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Cambridge IGCSE · International Maths Extended
Sequences
🧠 The Big Idea: A sequence is just a list of numbers following a rule — and once you find that rule as a formula in terms of position n, you can jump straight to any term (the 1st, the 100th, even the 10,000th) without ever writing out the whole list.
📋 Summary — What This Chapter Covers
Sequences are ordered lists of numbers (terms) that follow a rule, and each term has a position, labelled with n or as a subscript like an.
Term-to-term rules tell you how to get from one term to the next (add, subtract, multiply, divide).
Position-to-term rules (the nth term formula) let you jump directly to any term without building the whole sequence.
Linear sequences have a constant difference between terms — formula is dn + b.
Quadratic sequences have a constant second difference — formula is an² + bn + c.
Cubic sequences have a constant third difference — formula is an³ + bn² + cn + d.
Exponential (geometric) sequences multiply by the same number each time — formula is a × rⁿ.
You can check if a value belongs to a sequence by setting the formula equal to that value and solving for n — if n is a positive whole number, it's in the sequence.
Harder problems combine sequences (fractions of two sequences, sums of two sequences) or require solving simultaneous equations.
1. Introduction to Sequences
What is a sequence?
A sequence is simply an ordered set of numbers that follow a rule. For example: 3, 6, 9, 12, ... — here the rule is "add 3 each time."
Each number in the list is called a term. The position of a term tells you where it sits in the list — the letter n is used for this. So:
n = 1 refers to the 1st term
n = 2 refers to the 2nd term
If you don't know the exact position, you just call it the "nth term"
You'll also see sequences written using subscripts: a general sequence is a₁, a₂, a₃, ... where a₁ is the 1st term, a₂ is the 2nd term, and aₙ represents the nth term (any term you like).
Think of it like...
A sequence is like a numbered row of houses on a street. The "position" is the house number, and the "term" is what's actually inside that house (the value). The rule is what decides what goes in each house as you move down the street.
Term-to-term rules
Term-to-term rules tell you how to get the next term from the term you're currently on. It's the operation you repeat each time. Common ones include:
Adding or subtracting a number — e.g. 3, 7, 11, 15, ... (add 4 each time)
Multiplying or dividing by a number — e.g. 2, 6, 18, 54, ... (multiply by 3 each time)
The catch with term-to-term rules: to find the 50th term, you'd have to work out all 49 terms before it! That's where position-to-term rules save the day.
Position-to-term rules (the nth term formula)
A position-to-term rule — also called the nth term formula — is an algebraic expression in n that lets you calculate any term directly, just by plugging in its position number. No need to work out every term before it!
How to use it
Substitute n = 1 for the 1st term, n = 2 for the 2nd term, n = 100 for the 100th term...
You can jump straight to the 100th term without finding the other 99 first.
Worked Example
The nth term is 8n + 2.
1st term: 8×1 + 2 = 10
2nd term: 8×2 + 2 = 18
100th term: 8×100 + 2 = 802
Worked Example
The nth term is n² + 1.
1st term: 1² + 1 = 2
2nd term: 2² + 1 = 5
100th term: 100² + 1 = 10,001
Does a value belong to the sequence?
If you know the nth term formula, you can check whether a given number appears anywhere in the sequence. Just set the formula equal to that value and solve for n.
Rule
If n comes out as a positive whole number → the value IS in the sequence.
If n is a fraction or negative → the value is NOT in the sequence.
Worked Example
A sequence has nth term formula 8n + 2. Is 98 in the sequence?
8n + 2 = 98 → 8n = 96 → n = 96 ÷ 8 = 12
n is a whole number → YES, it's the 12th term ✓
Is 124 in the sequence?
8n + 2 = 124 → 8n = 122 → n = 122 ÷ 8 = 15.25
n is NOT a whole number → NOT in the sequence ✗
Practice Question 1
A sequence has the nth term formula 5n − 3. Is the number 47 a term in this sequence? If so, which term is it?
Practice Question 2
Using the same formula 5n − 3, determine whether 100 is a term in the sequence.
2. Linear (Arithmetic) Sequences
What makes a sequence "linear"?
A linear sequence goes up (or down) by the same amount every single time. That fixed amount is called the common difference, given the letter d.
1, 4, 7, 10, 13, ... → adding 3 each time, so d = 3
15, 10, 5, 0, -5, ... → subtracting 5 each time, so d = -5
Linear sequences are also called arithmetic sequences — same thing, different name.
Finding the nth term formula
Formula
nth term = dn + b
d = common difference (how much it goes up by each time) | b = the value that would come before the first term (the "zero term")
The trick to finding b: imagine going backwards one step from the first term. Whatever number lands there is b.
Worked Example
Sequence: 5, 7, 9, 11, ...
The sequence adds 2 each time → d = 2
Go backwards one step from 5: 5 − 2 = 3, so b = 3
nth term = 2n + 3
Worked Example
Sequence: -7, -3, 1, 5, 9, ...
Goes up by 4 each time → d = 4
Go backwards from -7: -7 − 4 = -11, so b = -11
nth term = 4n − 11
Common Mistake
Students often forget the "going backwards" step and just guess b from the first term itself. Remember: b is the term before the sequence starts (the "0th" term), not the first term!
Practice Question
Find the nth term formula for the sequence: 2, 9, 16, 23, 30, ...
3. Quadratic Sequences
What is a quadratic sequence?
A quadratic sequence has an nth term formula that involves n². Unlike linear sequences, the first differences (term-to-term gaps) are not constant — but the second differences (the differences between the first differences) are constant.
