Generating sequences from term-to-term rules

Maths · Sequences

Same start. Different sequence.

Pick an opening, then pick a rule that fits it. Every rule you can choose here turns the opening into a different sequence.

Opening terms → A rule that fits them

On Pick a pair of opening terms. 3 branches to choose from.

Same opening terms, different rules, different sequences. Stating the rule is what pins a sequence down.

Predict, then check

You've been going forwards. This one goes backwards, so think about undoing the rule.

A sequence uses the rule 'multiply by 0.1' and its second term is 4. What is the first term?

What you need to know

• A sequence is a run of terms, usually built by a rule. Each number in it is one term.
• A term-to-term rule tells you how to get the next term from the previous one.
• T1 is the first term, T2 the second, T10 the 10th. Capital T means 'a specific term'; the small number says which.
• Start on 124 with 'add five' and you get 129, 134, 139. Start on 6 with 'multiply by five': 30, 150, 750.
• Same rule, different first term, different sequence. The first term matters just as much as the rule.
• Rules needn't add. 'Multiply by 4' works, and so does 'multiply by −1/2', which is the same as divide by −2.
• Have a goStart on 8 and use the rule 'multiply by −1/2'. What are T2 and T3?

  T2 = −4 and T3 = 2

  Multiplying a negative by −1/2 flips the sign back: −4 × −1/2 = 2, not −2.

• To go backwards, use the inverse of the rule. 'Add three' with T3 = 7 gives T2 = 4, then T1 = 1.
• Have a goA sequence uses 'multiply by 2' and T3 = 20. Work backwards: what are T2 and T1?

  T2 = 10 and T1 = 5

  Undoing 'multiply by 2' means dividing by 2, not subtracting anything.

• For an adding sequence, count the additions, not the terms. T1 to T10 is nine additions; T2 to T5 is three.
• Have a goA sequence starts at 5 and adds 3 each time. How many additions take you from T1 to T6, and what is T6?

  Five additions, so T6 = 5 + 5 × 3 = 20

  T1 is where you start, so T6 is five jumps on. Six jumps would mean counting the terms instead.

• That shortcut only works when the rule adds the same amount every time. Doubling, for example, breaks it.
• Missing terms in a constant adding sequence? Divide the difference between known terms by the number of additions between them.
• A few terms don't fix the rule. Starting 1, 2 could be add 1, multiply by 2, or multiply by 5 then subtract 3.
• Even 1, 2, 4 has two rules: multiply by 2, or add 1, then add 2, then add 3.
• For a picture pattern, name the part that grows: white squares +1, black edge squares +2, all squares together +3 each time.
• Write a pattern's amounts as numbers to spot the rule. The same jump between every pair of terms is a constant adding rule.

The big picture

A term-to-term rule builds each term from the one before it. A first term plus a rule gives you a sequence, but a few terms on their own can fit several rules.

⚠ Watch out

Counting terms instead of jumps, so T10 becomes the first term plus ten additions. T1 to T10 is nine additions, and even that shortcut only works when the rule adds the same amount every time.

Memory hook

Think stepping stones: to get from the first stone to the tenth, you make nine jumps. Count the jumps, not the stones.

Check yourself

Try it cold. T1 = 4 and the rule is 'multiply by 3, then subtract 2'. Write T1 to T4, then redo it starting at 5. What stayed the same, and what changed?