10 votes)
Answer
- Austin Bruner
  Posted 6 years ago. Direct link to Austin Bruner's post “Because of the 8-1=7 and ...”
  Because of the 8-1=7 and then you do 17*7 to get 119 and then subtract 119 from 20 to get -99
  Button navigates to signup page
  (5 votes)
alexa.pomerantz
Posted 7 years ago. Direct link to alexa.pomerantz's post “How are recursive formula...”
How are recursive formulas still functions? The second part of the recursive formula seems to me like using a word in that word's definition (because the letter used to show the function is used in the definition of the function, if that makes sense...?).
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Michael Laman
  Posted 7 years ago. Direct link to Michael Laman's post “A function specifies a re...”
  A function specifies a relationship between two variables. While a recursive formula does use itself to sequence back to its initial value, the result is still specifying the relationship between two variables.
  
  Also notice that one can change arithmetic sequences to recursive formulas and vice versa. Both are functions.
  
  Recursion itself, whether in math or in other places such as computer programming, can be a very different way of looking at a problem. Recursion is not limited to mathematics, but that's a whole different topic all together.
  Comment on Michael Laman's post “A function specifies a re...”
  (7 votes)
John Paul Dominic Baylosis
Posted 5 years ago. Direct link to John Paul Dominic Baylosis's post “Is a(n) more confusing th...”
Is a(n) more confusing than "a" sub "n"?
It makes me think "a multiplied to n" sometimes.
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Alex
  Posted 5 years ago. Direct link to Alex's post “a_n (which is a sub n) ty...”
  a_n (which is a sub n) typically means the nth term of a sequence. It can also be found in the definition of the sequence.
  
  a(n) (a of n) is definitely less used; if one wanted to define the sequence as a function, one would use f(n) or even f(x).
  Button navigates to signup page
  (4 votes)
sierra
Posted 7 years ago. Direct link to sierra's post “I still don't understand ...”
I still don't understand what you mean. I tried clicking 'I need help' but that didn't explain it well enough for me to understand
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- HawkeHPP
  Posted 7 years ago. Direct link to HawkeHPP's post “Don't understand what? Ma...”
  Don't understand what? Mabye I could help you.
  Button navigates to signup page
  (7 votes)
sadaf seemeen
Posted 7 years ago. Direct link to sadaf seemeen's post “The sum of the first 7 te...”
The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P.
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- David Severin
  Posted 7 years ago. Direct link to David Severin's post “assume that the common di...”
  assume that the common difference is y and the first term is x, so
  x + x + y + x + 2y + x + 3y + x + 4y + x + 5y + x + 6y = 63 which means 7x + 21y = 63
  x + 7y + x + 8y + x + 9y + x + 10y + x + 11y + x + 12y + x + 13y = 161 so 7x + 70y = 161, subtract the first equation from the second and get 49y = 98, so y =2 and substitute this into either equation,
  7x + 21(2) = 63, 7x = 21, x = 3. x is the first term, and two is the common difference, so f(n) = 2(n-1) + 3. f(28)= 2(27) + 3 = 57.
  Comment on David Severin's post “assume that the common di...”
  (7 votes)