Quadratic sequences

#### Tier: #Higher

Description

A quadratic sequence has a constant second difference (the differences of the differences are equal). The $n$th term contains an $n^2$ term.

Method to find the $n$th term:

1. Find the first and second differences
2. The coefficient of $n^2$ is half the second difference
3. Subtract the $an^2$ sequence from the original to get a linear remainder
4. Find the $n$th term of the linear part

Example: Sequence: 3, 8, 15, 24, 35, …

• First differences: 5, 7, 9, 11 — not constant
• Second differences: 2, 2, 2 — constant → coefficient of $n^2 = \frac{2}{2} = 1$
• Subtract $n^2$ sequence (1, 4, 9, 16, 25): remainders are 2, 4, 6, 8, 10 → $n$th term is $2n$
• Full $n$th term: $n^2 + 2n$

Check: when $n=1$: $1+2=3$ ✓

Common error: forgetting to halve the second difference when finding the $n^2$ coefficient.

Links

Arithmetic sequences Factorising quadratics