IGCSE Maths / Sequences and series
Finding a rule for a sequence
Edexcel International GCSE Maths (4MA1), chapter 9: Sequences and series, specification point 3.1b-f. Free lesson.
Read the revision notes · IGCSE Maths past papers
A position-to-term rule gives any term in a sequence from its position, n. For an arithmetic sequence, use the common difference as the multiplier of n, then compare with the terms to find the adjustment.
Use the common difference as the multiplier
The common difference is the amount added each time to get the next term. It tells you the number multiplying n in the position rule. For example, a gap of 4 means the rule starts with 4n. For 3, 7, 11, 15, the common difference is 4, so start with 4n. The multiples of 4 are 4, 8, 12, 16. Compare these with the sequence to find the adjustment.
Compare the multiples with the sequence
The multiplier alone is not usually the full rule. Compare its multiples with the actual terms. If each term is lower, subtract the difference; if each is higher, add the difference. The amount added or subtracted is called the adjustment. For 3, 7, 11, the multiples of 4 are 4, 8, 12. Each sequence term is 1 less, so the rule is 4n − 1. The adjustment is not always the first term. It is the amount needed to turn the multiples of n into the sequence.
Check the rule on a known term
Substitute a position you already know into the rule. This checks that both the multiplier and adjustment are correct. For 3, 7, 11, 15, the rule should give 15 when n = 4.
Use the rule to find a far term
Once you have the nth term, substitute the position you need for n. This is quicker than listing every term and avoids errors from counting on.
Apply the method to another sequence
For 5, 8, 11, 14, the gap is 3, so start with 3n. The multiples of 3 are 3, 6, 9, 12, which are each 2 less than the sequence terms. Therefore, add 2. The rule is 3n + 2. To find the 20th term, substitute n = 20: 3(20) + 2 = 62.