Sequences: nth Term
The Skill
What is a Sequence?
A sequence is a list of numbers that follow a pattern. Each number is called a term.
Example: 3, 7, 11, 15, 19, ... (common difference = 4)
Linear (Arithmetic) Sequences
A linear sequence has a constant difference between consecutive terms.
Finding the nth Term
The nth term of a linear sequence has the form: $$\text{nth term} = dn + c$$
where $d$ is the common difference.
Method:
- Find the common difference ($d$)
- Write $dn$ (this gives you 0th term wrong)
- Adjust by comparing to the first term
Example: Find the nth term of 5, 8, 11, 14, ...
- Common difference = $8 - 5 = 3$
- Start with $3n$: when $n=1$, $3n = 3$
- But the first term is 5, so add 2
- nth term = $3n + 2$
Check: $n=1$: $3(1)+2=5$ ✓, $n=2$: $3(2)+2=8$ ✓
Quadratic Sequences (Higher)
A quadratic sequence has a second difference that is constant.
Example: 2, 5, 10, 17, 26, ...
| Terms | 2 | 5 | 10 | 17 | 26 |
|---|---|---|---|---|---|
| 1st diff | 3 | 5 | 7 | 9 | |
| 2nd diff | 2 | 2 | 2 |
The nth term has the form $an^2 + bn + c$
If 2nd difference = 2, then $a = 1$ (half of 2nd difference)
Using the nth Term
To find a specific term, substitute the term number.
If nth term = $4n - 3$, find the 20th term: $$20\text{th term} = 4(20) - 3 = 80 - 3 = 77$$
Is a Number in the Sequence?
Set the nth term equal to the number and solve for $n$. If $n$ is a positive whole number, the value is in the sequence.
The Traps
Common misconceptions and how to avoid them.
Adding first term to dn without checking "The First Term Fumble"
The Mistake in Action
Find the nth term of 5, 9, 13, 17, ...
Wrong: Difference = 4, first term = 5 nth term = $4n + 5$
Why It Happens
Students find the common difference correctly and then just add the first term, without checking if this actually gives the right values.
The Fix
After writing $dn$, compare it to the actual first term to find the adjustment.
Sequence: 5, 9, 13, 17, ... (difference = 4)
$4n$ gives: 4, 8, 12, 16, ... (when $n$ = 1, 2, 3, 4)
Compare first terms: sequence has 5, but $4n$ gives 4. Adjustment: $5 - 4 = 1$
nth term = $4n + 1$
Check: $4(1)+1=5$ ✓, $4(2)+1=9$ ✓
Spot the Mistake
Sequence: 5, 9, 13, 17, ...
Difference = 4
nth term = $4n + 5$
Click on the line that contains the error.
Using first term instead of common difference "The Difference Confusion"
The Mistake in Action
Find the nth term of 3, 7, 11, 15, ...
Wrong: nth term = $3n + 4$
Why It Happens
Students swap the roles of the first term and common difference in the formula.
The Fix
In $dn + c$:
- $d$ is the common difference (the gap between terms)
- The coefficient of $n$ is always the difference
For 3, 7, 11, 15, ...:
- Common difference = $7 - 3 = 4$
- So we start with $4n$: gives 4, 8, 12, 16, ...
- First term should be 3, but $4n$ gives 4 when $n=1$
- Adjustment: $3 - 4 = -1$
nth term = $4n - 1$
Spot the Mistake
Sequence: 3, 7, 11, 15, ...
nth term = $3n + 4$
Click on the line that contains the error.
Using the full second difference instead of half "The Half Slip"
The Mistake in Action
Find the nth term of 1, 4, 9, 16, 25, ...
Second differences = 2
Wrong: nth term = $2n^2 + ...$
Why It Happens
Students correctly find the second difference is 2, but forget to halve it to get the coefficient of $n^2$.
The Fix
For a quadratic sequence $an^2 + bn + c$:
$$a = \frac{\text{second difference}}{2}$$
If second difference = 2, then $a = \frac{2}{2} = 1$
For 1, 4, 9, 16, 25, ...: This is actually $n^2$ (the square numbers)!
nth term = $n^2$ (or $1n^2 + 0n + 0$)
Spot the Mistake
Second differences = 2
nth term = $2n^2 + ...$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Find the nth term of the sequence 7, 10, 13, 16, ...
