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Recursive formulas for arithmetic sequences

Learn how to find recursive formulas for arithmetic sequences. For example, find the recursive formula of 3, 5, 7,...
Before taking this lesson, make sure you are familiar with the basics of arithmetic sequence formulas.

How recursive formulas work

Recursive formulas give us two pieces of information:
  1. The first term of the sequence
  2. The pattern rule to get any term from the term that comes before it
Here is a recursive formula of the sequence 3, comma, 5, comma, 7, comma, point, point, point along with the interpretation for each part.
{a(1)=3the first term is 3a(n)=a(n1)+2add 2 to the previous term\begin{cases}a(1) = 3&\leftarrow\gray{\text{the first term is 3}}\\\\ a(n) = a(n-1)+2&\leftarrow\gray{\text{add 2 to the previous term}} \end{cases}
In the formula, n is any term number and a, left parenthesis, n, right parenthesis is the n, start superscript, start text, t, h, end text, end superscript term. This means a, left parenthesis, 1, right parenthesis is the first term, and a, left parenthesis, n, minus, 1, right parenthesis is the term before the n, start superscript, start text, t, h, end text, end superscript term.
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 #0d923f, 3, end color #0d923f
a, left parenthesis, 2, right parenthesisequals, a, left parenthesis, 1, right parenthesis, plus, 2equals, start color #0d923f, 3, end color #0d923f, 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 #11accd, 7, end color #11accd
a, left parenthesis, 4, right parenthesisequals, a, left parenthesis, 3, right parenthesis, plus, 2equals, start color #11accd, 7, end color #11accd, 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, comma, 5, comma, 7, comma, point, point, point

Check your understanding

1) 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

Writing recursive formulas

Suppose we wanted to write the recursive formula of the arithmetic sequence 5, comma, 8, comma, 11, comma, point, point, point
The two parts of the formula should give the following information:
  • The first term left parenthesiswhich is start color #0d923f, 5, end color #0d923f, right parenthesis
  • The rule to get any term from its previous term left parenthesiswhich is "add start color #ed5fa6, 3, end color #ed5fa6"right parenthesis
Therefore, the recursive formula should look as follows:
{c(1)=5c(n)=c(n1)+3\begin{cases}c(1)=\greenE 5\\\\ c(n)=c(n-1)\maroonC{+3} \end{cases}

Check your understanding

2) What is the recursive formula of the sequence 12, comma, 7, comma, 2, comma, point, point, point ?
Choose 1 answer:
Choose 1 answer:

3) Complete the missing values in the recursive formula of the sequence 2, comma, 8, comma, 14, comma, point, point.
{e(1)=Ae(n)=e(n1)+B\begin{cases}e(1)=A\\\\ e(n)=e(n-1)+B \end{cases}
A, 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
B, 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) Complete the missing values in the recursive formula of the sequence minus, 1, comma, minus, 4, comma, minus, 7, comma, point, point, point.
{f(1)=Af(n)=f(n1)+B\begin{cases}f(1)=A\\\\ f(n)=f(n-1)+B \end{cases}
A, 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
B, 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

Reflection question

5) Here is the general recursive formula for arithmetic sequences.
{g(1)=Ag(n)=g(n1)+B\begin{cases}g(1)=A\\\\ g(n)=g(n-1)+B \end{cases}
What is the common difference of the sequence?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • blobby green style avatar for user yk
    Do we have to find the term number before the other ones to find a certain term number?
    (16 votes)
    Default Khan Academy avatar avatar for user
    • orange juice squid orange style avatar for user graciousartist
      Yes, when using the recursive form we have to find the value of the previous term before we find the value of the term we want to find. For example, if we want to find the value of term 4 we must find the value of term 3 and 2. We are already given the value of the first term.

      In other words to find any term beyond the first term we have to start at the beginning which would be the 2nd term and continue to calculate the value of each proceeding term until we have reached the term we want to find.

      Makes sense?
      (23 votes)
  • blobby green style avatar for user Damon Lam
    I don't quite understand the purpose of the recursive formula. I understand how it works, and according to my understanding, in order to find the nth term of a sequence using the recursive definition, you must extend the terms of the sequence one by one. But doesn't this defeat the purpose of it? Isn't the purpose of a formula to find out the nth term of the sequence without computing all the terms before it?

    Am I missing something critical here?
    (17 votes)
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    • leaf green style avatar for user kubleeka
      Formulas are just different ways to describe sequences. Each description emphasizes a different aspect of the sequence, which may or may not be useful in different contexts. For example, we may be comparing two arithmetic sequences to see which one grows faster, not really caring about the actual terms of the sequences. In this case, the recursive definition gives the rate of change a little more directly than the standard formula.

