# Number Sequences – Comprehensive Explanations and Clear Examples

In $\textit{sequences}$, it is important that we can;

- recognise a pattern in a set of numbers
- describe the pattern in words
- continue the pattern.

A $\textit{number sequence}$ is an ordered list of numbers defined by a rule.

- The sequence starts at $4$ and adds $5$ each time.
- $4, 9, 14, 19, \cdots$

The numbers in the sequence are said to be its $\textit{terms}$.

- The first term is $4$.
- The second term is $9$.

A sequence that continues forever is called an $\textit{infinite sequence}$.

- $4, 9, 14, 19, \cdots$

A sequence that terminates is called a $\textit{finite sequence}$.

- $4, 9, 14, 19, 24, 29, 34$

### Example 1

Write down the first four terms of the sequence if you start with \(5\) and add \(3\) each time.

\( 5, 8, 11, 14 \)

### Example 2

Write down the first four terms of the sequence if you start with \(99\) and subtract \(4\) each time.

\( 99, 95, 91, 87 \)

### Example 3

Write down the first four terms of the sequence if you start with \(4\) and multiply \(3\) each time.

\( 4, 12, 36, 108 \)

### Example 4

Describe the sequence: \( 3, 7, 11, 14, \cdots \).

The sequence starts at \( 3 \), and each term is \( 4 \) more than the previous term.

### Example 5

Describe the sequence: $76, 73, 70, 67, \cdots$.

The sequence starts at $76$, and each term is $3$ less than the previous term.

### Example 6

Describe the sequence: \( 2, 6, 18, 54, \cdots \).

The sequence starts at \( 2 \); each term is \( 3 \)$ times the previous term.

### Example 7

Describe the sequence: $1, 4, 9, 16, \cdots$.

\( 1^2=1,2^2=4,3^2=9,4^2=16 \)

Each term is the square of the term number.

### Example 8

Find the next two terms of \( 1, 8, 27, 64 \).

Each term is the cube of the term number.

\( \begin{align}5^3 &= 125 \\ 6^3 &= 216 \end{align} \)

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