Sequences and nth Terms
A sequence is an ordered list of numbers or patterns. You will learn to describe how it changes and find a term without writing out every earlier term.
Foundation & Higher: start with patterns, term-to-term rules and linear nth terms. Recognising quadratic sequences and using a supplied quadratic rule are shared skills. Deriving a quadratic nth term and geometric sequences with surd multipliers are Higher.
You should be comfortable with negative numbers and substituting into expressions. The Introduction to Algebra lesson explains substitution.
Terms, positions and rules
Each number in a sequence is called a term. Its position tells you where it is in the list. In this lesson, the first position is 1.
For 5, 9, 13, 17, … the first term is 5, the second is 9 and the fourth is 17. The dots mean the sequence continues.
A term-to-term rule tells you how to get the next term from earlier terms. A position-to-term rule uses the position to find a term directly.
When you describe a sequence, give its starting term as well as its rule. “Add 4” alone could describe many different sequences.
| Position | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Term | 5 | 9 | 13 | 17 |
Recognising and continuing patterns
Worked example 1 — A constant difference
Continue 5, 9, 13, 17, … by two terms.
The differences are 9 − 5 = 4, 13 − 9 = 4 and 17 − 13 = 4.
Add 4 each time: 17 + 4 = 21, then 21 + 4 = 25.
Next terms: 21, 25. A sequence with a constant difference is called arithmetic or linear.
Worked example 2 — Not every sequence adds the same amount
Continue 3, 6, 12, 24, … using the rule “multiply by 2”.
24 × 2 = 48, then 48 × 2 = 96.
Next terms: 48, 96. Multiplying by the same number each time gives a geometric sequence. Halving each time also gives a geometric sequence.
For a different rule, start 2, 3 and add the previous two terms: 2, 3, 5, 8, 13, 21, … . This is a Fibonacci-type sequence. You need both starting terms.
| Useful pattern | First five terms | How it grows |
|---|---|---|
| Square numbers | 1, 4, 9, 16, 25 | 1², 2², 3², 4², 5² |
| Cube numbers | 1, 8, 27, 64, 125 | 1³, 2³, 3³, 4³, 5³ |
| Triangular numbers | 1, 3, 6, 10, 15 | Add 2, then 3, then 4, then 5 |
A few terms can fit more than one possible rule. Use the rule or pattern described in the question; do not assume that the first difference alone determines every later term.
Finding a linear nth term
The letter n means the position: 1, 2, 3 and so on. The nth term is a rule for the number in position n.
For example, 4n means 4 × n. It gives the multiples of 4: 4, 8, 12, 16, … .
To find a linear rule, find the common difference, multiply n by that difference, then adjust to match the terms. Check at least the first two positions.
Worked example 3 — Compare with a times table
Find the nth term of 5, 9, 13, 17, … .
The common difference is 4, so begin with 4n.
The terms are all 1 more than the corresponding multiples of 4.
nth term = 4n + 1.
Check n = 1: 4 × 1 + 1 = 5. Check n = 2: 4 × 2 + 1 = 9.
| Position n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 4n | 4 | 8 | 12 | 16 |
| 4n + 1 | 5 | 9 | 13 | 17 |
Worked example 4 — A decreasing sequence
Find the nth term of 18, 15, 12, 9, … .
The common difference is −3. Start with −3n.
At n = 1, −3n gives −3. Add 21 to get 18.
nth term = 21 − 3n.
Check n = 4: 21 − 3 × 4 = 9. A negative common difference is allowed.
Using and interpreting a rule
Worked example 5 — Find a distant term
The nth term is 4n + 1. Find the 20th term.
Substitute n = 20: 4 × 20 + 1 = 81.
To generate the first three terms, substitute n = 1, 2 and 3 separately: 5, 9 and 13. Do not put the previous term in place of n.
Worked example 6 — Is a number in the sequence?
Does 93 occur in the sequence with nth term 4n + 1?
Set 4n + 1 = 93.
Subtract 1: 4n = 92.
Divide by 4: n = 23. This is a positive whole-number position, so 93 is the 23rd term.
For 94, the same steps give n = 93 ÷ 4 = 23.25. There is no position 23.25, so 94 is not a term. A zero or negative position would also be invalid here.
Worked example 7 — Build a rule from a pattern
A row of joined squares uses matchsticks. One square needs 4 sticks; each extra square shares one side and adds 3 sticks.
For 1, 2, 3, 4 squares, the stick totals are 4, 7, 10, 13.
Start with the first 4 sticks and add 3 for each of the remaining n − 1 squares: 4 + 3(n − 1).
Expand and simplify: 3n + 1 sticks.
For 12 squares: 3 × 12 + 1 = 37 sticks. Here n counts squares, not sticks.
