Special sequences

In this section, we’ll learn how to identify common types of sequences: arithmetic, geometric, Fibonacci, quadratic, and triangular numbers. Each sequence type has a distinct pattern.

1. Arithmetic Sequences

Arithmetic sequences add or subtract the same amount each time. For example:

\[ 2, 5, 8, 11, 14 \]

Check the difference between terms. Here, each term increases by 3, so this is an arithmetic sequence.

2. Geometric Sequences

Geometric sequences multiply or divide by the same number each time. For example:

\[ 3, 6, 12, 24, 48 \]

Divide one term by the previous term. Here, \( \frac{6}{3} = 2 \), so each term is multiplied by 2. This is a geometric sequence.

3. Fibonacci Sequences

The Fibonacci sequence starts with 1 and 1 (or sometimes 0 and 1), and each term is the sum of the two previous terms. For example:

\[ 1, 1, 2, 3, 5, 8 \]

Check by adding two consecutive terms. Here, \( 1 + 1 = 2 \), \( 1 + 2 = 3 \), \( 2 + 3 = 5 \), and so on. This is a Fibonacci sequence.

4. Quadratic Sequences

Quadratic sequences have terms that follow a pattern based on square numbers, and the differences between terms change at a constant rate. For example:

\[ 2, 6, 12, 20, 30 \]

Find the differences between terms: \( 6 - 2 = 4 \), \( 12 - 6 = 6 \), \( 20 - 12 = 8 \), \( 30 - 20 = 10 \). The second differences are constant (\( 6 - 4 = 2, 8 - 6 = 2 \)), so this is a quadratic sequence.

5. Triangular Numbers

Triangular numbers are formed by arranging dots into triangles. The sequence is:

\[ 1, 3, 6, 10, 15 \]

Each term is the sum of all previous integers. For example, \( 3 = 1 + 2 \), \( 6 = 1 + 2 + 3 \), and so on. These are triangular numbers.

Summary

• Arithmetic sequences: Add or subtract the same amount each time.
• Geometric sequences: Multiply or divide by the same number each time.
• Fibonacci sequences: Add the two previous terms to get the next term.
• Quadratic sequences: The second differences are constant.
• Triangular numbers: Formed by summing consecutive integers.

Recognising these patterns makes it easy to classify and work with sequences.