What is a Quadratic Sequence?

In science and mathematics, we often look for patterns in data. Some patterns are simple and grow by the same amount each time — these are linear sequences. But many real-world patterns grow faster and faster, and these often follow a quadratic sequence.

Quadratic Sequence

A number sequence in which the second differences are constant (always the same value). The general term is given by a formula containing $n^{2}$ , making it a second-degree (quadratic) expression.

Think about how the area of a square grows as its side length increases: a $1 \times 1$ square has area 1, a $2 \times 2$ has area 4, a $3 \times 3$ has area 9. The areas form the sequence $1, 4, 9, 16, 25, \dots$ — a classic quadratic sequence!

Notice how the sequence grows faster and faster: the gaps between terms keep increasing. This "speeding up" is the hallmark of a quadratic pattern. Later in these notes, you will see how this connects to real science — like a ball rolling downhill or an object in free fall. For now, just hold onto the idea: quadratic sequences grow by increasing amounts each step.

Analogy

Imagine stacking squares of tiles. A $1 \times 1$ square uses 1 tile, a $2 \times 2$ uses 4, a $3 \times 3$ uses 9. Each time you go up one size, you need more extra tiles than before. That ever-growing increase is exactly what makes the pattern quadratic!

First and Second Differences

The key tool for identifying and working with quadratic sequences is the idea of differences.

First differences are found by subtracting each term from the next:
$First difference = T_{n + 1} - T_{n}$

Second differences are found by subtracting each first difference from the next:
$Second difference = D_{n + 1} - D_{n}$

For a quadratic sequence, the second differences are always equal (constant).

Example

Example: Analyse the sequence $3, 6, 11, 18, 27, \dots$

Step 1 — Write out the terms:
$36111827$

Step 2 — Find first differences:
$+ 3 + 5 + 7 + 9$

Step 3 — Find second differences:
$+ 2 + 2 + 2$

The second differences are constant (+2), so this is a quadratic sequence. ✓

Warning

If the first differences are constant, the sequence is linear, not quadratic. Always check second differences before concluding a sequence is quadratic.

Free preview

11 more sections in this topic

Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions.