Sequences and series

Spot the rule of an arithmetic, geometric or quadratic sequence, then find the n-th term and the sum. About 10 minutes, GCSE / A-level.

What you will do

1. Spot the pattern. A sequence follows a rule. First check the differences between neighbours: if they are all equal it is arithmetic. If instead each term is the previous one times the same number, it is geometric.
2. Arithmetic sequences. Start at a and add the common difference d each time: u_n = a + (n-1)d. The sum of the first n terms is n2(2a + (n-1)d).
3. Geometric sequences. Multiply by the common ratio r: u_n = a r^n-1. If |r| < 1 the terms shrink and the sum can settle on a fixed value.
4. Quadratic sequences. When the first differences are not constant but the second differences are, the rule is a quadratic an^2 + bn + c with a equal to half the second difference.
5. Which kind?. Always test: constant difference, constant ratio, constant second difference. The engine tests them in that order too.