# 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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