Term-to-Term Rules

Count your pocket money as it grows by £2 a week, or the sweets left in the bag as you eat two a day, or the seats filling row by row at a show — each one is a list of numbers that changes by the same step each time. Once you know that step, you can carry the list on as far as you like.

A sequence is a list of numbers in order — each one is a term. A term-to-term rule is the single operation that takes you from one term to the next: the same step, every single time. It might be +3 each time, or -5 each time, or \times 2 each time.

Once you know the rule, you can keep the sequence going forever — just apply it again. For 2,\ 5,\ 8,\ 11,\ \dots the rule is +3, so the next term is 11 + 3 = 14, and the one after that is 14 + 3 = 17. This is the same equal-step idea as skip counting, now written as a rule. Naming the rule is a small step into turning words into algebra.

Finding the rule: look at the gaps

To discover the rule, look at the gap between each term and the one before it. If every gap is the same, that gap is your rule. Take 4,\ 7,\ 10,\ 13,\ \dots

4 \xrightarrow{+3} 7 \xrightarrow{+3} 10 \xrightarrow{+3} 13

Every jump is +3, so the rule is "add 3". Apply it once more to continue: 13 + 3 = 16.

When the terms get bigger each time, the sequence is growing (the rule adds, or multiplies by more than one). When they get smaller, it is shrinking (the rule subtracts, or halves). For 20,\ 16,\ 12,\ 8,\ \dots each gap is -4, so the rule is "subtract 4" and the next term is 8 - 4 = 4.

A multiply rule

Not every rule adds. Sometimes each term is the one before multiplied by the same number. Take 3,\ 6,\ 12,\ 24,\ \dots

3 \xrightarrow{\times 2} 6 \xrightarrow{\times 2} 12 \xrightarrow{\times 2} 24

Here the gaps are not equal (+3, then +6, then +12) — so it is not an add rule. But each term is double the last, so the rule is \times 2, and the next term is 24 \times 2 = 48. A multiply rule makes a sequence grow much faster than an add rule.

Asha starts with 5 coins and earns 5 more every week. Her savings are 5,\ 10,\ 15,\ 20,\ \dots — the rule is +5 each week, a growing sequence.

coin → coin coin → coin coin coin

Each week the pile is a little taller by the same amount — that steady step is the term-to-term rule.

There are 10 cookies on the plate, and the family eats 2 every day. The count goes 10,\ 8,\ 6,\ 4,\ \dots — the rule is -2 each day, a shrinking sequence.

cookie cookie cookie cookie cookie → cookie cookie cookie

Growing or shrinking, the idea is the same: the same step happens at every stage.

See it: hop along the number line

Start on the first term and hop to the next, again and again. Each hop is exactly the same length, and that length is the rule. Read the size of one hop and you know how to continue forever. Press Refresh for a new starting point and a new rule.

Here is the same idea as an animation. Press play: each term appears in turn, and the jump between consecutive terms is labelled with the rule. Watch how the same operation builds the whole sequence.

A first term and a term-to-term rule together pin down the whole sequence: start at the first term, then apply the rule again and again. Change either one and you get a different sequence.

Two traps that catch people out:

Check the rule works for every step, not just the first two. If 2,\ 4,\ 8,\ 16 looks like +2 from the first jump, test the next: 4 to 8 is +4, so it is not an add rule at all — it is \times 2. A real rule must fit all the gaps.
"+3 each time" means add, not multiply. From 9 the next term is 9 + 3 = 12, not 9 \times 3. Read the rule carefully: a + adds and a \times multiplies.

Khan Academy extends arithmetic sequences here: