A linear sequence is a sequence where the difference between consecutive terms is constant. Finding the nth term means writing a formula that allows any term in the sequence to be found by substituting a value for \( n \).

 

The nth term of a linear sequence has the form:

$$
an + b
$$

 

where \( a \) is the common difference and \( b \) is a constant.

 

 

Finding the nth Term from a Numerical Sequence

Start by finding the common difference.

 

For example, consider the sequence:

$$
4,\ 7,\ 10,\ 13
$$

 

The difference between terms is \( 3 \), so \( a = 3 \).

 

Write the nth term as:

$$
3n + b
$$

 

Now find \( b \) by substituting a known term.

 

When \( n = 1 \), the term is \( 4 \):

$$
3(1) + b = 4
$$

$$
b = 1
$$

 

So the nth term is:

$$
3n + 1
$$

 

 

Another Numerical Example

Consider the sequence:

$$
10,\ 7,\ 4,\ 1
$$

 

The difference is \( -3 \), so \( a = -3 \).

 

Write:

$$
-3n + b
$$

 

Substitute \( n = 1 \), term \( = 10 \):

$$
-3(1) + b = 10
$$

$$
b = 13
$$

 

So the nth term is:

$$
-3n + 13
$$

 

 

Finding the nth Term from a Diagrammatic Sequence

Sometimes a sequence is shown using a diagram, such as shapes or patterns that increase regularly.

 

For example, suppose a pattern shows:

• 1st diagram has 5 squares
• 2nd diagram has 8 squares
• 3rd diagram has 11 squares

 

This gives the numerical sequence:

$$
5,\ 8,\ 11
$$

 

The difference is \( 3 \), so start with:

$$
3n + b
$$

 

Substitute \( n = 1 \), term \( = 5 \):

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

$$
b = 2
$$

 

So the nth term is:

$$
3n + 2
$$

 

 

Checking the nth Term Rule

Always check the rule using another value of \( n \).

 

For example, using \( 3n + 2 \) when \( n = 3 \):

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

 

This matches the third term, so the rule is correct.

 

 

Key Points to Remember

Linear sequences have a constant difference.
The nth term has the form \( an + b \).
The value of \( a \) is the common difference.
Find \( b \) by substituting a known term.
Always check the rule against the sequence.

 

Being able to find the nth term of a linear sequence links patterns to algebra and allows any term in the sequence to be found efficiently.