How to Write a Quadratic Sequence

A quadratic sequence is a series of numbers where the second difference between consecutive terms remains constant. It's defined by a formula that involves a squared term.

Understanding the Formula

The general form of a quadratic sequence is represented by the equation:

Tn = an² + bn + c

Where:

Tn represents the nth term in the sequence.
n represents the position of the term in the sequence (1st, 2nd, 3rd, etc.).
a, b, and c are constants. These values determine the specific quadratic sequence.

Finding the Constants (a, b, c)

There are several methods to determine the constants a, b, and c:

Method 1: Using the First Three Terms: This method involves solving a system of simultaneous equations. You substitute the first three terms of the sequence into the general formula (T1, T2, T3) and solve for a, b, and c.
Method 2: Using Differences: This is a more intuitive approach. Calculate the first differences (the difference between consecutive terms) and then the second differences. The second difference will be constant and equal to 2a. From there, you can find a, and use that information to deduce b and c. This method is explained in detail on resources such as Radford Mathematics.

Examples

Let's illustrate with an example using the second method. Consider the sequence: 1, 3, 7, 13, 21...

First Differences: 2, 4, 6, 8
Second Differences: 2, 2, 2 (constant, so it's quadratic)
Find 'a': The second difference is 2a, therefore, 2a = 2, so a = 1.
Find 'b' and 'c': Substitute the first three terms into the equation T_n = an² + bn + c using the value of a (a = 1):
- T₁ = 1² + b(1) + c = 1
- T₂ = 1² + b(2) + c = 3
- T₃ = 1² + b(3) + c = 7
Solve for b and c: Solving this system of equations (usually through substitution or elimination) yields b = 1 and c = -1.
Final Equation: The nth term of this quadratic sequence is T_n = n² + n -1.

Writing Your Own Quadratic Sequence

Choose your constants (a, b, c): Select any three numbers for a, b, and c.
Substitute into the formula: Plug these constants into the general formula, Tn = an² + bn + c.
Generate terms: Substitute integer values of n (1, 2, 3, 4, etc.) into the formula to generate the terms of your sequence.