PassCCAT
Number series

How to solve number series

Math & Logic

Find the pattern in a sequence of numbers and supply the next term.

How to solve these

Mini-lesson

A number series hides one rule that turns each term into the next. Find the rule from the gaps between numbers, then apply it one more time.

  1. 1

    Write the gaps between terms

    Look at how you get from each number to the next: +3, +6, +12. Seeing the gaps as their own little series is usually where the pattern appears.

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  2. 2

    Decide the operation

    Steady gaps mean adding or subtracting. Gaps that grow fast mean multiplying. A mix (times 2, then plus 1) is common at higher difficulty, so test 'x then +' if a single step does not fit.

  3. 3

    Apply the rule once more

    Once the rule fits every step, use it on the last number to get the answer, then sanity-check it against the options.

Hacks

  • If numbers shrink, the rule is subtraction or division. 81, 27, 9, 3 is divide by 3 each time.
  • If numbers grow much faster than adding would, suspect multiplication.
  • When nothing simple fits, check the gaps-of-the-gaps, or whether two interleaved series are at play.

Avoid these

  • Assuming a constant difference when the gaps are actually changing.
  • Stopping at the rule and forgetting to do the final step to reach the next term.

Worked examples

Answers shown

Follow the steps on real items before you practise.

Example 1Difficulty 1

Find the next number: 1, -2, -5, -8, -11, ?

Answer: A. Here is why:

  1. The rule is: add -3.
  2. Applying it to -11 gives -14.
Example 2Difficulty 1

Find the next number: 10, 15, 20, 25, 30, ?

Answer: C. Here is why:

  1. The rule is: add 5.
  2. Applying it to 30 gives 35.
Example 3Difficulty 1

Find the next number: 8, 11, 14, 17, 20, ?

Answer: E. Here is why:

  1. The rule is: add 3.
  2. Applying it to 20 gives 23.