Quadratic Sequences and the nth Term Formula

A quadratic sequence is a list of numbers where the rule connecting each term to its position \(n\) involves \(n^2\). Instead of growing by the same amount each time (like an arithmetic sequence), a quadratic sequence grows by an amount that itself increases steadily. This is what gives quadratic patterns their curved, accelerating look when you graph them.

Here is a typical quadratic sequence: 3, 8, 15, 24, 35, ... To confirm it is quadratic, look at how the terms change.

The fastest way to recognize a quadratic sequence is the second differences test:

• Find the first differences (subtract each term from the next).
• Find the second differences (subtract each first difference from the next).
• If the second differences are all the same constant, the sequence is quadratic.

This matters because it is the opposite test used for an arithmetic sequence, where it is the first differences that stay constant. A quadratic sequence needs one extra layer of differencing before things settle down.

Every quadratic sequence has an nth term rule of the form \(T_n = an^2 + bn + c\). Once the second differences are confirmed constant, use this method:

1. Find a. The constant second difference is always equal to \(2a\), so \(a = \frac{\)second difference\(}{2}\).
2. Subtract \(an^2\) from the sequence. This removes the quadratic part and leaves a simple linear sequence.
3. Find b and c. Treat the leftover linear sequence the same way you would find the nth term of an arithmetic sequence: the coefficient of \(n\) is the constant first difference, and \(c\) is whatever constant is left over.

Worked Example

Use the sequence 3, 8, 15, 24, 35 from above.

Step 1: The second difference is 2, so \(a = \frac{2}{2} = 1\).

Step 2: Subtract \(n^2\) from each term:

\(3 - 1 = 2\), \(8 - 4 = 4\), \(15 - 9 = 6\), \(24 - 16 = 8\), \(35 - 25 = 10\)

This leaves the linear sequence 2, 4, 6, 8, 10.

Step 3: That linear sequence has first difference 2 and no constant left over, so it is simply \(2n\). That means \(b = 2\) and \(c = 0\).

Formula: \(T_n = n^2 + 2n\).

Check it: \(T_1 = 1 + 2 = 3\), \(T_4 = 16 + 8 = 24\), \(T_5 = 25 + 10 = 35\). Both match the original sequence.

Plotting the terms against their position \(n\) shows why these are called quadratic: the points sit on a parabola. Below is the graph of \(T_n = n^2 + 2n\) for the first several positions.

It helps to see quadratic sequences alongside the sequence types you already know. Arithmetic sequences increase by the same fixed amount every step, which is why their graphs form a straight line rather than a curve. Geometric sequences grow (or shrink) by a constant ratio, producing exponential-style curves that are shaped very differently from the smooth, gradually steepening curve of a quadratic pattern. Recognizing which differences (or ratios) stay constant is the quickest way to tell all three apart.

• Always compute at least four or five terms of differences before concluding the second differences are constant, since one or two matching values can be a coincidence.
• If a problem gives you the formula and asks for a specific term, just substitute the value of \(n\) directly, there is no need to redo the differencing.
• If a problem asks you to find a missing term, use the pattern in the second differences to work backward or forward before checking with the full formula.
• Double-check your final formula against at least two terms from the original sequence, not just one.

Related lessons

• Algebra Help