Constant gaps and growing gaps

Learning outcome

You can find the next term when the gap between terms is constant, and when that gap itself grows by a constant.

Concepts and reasons

Write the gap between neighbours before you name the next term. If every gap is the same, add that gap once more.

If the gaps are not equal, write a second row: the gap between those gaps. When the second row is constant, add that constant to the last gap, then add the new gap to the last term.

Squares are a growing-gap series you can recognise faster. 4, 9, 16, 25 are 2 squared, 3 squared, 4 squared and 5 squared. The gaps are 5, 7, 9, which grow by 2. The next term is 36, then 49.

Worked examples

1. 6, 11, 16, 21, ? The gaps are 5, 5, 5. Next is 26.
2. 3, 8, 15, 24, 35, ? The gaps are 5, 7, 9, 11. They grow by 2, so the next gap is 13 and the next term is 48. The same series is n squared minus 1 for n = 2, 3, 4, 5, 6, and 7 squared minus 1 is also 48.
3. 2, 6, 12, 20, 30, ? The gaps are 4, 6, 8, 10. The next gap is 12, so the next term is 42. These terms are also 1 times 2, 2 times 3, 3 times 4, 4 times 5 and 5 times 6. The next is 6 times 7, which is 42.

Common traps

Do not invent a multiplication rule for a series that is only addition. 6, 11, 16 is plus 5, not a doubling. The rule has to fit every step you were given, not only the jump into the answer. 1, 4, 9, 16 starts at 1 squared. The next term is 25, even though the first gap is only 3.

Practice questions

1. 9, 14, 19, 24, ?
2. 5, 10, 17, 26, 37, ?
3. 4, 9, 16, 25, ?

Answers and explanations

1. 29. The gap is plus 5. 24 + 5 = 29.
2. 50. The gaps are plus 5, 7, 9 and 11, so the next gap is 13. 37 + 13 = 50. These are also one more than the squares from 2 squared onward, and 7 squared plus 1 is 50.
3. 36. These are squares of 2, 3, 4 and 5. The next is 6 squared.

Analogy

A staircase with equal risers is a constant gap. A staircase whose risers grow by the same amount each time is a second-difference series. Measure the riser before you predict the next step.

Quick reference

• Equal gaps: add the same number again.
• Gaps that grow by a fixed amount: grow the last gap, then add it.
• Try squares when the gaps grow by 2.
• The same rule must fit every given step.