The Essence of Quadratic Sequences

Characterizing Quadratic Sequences

• A quadratic sequence is a sequence of numbers where the second differences between consecutive terms are constant, pointing to a quadratic relationship among the terms.
• Unlike arithmetic sequences, where the first differences are constant, quadratic sequences involve changes that accelerate or decelerate in a quadratic manner.

Forming Quadratic Sequences

Quadratic sequences can be represented generalistically as:

$$an^2 + bn + c$$

Here, $n$ represents the term's position in the sequence, and $a$, $b$, and $c$ are constants that determine the exact nature of the sequence.

Identifying Quadratic Sequences

1. Calculate First Differences: Subtract each term from the next to find the first differences.

2. Calculate Second Differences: If the sequence is quadratic, the second set of differences (subtracted from the first differences) will be constant.

Solving Quadratic Sequences

To find the nth term of a quadratic sequence, you need to determine the constants $a$, $b$, and $c$ that fit the pattern of the sequence. This often involves setting up equations based on known terms of the sequence and solving for these constants.

Example: Finding the nth Term

Given the sequence $1, 4, 9, 16, \ldots$ (the square numbers), we can see that the second differences are constant ($4$), indicating a quadratic sequence of the form $an^2$. Since the sequence is the squares of $n$, it follows the form $n^2$.