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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 3,5,7,
a(n)=3+2(n1)
In the formula, n is any term number and a(n) is the nth 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=5 into the explicit formula.
a(5)=3+2(51)=3+24=3+8=11
Cool! This is in fact the fifth term of 3,5,7,

Check your understanding

1) Find b(10) in the sequence given by b(n)=5+9(n1).
b(10)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Writing explicit formulas

Consider the arithmetic sequence 5,8,11, The first term of the sequence is 5 and the common difference is 3.
We can get any term in the sequence by taking the first term 5 and adding the common difference 3 to it repeatedly. Check out, for example, the following calculations of the first few terms.
nCalculation for the nth term
15=5+03=5
25+3=5+13=8
35+3+3=5+23=11
45+3+3+3=5+33=14
55+3+3+3+3=5+43=17
The table shows that we can get the nth term (where n is any term number) by taking the first term 5 and adding the common difference 3 repeatedly for n1 times. This can be written algebraically as 5+3(n1).
In general, this is the standard explicit formula of an arithmetic sequence whose first term is A and common difference is B:
A+B(n1)

Check your understanding

2) Write an explicit formula for the sequence 2,9,16,.
d(n)=

3) Write an explicit formula for the sequence 9,5,1,.
e(n)=

4) The explicit formula of a sequence is f(n)=6+2(n1).
What is the first term of the sequence?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the common difference?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Equivalent explicit formulas

Explicit formulas can come in many forms.
For example, the following are all explicit formulas for the sequence 3,5,7,
  • 3+2(n1) (this is the standard formula)
  • 1+2n
  • 5+2(n2)
The formulas may look different, but the important thing is that we can plug an n-value and get the correct nth 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 2,8,14, has a first term of 2 and a common difference of 6.
The explicit formula 2+6(n1) describes this sequence, but the explicit formula 2+6n describes a different sequence.
In order to bring the formula 2+6(n1) to an equivalent formula of the form A+Bn, we can expand the parentheses and simplify:
=2+6(n1)=2+6n6=4+6n
Some people might prefer the formula 4+6n over the equivalent formula 2+6(n1), 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 12,7,2,
Choose all answers that apply:

Challenge problems

6*) Find the 124th term of the arithmetic sequence 199,196,193,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to join the conversation?

  • blobby green style avatar for user 😊
    what dose it mean to create an explicit formula for a geometric
    (22 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?
    (6 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 :~:
    (11 votes)
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  • 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?
    (3 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".
      (8 votes)
  • blobby green style avatar for user sop
    The first term of an arithmetic sequence is
    5 and the tenth term is 59
    How do we make sure the common difference is positive 6 and not negative 6?
    (2 votes)
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  • starky seed style avatar for user 19.amber.broyhill
    what is the recursive formula for airthmetic formula
    (3 votes)
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  • blobby green style avatar for user ca9034266
    do we need to know arithmetic sequences for the SAT?
    (2 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
    (0 votes)
    Default Khan Academy avatar avatar for user
    • 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
      (8 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?
    (2 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)
  • 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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