IGCSE Maths · Algebra and Graphs · Core & Extended

Sequences: Patterns, Term-to-Term Rules and the nth Term

A sequence is just a list of numbers following a rule. Linear, simple quadratic and simple cubic nth-term work is shared by both tiers. Exponential sequences, and combining or comparing sequences, are Extended-only — tagged throughout.

Why this matters Being able to describe a pattern with a single formula — rather than listing terms one by one — is a small taste of what makes algebra powerful: one nth-term rule can tell you the 500th term just as easily as the 5th.

1. Continuing Sequences and Finding the Term-to-Term Rule

The term-to-term rule describes how to get from one term to the next. Look at the differences between consecutive terms first — if they’re constant, it’s linear; if the differences themselves change by a constant amount, it’s quadratic.

Worked example — triangular numbers Find the next two terms of 1, 3, 6, 10, 15, …
Differences between terms: +2, +3, +4, +5 — the amount added increases by 1 each time.
Next difference is +6, then +7: 15 + 6 = 21, 21 + 7 = 28
Sequence continues: 1, 3, 6, 10, 15, 21, 28
Worked example — spotting a quadratic pattern Find the next term of 2, 5, 10, 17, 26, …
First differences: 3, 5, 7, 9 (not constant)
Second differences: 2, 2, 2 (constant — this confirms a quadratic sequence)
Next first difference: 9 + 2 = 11, so the next term is 26 + 11 = 37
Try it — Continuing Sequences
  1. Find the next two terms of 4, 7, 10, 13, …
  2. Find the next two terms of 1, 4, 9, 16, 25, …
  3. Find the next term of 3, 6, 11, 18, 27, … (check the second differences first).
Show answers
  1. 16, 19 (add 3 each time)
  2. 36, 49 (square numbers)
  3. First differences: 3, 5, 7, 9 (increasing by 2 — quadratic). Next difference is 11, so next term = 27 + 11 = 38

2. The nth Term of a Linear Sequence

nth term = dn + (a − d), where d = common difference, a = first term

In practice: find the common difference first, then check what needs to be added or subtracted to match the actual first term.

Worked example Find the nth term of 5, 8, 11, 14, 17, …
Common difference d = 3, so the formula starts as 3n + c.
When n = 1: 3(1) + c = 5 → c = 2
Answer: nth term = 3n + 2
Check: n=1 gives 5 ✓, n=2 gives 8 ✓
Try it — Linear nth Term
  1. Find the nth term of 2, 6, 10, 14, …
  2. Find the nth term of 20, 17, 14, 11, … (a decreasing sequence).
  3. Find the nth term of 7, 12, 17, 22, …, and use it to find the 20th term.
Show answers
  1. d = 4; 4(1)+c=2 → c=−2. nth term = 4n − 2
  2. d = −3; −3(1)+c=20 → c=23. nth term = 23 − 3n
  3. d = 5; 5(1)+c=7 → c=2. nth term = 5n + 2. 20th term: 5(20)+2 = 102

3. The nth Term of a Simple Quadratic Sequence

If the second differences are constant, the sequence is quadratic, in the form an² + bn + c. Find a first: it’s always half the (constant) second difference.

Worked example Find the nth term of 2, 5, 10, 17, 26, …
First differences: 3, 5, 7, 9. Second differences: 2, 2, 2 — constant, so a = 2 ÷ 2 = 1.
Compare n² to the actual sequence:
n1234
n²14916
Actual term251017
Difference1111

The difference is a constant +1 every time, so add 1 to n².
Answer: nth term = n² + 1

Try it — Quadratic nth Term
  1. Find the nth term of 3, 6, 11, 18, 27, …
  2. Find the nth term of 0, 3, 8, 15, 24, …
  3. Find the nth term of 5, 8, 13, 20, 29, …
Show answers
  1. Second differences = 2, so a = 1. n² gives 1,4,9,16; actual is 3,6,11,18 — difference is always +2. nth term = n² + 2
  2. Second differences = 2, so a = 1. n² gives 1,4,9,16; actual is 0,3,8,15 — difference is always −1. nth term = n² − 1
  3. Second differences = 2, so a = 1. n² gives 1,4,9,16; actual is 5,8,13,20 — difference is always +4. nth term = n² + 4

4. The nth Term of a Simple Cubic Sequence

A cubic sequence has a constant third difference. The most straightforward simple cubic sequences are built directly from n³.

