IGCSE Maths / Sequences and series
Number sequences and term-to-term rules revision notes
Edexcel International GCSE Maths (4MA1), chapter 9: Sequences and series, specification point 3.1a-f. Free lesson.
Watch the video lesson · IGCSE Maths past papers
A term-to-term rule explains how to move from one term to the next, while a position-to-term rule gives a term directly from its position. Checking gaps and ratios helps you identify a pattern, and a position rule is much quicker for finding a far-off term.
What the two types of rule tell you
A term-to-term rule tells you how to get from one term to the next, such as adding 4 each time. A position-to-term rule gives the term directly from its position number. For example, 4n − 1 gives the term at position n. Counting on works for nearby terms, but using a position rule is quicker and less error-prone for a far-off term. Use the term-to-term rule to continue a sequence. Use the position-to-term rule to find a term directly from its position.
How to spot an adding rule
Find the differences by subtracting each term from the next. If the differences are equal, the sequence follows an adding rule. You can then keep adding that same amount to continue it.
How to spot a multiplying rule
If the differences are not equal, check the ratios by dividing each term by the one before it. Equal ratios show that each term is made by multiplying the previous term by the same number. For 2, 6, 18, 54, the differences are 4 and 12, so this is not an adding rule. The ratios 6 ÷ 2 and 18 ÷ 6 are both 3, so multiply by 3 each time.
Why you must test a rule
Two terms can fit more than one possible rule, so they are not enough to identify a pattern confidently. Check your suggested rule against a third term before using it. Not every sequence has constant differences: for example, the square numbers 1, 4, 9, 16 have gaps of 3, 5 and 7. For 2, 4, 8, the first gap suggests adding 2, but the next gap is 4. The sequence doubles each time, so its next term is 16, not 10. When the gaps change, do not force an adding rule. Look for another pattern, such as multiplication.
Continue each sequence using its pattern
For a sequence with equal gaps, keep adding the gap. For a multiplying sequence, keep multiplying by the same ratio. For a term far along the sequence, use a position-to-term rule rather than listing every term. 5, 12, 19, 26 has a constant difference of 7. 1, 3, 9, 27 has a constant ratio of 3.