Cracking the Pattern: Quadratic Sequences

Look at these three sequences. Two of them have a constant difference between consecutive terms (like +3, +3, +3). One of them does NOT — its differences keep changing. Can you spot which one is different? Sequence A: 2, 5, 8, 11, 14 … Sequence B: 1, 4, 9, 16, 25 … Sequence C: 10, 13, 16, 19, 22 … Write down the differences between each pair of consecutive terms for all three sequences. What do you notice about Sequence B?

A linear sequence goes up by the same amount each time — the first differences are constant. A quadratic sequence is different: its first differences change, but if you subtract consecutive first differences you get a constant number. Those are called the second differences. If the second differences are constant (and not zero), the sequence is quadratic. The word quadratic comes from the Latin for 'square', which is a clue — quadratic sequences are always linked to n² (n squared, meaning n multiplied by itself). Finding the nth term of a quadratic sequence The nth term always has the form: an² + bn + c Here, n is the position number (1st term, 2nd term, etc.), and a, b, c are numbers you calculate. Step 1 — Find a The second difference equals 2a. So: a = (second difference) ÷ 2. Step 2 — Subtract an² from the original sequence This gives you a new, simpler sequence. Find its nth term in the usual linear way (it will be bn + c). Step 3 — Combine Write the full nth term: an² + bn + c. Worked example below shows exactly how.