M11 · Sequences | Edexcel GCSE Maths

Revise the key ideas

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.

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.

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.

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.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.