Solution
Step 1: Find the common difference. $$10 - 7 = 3$$
Step 2: Write down $3n$ and find what it gives. When $n = 1$: $3(1) = 3$ When $n = 2$: $3(2) = 6$ So $3n$ gives: 3, 6, 9, 12, ...
Step 3: Compare to the actual sequence and adjust. Actual sequence: 7, 10, 13, 16, ... $3n$ gives: 3, 6, 9, 12, ... Each term is 4 more than $3n$.
nth term = $3n + 4$
Check:
- $n=1$: $3(1)+4 = 7$ ✓
- $n=2$: $3(2)+4 = 10$ ✓
Question
Find the nth term of the sequence 2, 9, 16, 23, ...
Solution
Step 1: Find the common difference. $$9 - 2 = 7$$
Step 2: Compare $7n$ to the sequence. $7n$ gives: 7, 14, 21, 28, ... (when $n$ = 1, 2, 3, 4) Actual: 2, 9, 16, 23, ...
Each term in the sequence is 5 less than $7n$. $$7 - 5 = 2$$ ✓ $$14 - 5 = 9$$ ✓
nth term = $7n - 5$
Check: $n=4$: $7(4) - 5 = 28 - 5 = 23$ ✓
Question
The nth term of a sequence is $4n + 3$. Is 83 a term in this sequence?
Solution
Set the nth term equal to 83 and solve for $n$: $$4n + 3 = 83$$ $$4n = 80$$ $$n = 20$$
Since $n = 20$ is a positive whole number, 83 is in the sequence.
It is the 20th term.
Check: $4(20) + 3 = 80 + 3 = 83$ ✓
Question
Find the nth term of 3, 6, 11, 18, 27, ...
Solution
Step 1: Find the first and second differences.
| Terms | 3 | 6 | 11 | 18 | 27 |
|---|---|---|---|---|---|
| 1st diff | 3 | 5 | 7 | 9 | |
| 2nd diff | 2 | 2 | 2 |
Second difference = 2 (constant), so this is quadratic.
Step 2: Find the coefficient of $n^2$. $$a = \frac{\text{2nd difference}}{2} = \frac{2}{2} = 1$$
So the nth term starts with $n^2$.
Step 3: Compare $n^2$ to the sequence.
| $n$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $n^2$ | 1 | 4 | 9 | 16 | 25 |
| Sequence | 3 | 6 | 11 | 18 | 27 |
| Difference | 2 | 2 | 2 | 2 | 2 |
The sequence is always 2 more than $n^2$.
nth term = $n^2 + 2$
Check: $n=3$: $9 + 2 = 11$ ✓
Level 2: Scaffolded
Fill in the key steps.
Question
The nth term of a sequence is $5n - 2$. Find the 30th term.
Level 3: Solo
Try it yourself!
Question
Pattern 1 uses 4 matchsticks. Pattern 2 uses 7 matchsticks. Pattern 3 uses 10 matchsticks. (a) Find a formula for the number of matchsticks in pattern $n$. (b) How many matchsticks are in pattern 50?
Show Solution
(a) Sequence: 4, 7, 10, ...
Common difference = 3
$3n$ gives: 3, 6, 9, ... Sequence: 4, 7, 10, ... Adjustment: $4 - 3 = 1$
Formula: $3n + 1$
(b) Pattern 50: $$3(50) + 1 = 150 + 1 = 151$$
Answers: (a) $3n + 1$ matchsticks, (b) 151 matchsticks
Examiner's View
Mark allocation: Finding nth term: 2-3 marks. Using nth term: 1-2 marks. Quadratic: 3-4 marks.
Common errors examiners see:
- Using the first term as the adjustment (not comparing $dn$ to first term)
- Confusing the common difference with the first term
- Sign errors in the formula
- For quadratics, forgetting to halve the second difference
What gains marks:
- Showing the difference calculation
- Checking with at least two terms
- Clear working for quadratic sequences
AQA Notes
AQA often asks "Is 100 in this sequence?" — set up an equation and solve.