What is a Quadratic Sequence?

In mathematics, sequences are ordered lists of numbers that follow a rule. You have probably already worked with linear sequences, where the difference between consecutive terms is always the same (constant). Quadratic sequences are the next level up — they are more interesting and more powerful.

Quadratic Sequence

A sequence in which the second differences (the differences of the differences) are constant and non-zero. The general term (nth term) is given by a formula involving $n^{2}$ .

The word quadratic comes from the Latin quadratus, meaning "square" — which is a hint that $n^{2}$ will always appear in our formula.

Analogy

Think of a linear sequence like a car travelling at a constant speed — it covers equal distances in equal times. A quadratic sequence is like a car that is accelerating — the distances covered keep increasing by a fixed extra amount each second. That fixed extra amount is the constant second difference.

Here is a classic example of a quadratic sequence:

$1, 4, 9, 16, 25, 36, \dots$

These are the square numbers — each term equals $n^{2}$ . They are the simplest quadratic sequence of all.

First Differences and Second Differences

The key skill in identifying and working with quadratic sequences is calculating differences.

First Differences

The values obtained by subtracting each term from the next term in a sequence: $d_{n} = T_{n + 1} - T_{n}$ .

Second Differences

The values obtained by subtracting each first difference from the next: the differences of the first differences.

For a quadratic sequence:

First differences are not constant (they increase or decrease by a fixed amount)
Second differences are constant (and non-zero)

Example

Example: Find the first and second differences of 3, 8, 15, 24, 35, …

Step 1 — Write out the sequence:
$3, 8, 15, 24, 35$

Step 2 — Calculate first differences (subtract consecutive terms):
$8 - 3 = 5, 15 - 8 = 7, 24 - 15 = 9, 35 - 24 = 11$
First differences: $5, 7, 9, 11$ — not constant, so this is not a linear sequence.

Step 3 — Calculate second differences:
$7 - 5 = 2, 9 - 7 = 2, 11 - 9 = 2$
Second differences: $2, 2, 2$ — constant! This confirms it is a quadratic sequence.

First Differences and Second Differences

Exam Tip

Always set out your differences in a clear table or staircase layout underneath the sequence. This makes it easy to spot the pattern and reduces errors in exams.

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