IGCSE Sequences and nth Term
Spot patterns, find an nth term, and distinguish arithmetic from geometric sequences.
🎯 Key idea: A sequence is a list with a rule. First ask whether the same amount is added each time or the same multiplier is used each time.
The notation $u_n$ means the term in position $n$.
🔢 1. Arithmetic sequences
An arithmetic sequence has a constant difference. For example, $7, 11, 15, 19, [[dots]]$ increases by $4$ each time.
Worked example: Find the next two terms in $-3, 2, 7, 12, [[dots]]$.
The common difference is $+5$.
So the next two terms are $17$ and $22$.
Worked example: Find the nth term of $6, 10, 14, 18, [[dots]]$.
Step 1
The common difference is $4$, so start with $4n$.
Step 2
When $n=1$, $4n=4$, but the first term is $6$.
Step 3
Add $2$. The nth term is $4n+2$.
Arithmetic nth term: for first term $a$ and common difference $d$,
$$u_n=a+(n-1)d$$
✖️ 2. Geometric sequences
A geometric sequence has a constant multiplier, called the common ratio.
Worked example: Describe the rule for $3, 12, 48, 192, [[dots]]$.
Each term is multiplied by $4$.
This is geometric, not arithmetic: the differences are not constant.
Quick check: “add or subtract the same number” means arithmetic. “multiply or divide by the same number” means geometric.
A sequence can have negative terms or fractions; the rule still matters more than the appearance.
🧭 3. Using an nth term
An nth term lets you jump straight to any position without listing every earlier term.
Worked example: The nth term is $5n-3$. Find the $20$th term.
Substitute $n=20$.
$5(20)-3=97$
So the $20$th term is $97$.
Worked example: Is $72$ a term of $5n-3$?
Step 1
Set $5n-3=72$.
Step 2
$5n=75$, so $n=15$.
Since $15$ is a positive whole-number position, $72$ is in the sequence.
Common mistake: Do not use $n=0$ as the first term unless the question explicitly starts at zero. In most IGCSE questions, the first term is when $n=1$.
Practice Problems
Attempt each question before opening the answer. Explain the rule in words as well as calculating.
1. Find the next two terms: $14, 9, 4, -1, [[dots]]$.
Show answer
The common difference is $-5$.
Answer: $-6, -11$.
2. Find the nth term of $9, 13, 17, 21, [[dots]]$.
Show answer
Step 1
The difference is $4$, so begin with $4n$.
Step 2
$4(1)=4$ and the first term is $9$, so add $5$.
Answer: $4n+5$.
3. The nth term is $7n-2$. Find the $12$th term.
Show answer
$7(12)-2=82$.
Answer: $82$.
4. Is $101$ a term of the sequence with nth term $6n+5$?
Show answer
Set $6n+5=101$.
$6n=96$, so $n=16$.
Answer: Yes, it is the $16$th term.
5. Is $2, 10, 50, 250, [[dots]]$ arithmetic or geometric? State the rule.
Show answer
Each term is multiplied by $5$.
Answer: Geometric; multiply by $5$.
Interactive Quiz
Choose the best answer, then submit the quiz.
1. What is the common difference of $4, 10, 16, 22, [[dots]]$?
2. What is the nth term of $2, 5, 8, 11, [[dots]]$?
3. Which sequence is geometric?
4. If $u_n=4n+7$, what is $u_{10}$?
Sequence Explorer
Choose a rule, generate a preview, then find any term in the sequence.
Choose settings, then select Generate & check.
Enter a sequence of numbers separated by commas, and the detective will try to find the rule.