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Explicit formulas for arithmetic sequences

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

How explicit formulas work

Here is an explicit formula of the sequence 3, comma, 5, comma, 7, comma, point, point, point
a, left parenthesis, n, right parenthesis, equals, 3, plus, 2, left parenthesis, n, minus, 1, right parenthesis
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 formula allows us to simply plug in the number of the term we are interested in, and we will 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}
Cool! This is in fact the fifth term of 3, comma, 5, comma, 7, comma, point, point, point

Check your understanding

1) 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

Writing explicit formulas

Consider the arithmetic sequence 5, comma, 8, comma, 11, comma, point, point, point The first term of the sequence is start color #0d923f, 5, end color #0d923f and the common difference is start color #ed5fa6, 3, end color #ed5fa6.
We can get any term in the sequence by taking the first term start color #0d923f, 5, end color #0d923f and adding the common difference start color #ed5fa6, 3, end color #ed5fa6 to it repeatedly. Check out, for example, the following calculations of the first few terms.
nCalculation for the n, start superscript, start text, t, h, end text, end superscript term
1start color #0d923f, 5, end color #0d923fequals, start color #0d923f, 5, end color #0d923f, plus, 0, dot, start color #ed5fa6, 3, end color #ed5fa6, equals, 5
2start color #0d923f, 5, end color #0d923f, start color #ed5fa6, plus, 3, end color #ed5fa6equals, start color #0d923f, 5, end color #0d923f, plus, 1, dot, start color #ed5fa6, 3, end color #ed5fa6, equals, 8
3start color #0d923f, 5, end color #0d923f, start color #ed5fa6, plus, 3, plus, 3, end color #ed5fa6equals, start color #0d923f, 5, end color #0d923f, plus, 2, dot, start color #ed5fa6, 3, end color #ed5fa6, equals, 11
4start color #0d923f, 5, end color #0d923f, start color #ed5fa6, plus, 3, plus, 3, plus, 3, end color #ed5fa6equals, start color #0d923f, 5, end color #0d923f, plus, 3, dot, start color #ed5fa6, 3, end color #ed5fa6, equals, 14
5start color #0d923f, 5, end color #0d923f, start color #ed5fa6, plus, 3, plus, 3, plus, 3, plus, 3, end color #ed5fa6equals, start color #0d923f, 5, end color #0d923f, plus, 4, dot, start color #ed5fa6, 3, end color #ed5fa6, equals, 17
The table shows that we can get the n, start superscript, start text, t, h, end text, end superscript term (where n is any term number) by taking the first term start color #0d923f, 5, end color #0d923f and adding the common difference start color #ed5fa6, 3, end color #ed5fa6 repeatedly for n, minus, 1 times. This can be written algebraically as start color #0d923f, 5, end color #0d923f, start color #ed5fa6, plus, 3, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis.
In general, this is the standard explicit formula of an arithmetic sequence whose first term is start color #0d923f, A, end color #0d923f and common difference is start color #ed5fa6, B, end color #ed5fa6:
start color #0d923f, A, end color #0d923f, plus, start color #ed5fa6, B, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis

Check your understanding

2) Write an explicit formula for the sequence 2, comma, 9, comma, 16, comma, point, point, point.
d, left parenthesis, n, right parenthesis, equals

3) Write an explicit formula for the sequence 9, comma, 5, comma, 1, comma, point, point, point.
e, left parenthesis, n, right parenthesis, equals

4) The explicit formula of a sequence is f, left parenthesis, n, right parenthesis, equals, minus, 6, plus, 2, left parenthesis, n, minus, 1, right parenthesis.
What is the first term of the sequence?
  • 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
What is the common difference?
  • 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

Equivalent explicit formulas

Explicit formulas can come in many forms.
For example, the following are all explicit formulas for the sequence 3, comma, 5, comma, 7, comma, point, point, point
  • 3, plus, 2, left parenthesis, n, minus, 1, right parenthesis (this is the standard formula)
  • 1, plus, 2, n
  • 5, plus, 2, left parenthesis, n, minus, 2, right parenthesis
The formulas may look different, but the important thing is that we can plug an n-value and get the correct n, start superscript, start text, t, h, end text, end superscript term (try for yourselves that the other formulas are correct!).
Different explicit formulas that describe the same sequence are called equivalent formulas.