Worked examples Find the nth term of 1, 8, 27, 64, 125, …
These are exactly the cube numbers: nth term = n³

Find the nth term of 2, 16, 54, 128, 250, …
Compare to n³ (1, 8, 27, 64, 125): each term is exactly double.
Answer: nth term = 2n³
Try it — Cubic nth Term
  1. Find the nth term of 3, 24, 81, 192, 375, …
  2. Find the nth term of 1, 9, 28, 65, 126, … (hint: compare to n³ term by term).
Show answers
  1. Compare to n³ (1,8,27,64,125): each term is exactly 3 times bigger. nth term = 3n³
  2. n³ gives 1,8,27,64,125; actual is 1,9,28,65,126 — each is exactly 1 more. nth term = n³ + 1

5. Exponential Sequences Extended only

In an exponential sequence, each term is found by multiplying the previous one by a constant ratio, rather than adding a constant difference.

nth term = a × r^(n−1), where a = first term, r = common ratio
Worked example Find the nth term of 3, 6, 12, 24, 48, …
Each term is double the last, so r = 2, and a = 3.
Answer: nth term = 3 × 2^(n−1)
Check: n=1 gives 3×1=3 ✓, n=3 gives 3×4=12 ✓
Worked example — a decreasing (fractional ratio) case Find the nth term of 100, 50, 25, 12.5, …
Each term is half the last, so r = 1/2, and a = 100.
Answer: nth term = 100 × (1/2)^(n−1)
Try it — Exponential Sequences Extended
  1. Find the nth term of 2, 6, 18, 54, …
  2. Find the nth term of 5, 20, 80, 320, …
  3. Find the nth term of 81, 27, 9, 3, …
Show answers
  1. r = 3, a = 2. nth term = 2 × 3^(n−1)
  2. r = 4, a = 5. nth term = 5 × 4^(n−1)
  3. r = 1/3, a = 81. nth term = 81 × (1/3)^(n−1)

6. Combinations of Sequences and Relationships Between Sequences Extended only

Sometimes a sequence is built by combining two simpler patterns — or two sequences are related to each other in a way you’re asked to spot and use.

Worked example — a combined sequence Find the nth term of 3, 8, 15, 24, 35, … (built from a quadratic and a linear part combined).
Second differences: first differences are 5, 7, 9, 11; second differences are 2, 2, 2 — quadratic, a = 1.
n² gives 1, 4, 9, 16, 25; the actual sequence is 3, 8, 15, 24, 35 — differences are 2, 4, 6, 8, 10… that’s not constant, so there’s a linear part too. Compare instead to n² + 2n: gives 3, 8, 15, 24, 35 — matches exactly.
Answer: nth term = n² + 2n
Worked example — relating two sequences Sequence A has nth term 3n + 1. Sequence B is formed by doubling every term of Sequence A. Find the nth term of Sequence B.
nth term of B = 2 × (3n + 1) = 6n + 2
Try it — Combinations & Relationships Extended
  1. Find the nth term of 4, 9, 16, 25, 36, … (hint: compare to (n+1)²).
  2. Sequence P has nth term 2n². Sequence Q is formed by adding 5 to every term of Sequence P. Find the nth term of Sequence Q.
  3. Sequence R has nth term n² + n. Sequence S has nth term n² − n. Find the nth term of the sequence formed by R minus S, term by term.
Show answers
  1. 4, 9, 16, 25, 36 are (n+1)²: n=1 gives 2²=4, n=2 gives 3²=9, etc. nth term = (n+1)²
  2. nth term of Q = 2n² + 5
  3. (n²+n) − (n²−n) = 2n
Common mistakes to watch for
  • Assuming a sequence is linear just because the first couple of differences look similar — always check at least 4–5 terms before deciding.
  • Forgetting that the coefficient a in a quadratic sequence is half the constant second difference, not the second difference itself.
  • Mixing up a term-to-term rule (how to get the next term from the previous one) with an nth-term rule (a direct formula for any term).
  • For exponential sequences, subtracting instead of dividing when finding the common ratio — always divide a term by the one before it.
  • Not checking a derived nth-term formula against at least two terms of the original sequence before finalising the answer.