      There are also sequences that are much easier to describe recursively than with a direct formula. For example, the Fibonacci sequence, which starts {0, 1, 1, 2, 3, 5, 8...}, with each successive term being the sum of the previous two. While this does have a closed formula, it's very complex and unwieldy.
      (9 votes)
  • piceratops ultimate style avatar for user Aidan C.
    What good would this stuff do us in the real world? PLZ tell me!
    (10 votes)
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    • male robot donald style avatar for user Rithvik
      Sequences are really important in real life, as they play a key part in areas such as statistics, finance and even in controlling the growth of a species!! One example can be you planning for a vacation. You would look at the temperature of your choosen vacation spot for each month and then decide which month is the apt time to visit the place. Invariably, these temperatures are a sequence and are stored in a set. Who would have known that to enjoy your vacation, you would have to brush up on your sequences first!!
      (12 votes)
  • aqualine seedling style avatar for user marianamamario
    Hi. I don't understand what "common difference" stands for.
    (9 votes)
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    • leaf green style avatar for user kubleeka
      For an arithmetic sequence, we add a number to each term to get the next term. That number is the common difference.
      So for {0, 3, 6, 9...}, we're adding 3 each time. So the common difference is 3.

      Note: only arithmetic sequences have a common difference.
      (18 votes)
  • duskpin ultimate style avatar for user raahiljain
    How would you solve something like:
    f(n)=f(n-1)+f(n-1)-f(n-2)+35
    f(1)=5
    f(2)=30
    f(n)= Some number in the 10thousands, not sure what numbers work in this particular scenario
    For n
    Is there any way to solve this without going through each and every step?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      Well, lets see what the first few terms are, f(1) = 5, f(2) = 30, f(3) = 30+30-5+35= 90, f(4) = 90 + 90 - 30+35 = 185, f(5) = 185 + 185 - 90 + 35 = 315, f(6) = 315 + 315 - 185 + 35 = 480. So we have a sequence of 5, 30, 90, 185,315, 480 ... We then can find the first difference (linear) which does not converge to a common number (30-5 = 25, 90-30=60, 185-90=95, 315-185=130, 480-315=165. Then the second difference (60 - 25 = 35, 95-60 = 35, 130-95=35, 165-130 = 35) gives a second common difference, so we know that it is quadratic. I do not know any good way to find out what the quadratic might be without doing a quadratic regression in the calculator, in the TI series, this is known as STAT, so plugging the original numbers in, I ended with the equation:
      f(x) = 17.5x^2 - 27.5x + 15. This gives us any number we want in the series. Maybe these having two levels of numbers to calculate the current number would imply that it would be some kind of quadratic function just as if I only had 1 level, it would be linear which is easier to calculate by hand.
      (13 votes)
  • aqualine seed style avatar for user 22oaubie
    if the sequence is 4,8,12,16... and arithmetic how could I write a recessive and explicit formula for that sequence?
    (5 votes)
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  • starky tree style avatar for user Abhishek Gahlaut
    When ever we are doing recursive formulas why do we add that x(n-1)+ something, why do we do that
    (5 votes)
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    • piceratops tree style avatar for user Karttikeya
      That would be the rule to get any term from its previous term

      For example,
      c(1)=5
      in order to find any term, we simply need to put the nth term into
      c(n)=c(n−1)+3
      ​where +3 is the common difference

      Only arithmetic sequences have a common difference
      The common difference of an A.P. can be positive, negative or zero
      (3 votes)
  • aqualine seed style avatar for user Tian McDonald
    What does the d mean in f(n) = f(n − 1) + d ?
    (3 votes)
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  • male robot hal style avatar for user jdfrakes
    I'm still confused on why people use recursive formulas. I know they give us the first term and the pattern for a sequence, but don't explicit formulas give us the same information, but without the need for the previous term? Is there any information that recursive formulas do that explicit formulas don't?
    (4 votes)
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  • hopper cool style avatar for user Eunice Zhang
    Can someone explain in #2, how it works/
    (3 votes)
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    • duskpin sapling style avatar for user Bonster03
      This is the way I understand it. First some basics. Since 12 is the starting number you have d(1)=12. d is basically saying this is a arithmetic sequence. (1) is saying this is the first number in the sequence and = 12 is saying that that number is 12. Now that that's out of the way, on to the more difficult stuff. You can look at the sequence and see a pattern. What pattern does 12,7,2,-3,-8,... have, well you probably already see that as each new number is added it is 5 less than the one before it. How would we write that ? Well d(n−1) basically means the number from the number before it's finished product. So like
      d(1)=12 then (d(n-1)-5) = (12-5). Its really simple when you think of it this way. I hope this helped If not tell me so I can try to explain better. : )
      (5 votes)