A common misconception

An arithmetic sequence may have different equivalent formulas, but it's important to remember that only the standard form gives us the first term and the common difference.
For example, the sequence 2, comma, 8, comma, 14, comma, point, point, point has a first term of start color #0d923f, 2, end color #0d923f and a common difference of start color #ed5fa6, 6, end color #ed5fa6.
The explicit formula start color #0d923f, 2, end color #0d923f, start color #ed5fa6, plus, 6, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis describes this sequence, but the explicit formula start color #0d923f, 2, end color #0d923f, start color #ed5fa6, plus, 6, end color #ed5fa6, n describes a different sequence.
In order to bring the formula 2, plus, 6, left parenthesis, n, minus, 1, right parenthesis to an equivalent formula of the form A, plus, B, n, we can expand the parentheses and simplify:
=2+6(n1)=2+6n6=4+6n\begin{aligned}&\phantom{=}2+6(n-1)\\\\ &=2+6n-6\\\\ &=-4+6n\end{aligned}
Some people might prefer the formula minus, 4, plus, 6, n over the equivalent formula 2, plus, 6, left parenthesis, n, minus, 1, right parenthesis, because it's shorter. The nice thing about the longer formula is that it gives us the first term.

Check your understanding

5) Find all correct explicit formulas of the sequence 12, comma, 7, comma, 2, comma, point, point, point
Choose all answers that apply:
Choose all answers that apply:

Challenge problems

6*) Find the 124, start superscript, start text, t, h, end text, end superscript term of the arithmetic sequence 199, comma, 196, comma, 193, comma, point, point, point
  • 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*) The first term of an arithmetic sequence is 5 and the tenth term is 59.
What is the common difference?
  • 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 😊
    what dose it mean to create an explicit formula for a geometric
    (23 votes)
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  • purple pi teal style avatar for user Franscine Garcia
    What's the difference between this formula and a(n) = a(1) + (n - 1)d? Is one better or something?
    (7 votes)
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  • aqualine tree style avatar for user Shelby Anderson
    Can you add a section on Simplifying Geometric and arithmetic equations? I have a test on this tomorrow, and there isn't a section to help me study... Wish me luck I guess :~:
    (8 votes)
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  • blobby green style avatar for user Dzeerealxtin
    Determine the next 2 terms of this sequence
    2,5,10,17.. then write the explicit form
    (1 vote)
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    • aqualine ultimate style avatar for user Timber Lin
      warning: long answer
      this isn't an arithmetic ("linear") sequence because the differences between the numbers are different (5-2=3, 10-5=5, 17-10=7)
      however, you might notice that the differences of the differences between the numbers are equal (5-3=2, 7-5=2). that means the sequence is quadratic/power of 2.
      since the sequence is quadratic, you only need 3 terms.
      let x=the position of the term in the sequence
      let y=the value of the term
      the 1st term is 2, so x=1 and y=2
      the 2nd term is 5, so x=2 and y=5
      the 3rd term is 10, so x=3 and y=10
      the function is y=ax^2+bx+c, so plug in each point to solve for a, b, and c.
      (1,2): 2=a(1^2)+b(1)+c
      (2,5): 5=a(2^2)+b(2)+c
      (3,10): 10=a(3^2)+b(3)+c

      simplify: 2=a+b+c
      5=4a+2b+c
      10=9a+3b+c

      solve this using any method, but i'll use elimination:
      10=9a+3b+c
      -(5=4a+2b+c)
      5=5a+b (equation 3 - equation 2)