Using supplied rules — Foundation & Higher
Worked example — A supplied quadratic rule
The nth term is n² + 2. Substituting n = 1, 2, 3, 4 gives 3, 6, 11, 18. The first differences are 3, 5, 7; the second differences are 2, 2. This is a quadratic pattern. Recognising it and using its supplied rule do not require deriving that rule.
Worked example — A fractional geometric multiplier
Start at 8 and multiply by 3/2 each time: 8, 12, 18, 27, 81/2, … . The multiplier is positive and rational even though it is not a whole number. For a supplied nth-term rule (3/2)ⁿ, positions 1, 2 and 3 give 3/2, 9/4 and 27/8.
Shared practice — A sequence has nth term 2n² + 1. Find its first four terms and its fifth term.
For n = 1, 2, 3, 4: 3, 9, 19, 33. For n = 5: 2 × 25 + 1 = 51.
The first differences of the first four terms are 6, 10, 14 and the second differences are 4, 4. Using a supplied quadratic rule is shared work.
Shared practice — Start at 16 and multiply by 3/4 each time. Find the next three terms.
16 × 3/4 = 12; 12 × 3/4 = 9; 9 × 3/4 = 27/4.
The next terms are 12, 9, 27/4. Keep the same multiplier.
Higher — quadratic nth terms
A quadratic sequence can be generated by a rule containing n². Its first differences change, but its second differences are constant and non-zero.
Second differences are the differences between neighbouring first differences. Use several terms before deciding a pattern is quadratic.
For a rule an² + bn + c, the second difference is 2a. The letters a, b and c stand for fixed numbers; n is still the position.
Why 2a? The first differences of n² are 3, 5, 7, … and their differences are 2. Multiplying n² by a multiplies these second differences by a. A linear part bn + c adds no second difference.
Find a by halving the second difference. Subtract an² from each term. The remainder is linear, so find its rule using the earlier method.
Worked example 8 — Start with n²
Find the nth term of 3, 7, 13, 21, 31, … .
First differences: 4, 6, 8, 10. Second differences: 2, 2, 2.
Half of 2 is 1, so start with n².
Subtract 1, 4, 9, 16, 25 from the terms. The remainders are 2, 3, 4, 5, 6.
The remainder rule is n + 1, so nth term = n² + n + 1.
Check n = 4: 16 + 4 + 1 = 21.
Worked example 9 — A coefficient before n²
Find the nth term of 4, 9, 18, 31, 48, … .
First differences: 5, 9, 13, 17. Second differences: 4, 4, 4.
Half of 4 is 2, so subtract 2n².
The remainders are 2, 1, 0, −1, −2. Their rule is 3 − n.
nth term = 2n² − n + 3.
Check n = 5: 2 × 25 − 5 + 3 = 48.
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Given term | 4 | 9 | 18 | 31 | 48 |
| 2n² | 2 | 8 | 18 | 32 | 50 |
| Remainder | 2 | 1 | 0 | −1 | −2 |
Higher — geometric sequences with surds
Worked example 10 — Keep the terms exact
A geometric sequence starts √3, 3, 3√3, 9, … . The symbol √ means square root.
The multiplier is √3: √3 × √3 = 3, and 3 × √3 = 3√3.
Continue from 9: 9 × √3 = 9√3.
Then 9√3 × √3 = 9 × 3 = 27.
Next terms: 9√3, 27. Keep root expressions exact. This sequence is not linear: its differences are not constant.
Higher — other supplied sequence rules
A question may define a different rule. Apply exactly that rule rather than assuming a constant difference or multiplier.
Worked example — A fractional position rule
For nth term n/(n + 1), the first four terms are 1/2, 2/3, 3/4, 4/5. At n = 10 the term is 10/11. Substitute n into both numerator and denominator.
Worked example — A defined recurrence
A recurrence defines a term from earlier terms. Let u₁ = 2 and uₙ₊₁ = 2uₙ + 1; read this as “double the current term, then add 1”. Thus u₂ = 5, u₃ = 11 and u₄ = 23. The subscript is the position: here the start is at 1, not 0.
Higher practice — Find the first three terms of 1/(n + 2). Also find u₂ and u₃ when u₁ = 3 and uₙ₊₁ = 2uₙ − 1.
Substitute n = 1, 2, 3: 1/3, 1/4, 1/5.
u₂ = 2 × 3 − 1 = 5; u₃ = 2 × 5 − 1 = 9. Each step uses the newest term.
Think Like a Mathematician
Why does a difference of 4 suggest 4n, but not always give the complete rule?
Every sequence with rule 4n + c increases by 4, whatever fixed number c represents. Changing c moves every term by the same amount.
The sequences 4, 8, 12, … and 5, 9, 13, … share a difference but have different starting terms. The starting term tells you which adjustment is needed.
Common Mistakes
Confusing a term with its position: in 4n + 1, replace n with 1, 2, 3, …, not with the previous term.
Writing only the difference: “add 4” continues a sequence; 4n + 1 finds its nth term. They answer different questions.
