M11 · Sequences | Edexcel GCSE Maths

Revise the key ideas

Recognising sequences

A term-to-term rule tells you how to get the next term. A position-to-term rule gives a term directly from its position n, starting with n = 1 unless stated otherwise.
An arithmetic sequence has a constant difference. For 5, 8, 11, 14, … the difference is 3, so the nth term is 3n + 2: compare with 3, 6, 9, 12, ….
Worked example: For 12, 8, 4, 0, … the nth term is 16 − 4n. The tenth term is 16 − 40 = −24. The difference can be negative.
To test whether a value belongs to a sequence, set its nth-term expression equal to that value. For 3n + 2 = 50, n = 16, so 50 is a term; n must be a positive integer.
Square numbers are 1, 4, 9, 16, …; cubes are 1, 8, 27, 64, …; triangular numbers are 1, 3, 6, 10, …, increasing by 2, 3, 4, ….
First and second differences
Enlarge diagram: First and second differences
A Fibonacci-type sequence adds the two previous terms. Starting 2, 3 gives 2, 3, 5, 8, 13, …. Its first two terms must be specified.
A geometric sequence multiplies by a constant ratio. Starting at 3 and multiplying by 2 gives 3, 6, 12, 24, …. Starting at 8 and multiplying by 12 gives 8, 4, 2, 1, ….
A quadratic sequence has constant non-zero second differences. Recognise it by finding first differences and then differences of those differences; the first differences are not constant.

Using rules carefully

Worked example: For a pattern made from adjoining squares, the first square needs four sticks and each extra square adds three. The nth pattern needs 3n + 1 sticks; pattern 20 needs 61.
A few terms can fit more than one rule. Use the stated pattern or enough structural information; do not claim a rule is uniquely determined by only two or three numbers.

Higher — quadratic nth terms

For an² + bn + c, the constant second difference is 2a. Divide it by 2 to find a, then subtract an² from each term to leave a linear sequence.
Worked example: 3, 8, 15, 24, … has differences 5, 7, 9 and second difference 2. Subtract n² to get 2, 4, 6, 8, …, so the rule is n² + 2n.

Higher — geometric formulae

For first term a and common ratio r, the nth term is arⁿ⁻¹. The first term uses power 0, so it equals a, not ar.
Worked example: 5, 15, 45, … has nth term 5 × 3ⁿ⁻¹. Its fifth term is 5 × 3⁴ = 405. Higher problems can use ratios such as √2; standard positive rational ratios are shared content.

Test yourself

50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.

M11 · Sequences