An nth term rule gives a formula that allows any term in a sequence to be found by substituting a value for \( n \). Generating a sequence means using this rule to find the first few terms.

 

The value of \( n \) starts at \( 1 \) for the first term, \( 2 \) for the second term and so on.

 

 

Generating Linear Sequences

A linear sequence has a constant difference between terms. Its nth term rule has the form:

$$
an + b
$$

where \( a \) and \( b \) are constants.

 

For example, consider the nth term rule:

$$
3n + 2
$$

 

Find the first four terms by substituting values of \( n \).

 

When \( n = 1 \):

$$
3(1) + 2 = 5
$$

 

When \( n = 2 \):

$$
3(2) + 2 = 8
$$

 

When \( n = 3 \):

$$
3(3) + 2 = 11
$$

 

When \( n = 4 \):

$$
3(4) + 2 = 14
$$

 

So the sequence is:

$$
5,\ 8,\ 11,\ 14
$$

 

The constant difference is \( 3 \), confirming that the sequence is linear.

 

 

Another Linear Example

Given the nth term rule:

$$
5n - 1
$$

 

Substitute values of \( n \):

$$
n = 1 \Rightarrow 4
$$

$$
n = 2 \Rightarrow 9
$$

$$
n = 3 \Rightarrow 14
$$

 

So the sequence is:

$$
4,\ 9,\ 14
$$

 

 

Generating Non Linear Sequences

A non linear sequence does not have a constant difference between terms. These sequences often involve powers of \( n \), such as \( n^2 \) or \( n^3 \).

 

For example, consider the nth term rule:

$$
n^2 + 1
$$

 

Generate the first four terms.

$$
n = 1 \Rightarrow 2
$$

$$
n = 2 \Rightarrow 5
$$

$$
n = 3 \Rightarrow 10
$$

$$
n = 4 \Rightarrow 17
$$

 

So the sequence is:

$$
2,\ 5,\ 10,\ 17
$$

 

The differences are not constant, so the sequence is non linear.

 

 

Another Non Linear Example

Given the nth term rule:

$$
2n^2
$$

 

Substitute values of \( n \):

$$
n = 1 \Rightarrow 2
$$

$$
n = 2 \Rightarrow 8
$$

$$
n = 3 \Rightarrow 18
$$

$$
n = 4 \Rightarrow 32
$$

 

The sequence is:

$$
2,\ 8,\ 18,\ 32
$$

 

 
Key Points to Remember

Start with \( n = 1 \) to find the first term.
Substitute increasing integer values of \( n \) to generate the sequence.
Linear sequences come from rules of the form \( an + b \).
Non linear sequences usually involve \( n^2 \), \( n^3 \) or higher powers.

 

Being able to generate sequences from the nth term rule helps link algebra with patterns and prepares for further work with sequences and graphs.