Term-to-Term Rules

From one term to the next

A sequence is a list of numbers in order, and each number in it is a term. In the sequence 3, 7, 11, 15, … the first term is 3, the second term is 7, and so on.

To find the rule, look at the gaps between neighboring terms: 7 − 3 = 4, 11 − 7 = 4 and 15 − 11 = 4. Every gap is 4, so the term-to-term rule is add 4, and the next term is 15 + 4 = 19. A gap that stays the same all the way along is called the common difference.

Each jump from one term to the next is 4, the common difference.

One handle sets the first term and the other sets the common difference. Every term after the first is the term before it plus the common difference.

A rule that does not add

Not every rule adds the same number. In 2, 4, 8, 16, 32, the gaps are 2, 4, 8 and 16, and they grow. Look at how each term compares with the one before instead: 4 is 2 × 2, 8 is 4 × 2, and 16 is 8 × 2.

Each term is double the one before, so the rule is multiply by 2, and the next term is 32 × 2 = 64. When the gaps change, check whether each term is the one before multiplied by the same number.

Each term doubles the one before, so each step adds more dots than the last: 2, then 4, then 8.

The usual mistakes

Check every gap, not just the first one, before you decide on a rule. Then take exactly one step: after 3, 7, 11, 15 the next term is 15 + 4 = 19. Adding one more than the gap gives 20, and jumping two gaps gives 23.