7. Quick-Fire Challenge Round

True or False?
  1. A sequence with a constant first difference is linear.
  2. A sequence with a constant second difference is cubic.
  3. In an exponential sequence, you find the next term by adding a constant amount.
  4. The coefficient of n² in a quadratic sequence’s nth term is always half the second difference.
Show answers
  1. True.
  2. False — a constant second difference means quadratic; cubic sequences have a constant third difference.
  3. False — exponential sequences multiply by a constant ratio, not add a constant difference.
  4. True. Ext.
Spot the Error
  1. “The sequence 2, 4, 8, 16, 32 has a common difference of 2, so it’s linear.”
  2. “For the sequence 1, 5, 13, 25, 41 with second difference 4, the coefficient of n² is 4.”
  3. “The nth term of 6, 12, 24, 48 is found by adding 6 each time, giving nth term = 6n.”
Show answers
  1. The terms don’t have a constant difference — they have a constant ratio (×2 each time), which makes this exponential, not linear.
  2. The coefficient of n² is half the second difference, not the second difference itself: 4 ÷ 2 = 2, so the coefficient is 2, not 4.
  3. These terms don’t have a constant difference (6, 12, 24, 48 — differences are 6, 12, 24, not constant); they double each time, making this exponential with nth term = 6 × 2^(n−1), not linear.

8. Mixed Practice — Bringing It All Together

Core & Extended
  1. Find the next two terms of 6, 11, 16, 21, …
  2. Find the nth term of 4, 9, 14, 19, …
  3. Find the nth term of 4, 7, 12, 19, 28, … (quadratic).
  4. Find the nth term of 5, 40, 135, 320, … (cubic — compare to n³).
Show answers
  1. 26, 31
  2. d=5; nth term = 5n − 1
  3. Second differences = 2, a=1. n² gives 1,4,9,16; actual is 4,7,12,19 — always +3. nth term = n² + 3
  4. n³ gives 1,8,27,64; actual is 5,40,135,320 — each is exactly 5 times bigger. nth term = 5n³
Extended-level mixed set Extended only
  1. Find the nth term of 4, 12, 36, 108, … (exponential).
  2. Sequence A has nth term n² + 3. Sequence B is formed by subtracting 1 from every term of A. Find the nth term of B.
Show answers
  1. r=3, a=4. nth term = 4 × 3^(n−1)
  2. nth term of B = n² + 2

Practice at Home

A large set again — take it steadily.

Part A — Continuing sequences
  1. Find the next two terms of 8, 13, 18, 23, …
  2. Find the next term of 2, 6, 12, 20, 30, … (check the second differences).
  3. Find the next term of 1, 3, 7, 13, 21, …
Part B — Linear nth term
  1. Find the nth term of 9, 14, 19, 24, …
  2. Find the nth term of 30, 26, 22, 18, …
  3. Find the nth term of 1, 4, 7, 10, …, and use it to find the 50th term.
Part C — Quadratic and cubic nth term
  1. Find the nth term of 6, 9, 14, 21, 30, …
  2. Find the nth term of 1, 6, 15, 28, 45, … (hint: second differences here are 4).
  3. Find the nth term of 4, 32, 108, 256, 500, … (cubic — compare to n³).
Part D — Exponential sequences Extended
  1. Find the nth term of 7, 14, 28, 56, …
  2. Find the nth term of 3, 15, 75, 375, …
  3. Find the nth term of 200, 100, 50, 25, …
Part E — Combinations and relationships Extended
  1. Find the nth term of 2, 7, 14, 23, 34, … (hint: try n² + 2n − 1).
  2. Sequence M has nth term 4n − 1. Sequence N is formed by squaring every term of Sequence M. Write an expression for the nth term of Sequence N (you don’t need to expand it).
Part F — Applied / challenge
  1. A pattern of dots is built in rows: row 1 has 1 dot, row 2 has 4 dots, row 3 has 9 dots, row 4 has 16 dots. Find a formula for the number of dots in row n, and use it to predict row 10.
  2. A ball is dropped and bounces to 80% of its previous height each time. If the first bounce reaches 200 cm, find an expression for the height of the nth bounce.
  3. Sequence X has nth term n + 1, and Sequence Y has nth term n² − 2n − 3. For which positive integer value of n are the two sequences’ terms equal? (Set the expressions equal and solve.)

Key Vocabulary Recap

Term-to-term rule — describes how to get from one term to the next
nth term — a formula giving any term directly, using its position n
Common difference — the constant amount added in a linear sequence
Common ratio — the constant multiplier in an exponential sequence Ext.
Second difference — the difference between consecutive first differences; constant for a quadratic sequence

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C2.7 (Core subject content) and section E2.7 (Extended subject content) — Sequences. Continuing sequences, term-to-term rules, and the nth term of linear, simple quadratic, and simple cubic sequences are shared by both tiers. Exponential sequences, and combinations of or relationships between sequences, are Extended-only, marked above.

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