Other Sequences: Fibonacci, Geometric, and More

Sequences are ordered lists of numbers following a particular rule. Apart from arithmetic sequences, there are other types of sequences, such as Fibonacci sequences, geometric sequences, and special number patterns.

 

The Fibonacci Sequence

The Fibonacci sequence is a special sequence where each term is found by adding the two previous terms.

Definition: $$F_n = F_{n-1} + F_{n-2}$$

 

where:

• $F_1 = 1$
• $F_2 = 2$

 

First Few Terms:

$$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...$$

 

Example:

Find the next term in the Fibonacci sequence after 21 and 34.

Solution: $$21 + 34 = 55$$

 

 

Diagram Suggestion:
A simple tree diagram showing how each term is formed by adding the previous two terms.

 

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant number called the common ratio, $r$.

General Formula: $$a_n = a_1 \dot r^(n-1)$$

 

where:

• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the position of the term 

 

Example:

Find the first 5 terms of the geometric sequence where $a_1 = 3$ and $r = 2$

Solution: $$3, 6, 12, 24, 48, ...$$

 

Diagram Suggestion:
A flowchart illustrating how each term is obtained by multiplying by $r$

 

Special Sequences

Triangular Numbers

A sequence of numbers forming triangular patterns: $$1, 3, 6, 10, 15, 21, ...$$

 

Each term is found by adding consecutive natural numbers.

Formula: $$T_n = \frac{n(n + 1)}{2}$$

 

Example: Find the 5th triangular number. $$T_5 = \frac{5(5 + 1)}{2} = \frac{30}{2} = 15$$

 

Square Numbers

A sequence of perfect squares $$1, 4, 9, 16, 25, 36, ...$$

 

Formula: $$n^2$$

 

Example: Find the 7th square number $$7^2 = 49$$

 

Cube Numbers

A sequence of perfect cubes: $$1, 8, 27, 64, 125, ...$$

Formula: $$n^3$$

 

Example: Find the 4th cube number. $$4^3 = 64$$