Forgetting the adjustment: compare your proposed rule with the first term and check another term.
Using a non-integer position: a membership answer such as n = 8.5 means the number is not in this sequence.
Treating every pattern as linear: look at several differences. A multiply-by rule is different from an add-on rule.
Higher — using the second difference as a: halve it first. Then find the remaining linear part.
Practice — build your confidence
Questions 1–10 are Foundation & Higher. Questions 11–12 derive quadratic rules and Question 14 uses a surd multiplier: these are Higher. Question 13 uses a supplied quadratic rule and is shared. Try each question before revealing the working.
1. Continue 7, 12, 17, 22, … by two terms. State the rule.
Each difference is 5. Add 5 to 22 to get 27, then add 5 again to get 32.
27, 32; add 5 each time, starting at 7.
2. Continue 64, 32, 16, 8, … by two terms.
Each term is half the previous term.
8 ÷ 2 = 4 and 4 ÷ 2 = 2. Next terms: 4, 2.
3. A Fibonacci-type sequence starts 3, 4. Each new term is the sum of the previous two. Find the next four terms.
3 + 4 = 7; 4 + 7 = 11; 7 + 11 = 18; 11 + 18 = 29.
7, 11, 18, 29. Each step uses the latest two terms.
4. Continue the triangular numbers 1, 3, 6, 10, 15 by two terms.
The additions are 2, 3, 4 and 5. Next add 6, then 7.
15 + 6 = 21; 21 + 7 = 28. 21, 28.
5. Find the nth term of 8, 13, 18, 23, … .
The difference is 5, so compare with 5n: 5, 10, 15, 20.
Every term is 3 greater. 5n + 3. Check n = 3: 15 + 3 = 18.
6. Find the nth term of 14, 10, 6, 2, … .
The difference is −4, so start with −4n.
At n = 1 this gives −4; add 18 to get 14. 18 − 4n.
Check n = 4: 18 − 16 = 2.
7. The nth term is 6n − 2. Find the first three terms and the 15th term.
Use n = 1, 2, 3: 6 − 2 = 4; 12 − 2 = 10; 18 − 2 = 16.
For n = 15: 6 × 15 − 2 = 88. The first terms are 4, 10, 16.
8. Is 103 in the sequence 5n + 3? If so, give its position. Is 104 in it?
For 103: 5n + 3 = 103, so 5n = 100 and n = 20. 103 is the 20th term.
For 104: 5n = 101, so n = 20.2. This is not a whole-number position, so 104 is not a term.
9. A pattern starts with 6 tiles and adds 4 tiles per stage. Write a rule for stage n and find the number at stage 9.
There are n − 1 additions after the first stage.
6 + 4(n − 1) = 6 + 4n − 4 = 4n + 2.
At stage 9: 4 × 9 + 2 = 38 tiles.
10. A student says the nth term of 6, 10, 14, 18, … is n + 4. Explain the error and correct it.
The rule n + 4 increases by 1 as n increases by 1. The given terms increase by 4.
Use 4n, then add 2 to match the first term. 4n + 2 gives 6, 10, 14, 18.
11. Higher: find the nth term of 2, 5, 10, 17, 26, … .
First differences: 3, 5, 7, 9. Second difference: 2. Start with n².
Subtract n²: the remainders are 1, 1, 1, 1, 1.
n² + 1. At n = 5, 25 + 1 = 26.
12. Higher: find the nth term of 5, 12, 23, 38, 57, … .
First differences: 7, 11, 15, 19. Second difference: 4, so a = 2.
Subtract 2n²: 3, 4, 5, 6, 7. This remainder has rule n + 2.
2n² + n + 2. Check n = 4: 32 + 4 + 2 = 38.
13. A sequence has nth term n² + n + 1. Is 43 a term?
Try nearby positive positions: n = 5 gives 31 and n = 6 gives 36 + 6 + 1 = 43.
Yes, 43 is the 6th term. Substituting one valid position is enough to establish membership.
14. Higher: continue √2, 2, 2√2, 4, … using the same multiplier. Give two more terms.
The multiplier is √2.
4 × √2 = 4√2. Then 4√2 × √2 = 4 × 2 = 8.
4√2, 8. Do not round √2 before calculating.
Frequently Asked Questions
Is an nth-term rule the same as a term-to-term rule?
No. “Add 5” tells you the next term once you know a term. A rule such as 5n + 3 gives a term directly from its position.
Can terms be negative or fractional?
Yes. A sequence might be 3, 1, −1, −3, … or 8, 4, 2, 1, 0.5, … . The positions in this lesson are still positive whole numbers.
Does a constant second difference prove every later term follows the same rule?
It identifies a quadratic pattern consistent with the terms shown. A finite list alone cannot determine every later term uniquely; use the stated type or intended pattern in the question.
Do I need a calculator?
These methods can be practised without one. A calculator can check a distant term, but show the rule and substitution rather than only its display.