Sequences

• Continue a number sequence or pattern and describe its term-to-term rule.
• Recognise relationships between sequences; subscript notation T_n means the nth term of sequence T.
• Find and use the nth term of linear, quadratic, cubic and exponential sequences, and simple combinations of these.

How to get from one term to the next

The term-to-term rule says how to get the next term from the previous one, e.g. “add 4” or “multiply by 3”. It is useful for continuing a sequence, but to find the 100th term you need the nth term (position-to-term rule).

nth term = (difference)n + (zero term)

If the first differences are constant, the sequence is linear. The nth term is dn + c, where d is the common difference and c is the term before the first (the “zero term”).

Constant second differences

If the second differences are constant, the nth term is an² + bn + c, and a = (second difference) ÷ 2. Subtract an² from each term; what is left is a linear sequence bn + c.

Compare with n³, or look for a common ratio

Compare a sequence with the cubes 1, 8, 27, 64, … to spot a cubic. If each term is the previous one multiplied by the same number r (the common ratio), the sequence is exponential, with nth term a × rⁿ⁻¹, where a is the first term.