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Intro to arithmetic sequence formulas

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, comma, 5, comma, 7, comma, point, point, point
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, left parenthesis, n, right parenthesis to represent the n, start superscript, start text, t, h, end text, end superscript term of the sequence. For example, here are the first few terms of the arithmetic sequence 3, 5, 7, ...
na, left parenthesis, n, right parenthesis
(The term number)(The n, start superscript, start text, t, h, end text, end superscript term)
13
25
37
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, left parenthesis, n, right parenthesis for any possible n.

Check your understanding

1) Find a, left parenthesis, 4, right parenthesis in the sequence 3, 5, 7, ...
a, left parenthesis, 4, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

2) For any term number n, what does a, left parenthesis, n, minus, 1, right parenthesis represent?
Choose 1 answer:
Choose 1 answer:

Recursive formulas of arithmetic sequences

Recursive formulas give us two pieces of information:
  1. The first term of a sequence
  2. 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.
{a(1)=3The first term is three.a(n)=a(n1)+2Add two to the previous term.\begin{cases}a(1) = 3&\leftarrow\gray{\text{The first term is three.}}\\\\ a(n) = a(n-1)+2&\leftarrow\gray{\text{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, left parenthesis, n, right parenthesisequals, a, left parenthesis, n, minus, 1, right parenthesis, plus, 2
a, left parenthesis, 1, right parenthesisequals, start color #11accd, 3, end color #11accd
a, left parenthesis, 2, right parenthesisequals, a, left parenthesis, 1, right parenthesis, plus, 2equals, start color #11accd, 3, end color #11accd, plus, 2equals, start color #aa87ff, 5, end color #aa87ff
a, left parenthesis, 3, right parenthesisequals, a, left parenthesis, 2, right parenthesis, plus, 2equals, start color #aa87ff, 5, end color #aa87ff, plus, 2equals, start color #1fab54, 7, end color #1fab54
a, left parenthesis, 4, right parenthesisequals, a, left parenthesis, 3, right parenthesis, plus, 2equals, start color #1fab54, 7, end color #1fab54, plus, 2equals, start color #e07d10, 9, end color #e07d10
a, left parenthesis, 5, right parenthesisequals, a, left parenthesis, 4, right parenthesis, plus, 2equals, start color #e07d10, 9, end color #e07d10, plus, 2equals, 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, left parenthesis, n, right parenthesis to represent the n, start superscript, start text, t, h, end text, end superscript term of the sequence 3, 5, 7, ..., we can use other letters to represent other sequences. For example, we can use b, left parenthesis, n, right parenthesis, c, left parenthesis, n, right parenthesis, or d, left parenthesis, n, right parenthesis.
3) Find b, left parenthesis, 4, right parenthesis in the sequence given by {b(1)=5b(n)=b(n1)+9\begin{cases}b(1)=-5\\\\ b(n)=b(n-1)+9 \end{cases}
b, left parenthesis, 4, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

4) Find c, left parenthesis, 3, right parenthesis in the sequence given by {c(1)=20c(n)=c(n1)17\begin{cases}c(1)=20\\\\ c(n)=c(n-1)-17 \end{cases}
c, left parenthesis, 3, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

5) Find d, left parenthesis, 5, right parenthesis in the sequence given by {d(1)=2d(n)=d(n1)+0.4\begin{cases}d(1)=2\\\\ d(n)=d(n-1)+0.4 \end{cases}
d, left parenthesis, 5, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Explicit formulas of arithmetic sequences

Here is an explicit formula of 3, 5, 7, ...
a, left parenthesis, n, right parenthesis, equals, 3, plus, 2, left parenthesis, n, minus, 1, right parenthesis
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, equals, 5 into the explicit formula.
a(5)=3+2(51)=3+24=3+8=11\begin{aligned}a(\greenE 5)&=3+2(\greenE 5-1)\\\\ &=3+2\cdot4\\\\ &=3+8\\\\ &=11\end{aligned}
Lo and behold, we get the same result as before!

Check your understanding

6) Find b, left parenthesis, 10, right parenthesis in the sequence given by b, left parenthesis, n, right parenthesis, equals, minus, 5, plus, 9, left parenthesis, n, minus, 1, right parenthesis.
b, left parenthesis, 10, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

7) Find c, left parenthesis, 8, right parenthesis in the sequence given by c, left parenthesis, n, right parenthesis, equals, 20, minus, 17, left parenthesis, n, minus, 1, right parenthesis.
c, left parenthesis, 8, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

8) Find d, left parenthesis, 21, right parenthesis in the sequence given by d, left parenthesis, n, right parenthesis, equals, 2, plus, 0, point, 4, left parenthesis, n, minus, 1, right parenthesis.
d, left parenthesis, 21, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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, left parenthesis, n, right parenthesis.
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, left parenthesis, 4, right parenthesis, for example, to represent the fourth term, but other sources sometimes write a, start subscript, 4, end subscript.
Both notations are fine to use. We prefer a, left parenthesis, 4, right parenthesis 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?
Choose 1 answer:
Choose 1 answer:

Challenge problem

10) The explicit formula of an arithmetic sequence is f, left parenthesis, n, right parenthesis, equals, 3, minus, 4, left parenthesis, n, minus, 1, right parenthesis.
Which term of the sequence is equal to -65?
Term number
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.

Want to join the conversation?

  • blobby green style avatar for user Mrs. Beck
    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?
    (12 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user hcomet2062
    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?
    (14 votes)
    Default Khan Academy avatar avatar for user
  • male robot hal style avatar for user farchettiensis
    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?
    (5 votes)
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  • starky seedling style avatar for user Tiya Sharma
    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?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      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!
      (15 votes)
  • aqualine sapling style avatar for user Frenchy Starfire (I'm offiline so don't try anything funny XD)
    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
    (10 votes)
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  • duskpin sapling style avatar for user alexa.pomerantz
    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...?).
    (4 votes)
    Default Khan Academy avatar avatar for user
    • hopper jumping style avatar for user Michael Laman
      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.
      (7 votes)
  • blobby green style avatar for user John Paul Dominic Baylosis
    Is a(n) more confusing than "a" sub "n"?
    It makes me think "a multiplied to n" sometimes.
    (5 votes)
    Default Khan Academy avatar avatar for user
    • leaf grey style avatar for user Alex
      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).
      (4 votes)
  • duskpin sapling style avatar for user sierra
    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
    (3 votes)
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  • aqualine seed style avatar for user sadaf seemeen
    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.
    (3 votes)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      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.
      (6 votes)
  • male robot donald style avatar for user Vinay Sharma
    If we were to graph this, it would make a line. Since every time we increase 1 term, our nth term increase by a certain amount. This is really a neat concept. But, Sal somewhere else on the topic of sequence said that this won't work all of the time. Can anybody tell me about this "not all of the time".
    (4 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      The graph for an arithmetic sequence is actually a series of points that would sit on a line. They don't themselves create a line because there are gaps (other values) between the numbers in the sequences.

      A graph for a geometric sequence would not look like a line. It would look like an exponential functions (again with gaps for the values not in the sequence).

      I hope this answers your question.
      (3 votes)