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Finding the Next Number in a Sequence:
     General Examples
(page 4 of 7)

Sections: Common differences, Recursions, General examples, Non-math "sequences"


What follows are just some additional examples, given so you can see the process at work.

  • Find the missing number in the sequence:  3, 4, 6, 9, ___, 18.

    First, I'll see if anything happens to pop out at me. To multiply from 3 to 4, I'd have to multiply by 4/3, but (4)(4/3) does not equal 6, so that must not be the rule. To add from 3 to 4, I'd have to add 1, but 4 + 1 is not 6; 4 + 2 = 6. Wait....

      3 + 1 = 4
      4 + 2 = 6

      6 + 3 = 9

    Hmm... What if the rule is "add the next bigger number to the last term"? Then I'd have:

      9 + 4 = 13

    Does this fit? Do I get "18" for the next value?

      13 + 5 = 18

    Yup; it worked! So it would appear that the rule is "add the next bigger number to the previous term", and:   Copyright Elizabeth Stapel 2002-2011 All Rights Reserved

      The missing number is 13.

Note that I could have gone straight to the differences:

 

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    difference rows

Since the second differences are the same, then the formula is a quadratic. Plugging in the first three data points, I get:

    a + b + c = 3
    4a + 2b + c = 4
    9a + 3b + c = 6

Solving this system of equations, I get:

    an = 0.5n2 0.5n + 3

Plugging in n = 5 for the missing fifth term, I get:

    a5 = 0.5(25) 0.5(5) + 3 = 12.5 2.5 + 3 = 10 + 3 = 13

So my previous answer was right, or has at least been confirmed as logical.

  • Find the formula for the n-th term in the sequence:  4, 12, 20, 28, 36,...

    To add from 4 to 12, I'd have to use 8. To add from 12 to 20, I'd also have to use 8. Let's check to see if "add 8" is the rule:

      4 + 8 = 12
      12 + 8 = 20

      20 + 8 = 28

      28 + 8 = 36

    It appears that the rule is "add 8". So what is the rule for the n-th term? Let's look at the terms:

      n = 1:  4
      n = 2:  4 + 8

      n = 3:  4 + 8 + 8 = 4 + 28

      n = 4:  4 + 8 + 8 + 8 = 4 + 38

      n = 5:  4 + 8 + 8 + 8 + 8 = 4 + 48

    Following this pattern, the rule for the n-th term will be:

      an = 4 + (n 1)8

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Cite this article as:

Stapel, Elizabeth. "Finding the Next Number in a Sequence: General Examples."
    Purplemath. Available from 
http://www.purplemath.com/modules/nextnumb4.htm.
    Accessed
 

 



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