5.2 Sequences & Function Rules

From Paper D (Year 6) onwards, ICAS moves past 'what comes next' and asks 'what is the rule?' — the real beginning of algebra. Papers E–F list algebra, linear equations and number sentences with all four operations as tested skills, and sequence questions are where those skills are introduced. The good news: because calculators are not expected here, the numbers stay small and the methods are pure reasoning.

Arithmetic Sequences and the Common Difference

A sequence is an ordered list of numbers, and an arithmetic sequence is one where you add (or subtract) the same amount every step. That fixed amount is called the common difference, usually written d.

For the sequence 5, 9, 13, 17, …:

• 9 − 5 = 4
• 13 − 9 = 4
• 17 − 13 = 4

The common difference is d = 4. If the difference is negative, the sequence decreases: 30, 26, 22, … has d = −4. Always subtract in the same direction: (any term) − (the term before it).

Term-to-Term vs Position-to-Term Rules

There are two ways to describe a sequence:

• Term-to-term rule: 'Start at 5, add 4 each time.' Easy to write, but to find the 50th term you must add 4 forty-nine times — slow and error-prone.
• Position-to-term rule (nth term): a formula using n, the position number, that gives the term directly. For 5, 9, 13, 17, … the rule is 4n + 1. Check: n = 1 gives 4(1) + 1 = 5 ✓; n = 2 gives 4(2) + 1 = 9 ✓.

The nth term rule is the one ICAS cares about in the middle papers, because it turns 'find the 100th term' into one line of arithmetic: 4(100) + 1 = 401.

Building the nth Term Rule

Follow these steps for any arithmetic sequence:

1. Find the common difference d. That is the number multiplying n.
2. Write the d times table: for d = 3, that is 3, 6, 9, 12, … (this is 3n).
3. Compare with your sequence and find what to add or subtract.

Worked example: Find the nth term of 5, 8, 11, 14, …

• d = 3, so start with 3n: 3, 6, 9, 12, …
• Each term of our sequence is 2 more than 3n (5 = 3 + 2, 8 = 6 + 2).
• Rule: 3n + 2. Verify: n = 4 gives 3(4) + 2 = 14 ✓.

For a decreasing sequence like 20, 17, 14, …, d = −3, so the rule starts with −3n: −3, −6, −9, …. Each term is 23 more (−3 + 23 = 20), giving −3n + 23, also written 23 − 3n.

Using the Rule for Far Terms

Once you have the rule, far terms are easy and need no calculator:

• 50th term of 3n + 2: 3 × 50 = 150, then 150 + 2 = 152
• 200th term: 3 × 200 = 600, plus 2 = 602

You can also run the rule backwards. 'Is 77 a term of 3n + 2?' Solve 3n + 2 = 77: subtract 2 to get 75, divide by 3 to get n = 25 — a whole number, so yes, 77 is the 25th term. If n had come out as a fraction, the number would not be in the sequence. ICAS loves this style: 'Which of these numbers appears in the sequence?'

Geometric Sequences (Extension)

A geometric sequence multiplies by the same number each step instead of adding: 2, 6, 18, 54, … multiplies by 3. The multiplier is found by dividing: 6 ÷ 2 = 3, 18 ÷ 6 = 3. These appear as extension questions — you do not need a formula for the nth term, just keep multiplying. Watch for divide-by patterns too: 800, 400, 200, 100, … halves each step, so the next term is 50.

Function Machines

A function machine takes an input, applies one or more operations in order, and produces an output.

For the machine '× 3, then + 2':

Two traps to avoid:

• Order matters. '× 3 then + 2' on input 4 gives 14, but '+ 2 then × 3' gives 18. Apply the operations exactly in the order shown.
• Finding an input means reversing the machine. If the output is 20, undo the steps in reverse order with inverse operations: 20 − 2 = 18, then 18 ÷ 3 = 6. The input was 6.

Finding a Rule from a Table

ICAS often gives a table of input/output pairs and asks for the rule. Use the difference method:

1. As x grows by 1, y grows by 4 — so the rule involves × 4.
2. Try y = 4x: that gives 4, 8, 12, 16 — each 3 too small.
3. Rule: y = 4x + 3. Check with x = 4: 4 × 4 + 3 = 19 ✓. Always check with a pair you did not use to build the rule.

If the outputs do not grow by a steady amount, the rule is not a simple × and +. Try squaring (1, 4, 9, 16) or doubling before assuming you made an arithmetic error.

Common Traps

• Using the first term instead of the common difference as the multiplier: the sequence 5, 9, 13, … has rule 4n + 1, not 5n.
• Forgetting to verify the rule against a second term — one check is never enough.
• Reversing a function machine in the wrong order: undo the last step first.
• Slipping on mental multiplication; keep the 3, 4 and 6 times tables sharp, because no calculator will appear for these.