A quick method
- Write the difference between each pair of numbers.
- If the differences are not constant, look at the differences of the differences, or check the ratios.
- If nothing fits, check squares, cubes and alternating patterns.
The eight patterns
- 1. Constant difference: 4, 9, 14, 19, __ → add 5, so the next number is 24.
- 2. Constant ratio: 2, 6, 18, 54, __ → multiply by 3, so the next number is 162.
- 3. Growing differences: 1, 2, 4, 7, 11, __ → add 1, 2, 3, 4, 5, so the next number is 16.
- 4. Even-step growing differences: 3, 5, 9, 15, 23, __ → add 2, 4, 6, 8, so the next number is 33.
- 5. Squares: 1, 4, 9, 16, 25, __ → the next number is 36.
- 6. Cubes: 1, 8, 27, 64, __ → the next number is 125.
- 7. Squares plus a number: 2, 5, 10, 17, 26, __ → each term is n² + 1, so the next number is 37.
- 8. Two operations in turn: 3, 6, 5, 10, 9, 18, __ → multiply by 2, then subtract 1, so the next number is 17.
Two series mixed together
If the numbers seem to jump around, split them into two series by taking every other number. For example, in 2, 10, 4, 20, 6, 30 the odd positions are 2, 4, 6 and the even positions are 10, 20, 30.
Check your answer
Once you find a rule, apply it to the whole sequence from the start. If it works for every term, you have the right pattern. If it breaks on one term, look for a different rule.
Practice
Time yourself on a set of number series and review each one you miss in our free Numerical Ability reviewer. Within a week of daily practice you will start recognizing the patterns on sight.