Sequence
3
9
19
33
51
1st differences
6
10
14
18
2nd differences
4
4
4
Since the second differences are constant (all 4), this confirms it's a quadratic sequence.
Simple quadratic sequences (comparing to square numbers)
The formula n² generates the square numbers: 1, 4, 9, 16, 25, 36, 49, ... (from 1², 2², 3², 4²...). Many quadratic sequences are just variations of these:
16, 25, 36, 49, ... has formula (n + 3)² — same square numbers, shifted start
2, 5, 10, 17, 26, ... has formula n² + 1 — each term is 1 more than a square number
2, 8, 18, 32, 50, ... has formula 2n² — each term is double a square number
3, 9, 19, 33, 51, ... has formula 2n² + 1 — a combination: double, then add 1
The full 5-step method for ANY quadratic sequence
The 5 Steps
1. Find the 1st and 2nd differences
2. Divide the 2nd difference by 2 → gives you a (coefficient of n²)
3. Write out the first terms of an² and subtract from the original sequence
4. Find the linear nth term of that difference row → gives you bn + c
5. Add: an² + bn + c = your final formula
Step 3: Compare to n² = 1, 4, 9, 16: differences are 4, 3, 2, 1
Step 4: The nth term of 4, 3, 2, 1, ... is −n + 5
nth term = n² − n + 5
Examiner Tip
Learn your square numbers from 1² to 15² by heart. This one memorisation trick makes quadratic sequence questions dramatically faster because you'll instantly spot the comparison.
Practice Question
Find the nth term formula for the sequence: 4, 7, 12, 19, 28, ...
4. Other Sequences (Cubic & Exponential)
Cubic sequences
A cubic sequence has an nth term formula involving n³. This time it's the third differences that stay constant.
Sequence
4
25
82
193
376
649
1st diff
21
57
111
183
273
2nd diff
36
54
72
90
3rd diff
18
18
18
Key relationship
a = (third difference) ÷ 6
This gives the coefficient of n³ in an³ + bn² + cn + d. Then subtract an³ from the sequence and find the nth term of what's left (usually linear).
Worked Example
Sequence: 5, 19, 57, 131, 253, 435, ...
1st diff: 14, 38, 74, 122, 182 — not constant
2nd diff: 24, 36, 48, 60 — not constant
3rd diff: 12, 12, 12 — constant! So a = 12 ÷ 6 = 2
Compare to 2n³ = 2, 16, 54, 128, 250: the difference is 3, 3, 3, 3, 3 each time
nth term = 2n³ + 3
Exponential (geometric) sequences
An exponential sequence (also called geometric) is one where you multiply each term by the same number to get the next term — rather than adding.
E.g. 3, 6, 12, 24, 48, ... — each term is multiplied by 2. This "2" is called the common ratio, r. Find it by dividing any term by the term right before it.
Formula
nth term = a × rⁿ
where r is the common ratio (multiplier)
If r > 1 → the sequence increases
If 0 < r < 1 → the sequence decreases (e.g. r = ½)
Worked Example
Sequence: 4, 20, 100, 500, 2500, ...
Check: 4×5=20, 20×5=100, 100×5=500 → common ratio r = 5
Compare to 5ⁿ = 5, 25, 125, 625, 3125...
Divide 5ⁿ by 5, then multiply by 4 to match the sequence
nth term = (4/5) × 5ⁿ or equivalently 4 × 5ⁿ⁻¹
Common Mistake
Don't confuse "multiply by a constant" (exponential) with "add a constant" (linear)! If the numbers are growing very fast — much faster than a straight-line increase — suspect exponential and test by dividing consecutive terms.
Harder combinations
Exam questions love to combine ideas. Two common twists:
Fraction sequences: if the numerators follow one sequence and denominators follow another, find each formula separately, then write as a fraction. E.g. ³⁄₁, ⁵⁄₈, ⁷⁄₂₇, ⁹⁄₆₄, ... → numerators are 2n+1, denominators are n³, so nth term = (2n+1)/n³
Sum of two sequences: if U is the prime numbers (2,3,5,7,...) and V has nth term 4n², then U + V is found by adding corresponding terms.
Setting up equations: some questions give you algebraic terms (like x−1, 2x, x²) and ask you to find x by setting up and solving simultaneous equations using the common ratio or difference.
Worked Example
The first three terms of an exponential sequence are x − 1, 2x, x². Find the common ratio r (given x ≠ 0).
From terms 1&2: r = 2x/(x−1). From terms 2&3: r = x²/2x
In the exam, write "1, 2, 3, 4..." above your sequence terms — this small habit prevents you from miscounting which position you're looking at, especially in quadratic/cubic difference tables.
Learn your squares and cubes cold
Recognising 1²–15² and 1³–7³ instantly is the single biggest time-saver for quadratic and cubic sequence questions. Don't calculate them each time — memorise them.
"Is it in the sequence?" always needs a whole number check
A common mark-scheme trap: students solve for n correctly but forget to state whether n is a whole number, and therefore whether the value is actually in the sequence. Always finish with a clear statement like "n = 15.25, which is not a whole number, so 124 is not in the sequence."
Don't mix up "b" with the first term
In dn + b, students often plug in the first term as b directly. Remember: b is the term that would come BEFORE position 1 — always step backwards once.
Check your formula by testing n = 1, 2, 3
After finding any nth term formula, always substitute n = 1, 2, 3 back in to confirm you get the original sequence terms. This catches arithmetic slips before you lose marks.
Second/third differences must be checked fully
Don't assume a sequence is linear after checking just two gaps — always compute enough terms (at least 4-5) to be sure the differences are genuinely constant before deciding linear vs. quadratic vs. cubic.