The nth term is a rule that allows you to find any term in a sequence without having to write out the whole sequence.
For GCSE Grade 4, the most important skill is finding the nth term of a linear sequence.
What is a sequence?
A sequence is a list of numbers that follows a pattern.
For example:
\[ 3,\ 7,\ 11,\ 15,\ 19,\ldots \]
Each number is called a term.
- 1st term = 3
- 2nd term = 7
- 3rd term = 11
- 4th term = 15
- 5th term = 19
The sequence goes up by 4 each time.
How to find the nth term
Follow these two steps.
1 Find the difference
Work out how much the sequence increases or decreases by each time.
Example: \[5,\ 8,\ 11,\ 14,\ldots\]
The difference is $+3$.
2 Put the difference in front of $n$
If the difference is 3, start with $3n$.
So the nth term starts as:
\[ 3n \]
But how do we get the final answer? We need to adjust the number at the end.
3 Finding the number at the end
Let’s use this sequence:
\[ 5,\ 8,\ 11,\ 14,\ldots \]
Step 1: The difference is 3, so start with $3n$.
Step 2: Write out the first term of $3n$.
When $n=1$:
\[ 3n=3\times1=3 \]
But the actual first term is 5.
Step 3: Work out the difference between 3 and 5.
\[ 5-3=2 \]
We need to add 2.
Therefore, the nth term is:
\[ \boxed{3n+2} \]
Check it:
| Term | Calculation | Answer |
|---|---|---|
| 1st | $3(1)+2$ | 5 |
| 2nd | $3(2)+2$ | 8 |
| 3rd | $3(3)+2$ | 11 |
It works!
Three examples to learn
EXAMPLE 1 — Increasing sequence
Find the nth term of:
\[ 4,\ 7,\ 10,\ 13,\ldots \]
Step 1: Difference = 3, so start with $3n$.
Step 2: The first term of $3n$ is 3.
Step 3: The actual first term is 4, so add 1.
Answer: \[\boxed{3n+1}\]
EXAMPLE 2 — A larger starting number
Find the nth term of:
\[ 7,\ 12,\ 17,\ 22,\ldots \]
Step 1: Difference = 5, so start with $5n$.
Step 2: The first term of $5n$ is 5.
Step 3: $7-5=2$, so add 2.
Answer: \[\boxed{5n+2}\]
EXAMPLE 3 — A decreasing sequence
Find the nth term of:
\[ 15,\ 12,\ 9,\ 6,\ldots \]
Step 1: Difference = $-3$, so start with $-3n$.
Step 2: The first term of $-3n$ is $-3$.
Step 3: $15-(-3)=18$, so add 18.
Answer: $\boxed{-3n+18}$
Remember this exam tip
For most nth-term questions, use this method:
- Find the difference between the numbers.
- Write that difference followed by $n$.
- Compare the first term with your starting expression.
- Add or subtract the difference needed to get the correct first term.
The key rule: The number in front of $n$ is the difference between consecutive terms.