      5=4a+2b+c
      -(2=a+b+c)
      3=3a+b (equation 2 - equation 1)

      then subtract the 2 equations just produced:
      5=5a+b
      -(3=3a+b)
      2=2a
      that means a=1.
      substitute a=1 into 3=3a+b: 3=3+b, b=0.
      substitute a and b into 2=a+b+c: 2=1+0+c, c=1

      so the equation becomes y=1x^2+0x+1, or y=x^2+1
      btw you can check (4,17) to make sure it's right
      (9 votes)
  • blobby green style avatar for user Donald Postema
    how do you do this -3,-1/3,5/9,23/27,77/81,239/243
    (2 votes)
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    • aqualine tree style avatar for user Judith Gibson
      The main thing to notice in your sequence is that there are actually 2 different patterns taking place --- one in the numerator and one in the denominator.
      Looking at the denominators first, we have 1, 3 9, 27, 81, 243, ... (a geometric sequence).
      Each successive term is multiplied by 3, so for any term n (where n>=1), its denominator would be 3^(n-1).
      Then notice that every numerator is 4 less than its denominator, so the formula for the numerator would be 3^(n-1) - 4.
      Putting these together, the terms of the sequence would be represented by the formula
      ( 3^(n-1) - 4 ) / 3^(n-1) where n>=1.
      Hope this helps!
      (6 votes)
  • starky tree style avatar for user Siegrid Pregartner
    To find the common difference between two terms, is taking the difference and dividing by the number of terms a viable workaround?

    For example, the first term is 5 and the tenth term is 59. So take the difference, 59-5=54, then divide by the number of terms between them, which is 5. And the average difference would be 6. Does this always work?
    (2 votes)
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    • marcimus orange style avatar for user Tim Nikitin
      Your shortcut is derived from the explicit formula for the arithmetic sequence like 5 + 2(n – 1) = a(n). Plug your numbers into the formula where x is the slope and you'll get the same result:
      5 + x(10 – 1) = 59
      5 + 9x = 59
      9x = 54
      x = 6
      To find the slope, you take the difference between the 10th and the 1st term and divide it by the "# of additions" or by the difference between the 10th term and the 1st term.
      So your shortcut works always, because it's the same thing as the explicit formula (5 + 6(10 – 1) = 59). Basically, it says to get the 10th number in the sequence, you have to start from the base of 5 then add 6 (the slope) to it not once, but in this case 9 times ("# of additions").

      P.S. You have a typo in "divide by the number of terms between them, which is 5". It should be 9, not 5. Also the "average difference" should be called "common difference" or "slope".
      (5 votes)
  • leaf grey style avatar for user Alex T.
    It seems to me that 'explicit formula' is just another term for iterative formulas, because both use the same form. Is this true? And is there another term for formulas using the m_ + _Bn form as opposed to the A_ + _B(n-1) form or are they both referred to as explicit formulas?
    (3 votes)
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    • leaf green style avatar for user Ken Burwood
      m + Bn and A + B(n-1) are both equivalent explicit formulas for arithmetic sequences. A + B(n-1) is the standard form because it gives us two useful pieces of information without needing to manipulate the formula (the starting term A, and the common difference B).

      An explicit formula isn't another name for an iterative formula. Even though they both find the same thing, they each work differently--they're NOT the same form.

      In the iterative formula, "a(n-1)" means "the value of the (n-1)th term in the sequence", this is not "a times (n-1)."

      In the explicit formula "d(n-1)" means "the common difference times (n-1), where n is the integer ID of term's location in the sequence."

      Thankfully, you can convert an iterative formula to an explicit formula for arithmetic sequences. Converting is usually less work.

      Take the iterative formula:
      a(1) = A
      a(n) = a(n-1) + B (here a(n-1) is this function for the previous term, not multiplication)

      Turn it into an explicit formula by taking the initial term's value and adding it to B times the integer (n-1):
      a(n) = A + B(n-1) (here B(n-1) is multiplication, not a function)
      (2 votes)
  • starky seed style avatar for user 19.amber.broyhill
    what is the recursive formula for airthmetic formula
    (3 votes)
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  • purple pi purple style avatar for user louisaandgreta
    How do you algebraically get
    5+2(n-2) from
    the standard form 3+2(n-1)?
    (1 vote)
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  • leafers tree style avatar for user ivan Baller
    There is a practice problem that I got for review for a test I have a few days that I don't really get. Basically it says:

    Her production follows the arithmetic sequence 299 + b, 222, 60b + 1, find:

    1) b
    2) the first term
    3) the common difference
    4) the formula for the general term


    So I don't really get what I am supposed to do with the given information.

    Any help would be appreciated.
    (2 votes)
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