Number sequences
Generating terms from a rule, finding the nth term of an arithmetic sequence, and using it to solve problems
About three lessons of 45 to 60 minutes
The pattern that predicts its own future
A market trader adds £3 to her takings target every week: £10, £13, £16, £19... She does not need to add £3 up 40 times to know week 40's target. If she can find 1 short rule that works for ANY week number, she can jump straight to week 40, or week 100, without ever writing out the numbers in between.
That short rule is called the nth term. Every arithmetic sequence, a list of numbers that goes up (or down) by the same amount each time, has one, and once you find it, you can predict any term instantly, check whether a particular number will EVER appear in the sequence, and model real growing patterns, from matchstick shapes to savings accounts, with a single piece of algebra.
- 10, 13, 16, 19, ... (add 3 each time)an arithmetic sequence: the nth term, 3n + 7, predicts week 40's target instantly: 3(40) + 7 = 127
- A growing matchstick pattern: 4, 7, 10, 13, ...the same 'add a constant amount' idea applied to a real shape, not just numbers
- 1, 4, 9, 16, 25, ... (the square numbers)NOT arithmetic: the gaps (3, 5, 7, 9) grow, so there is no single 'add the same amount' rule
- A savings account: £50 to open, plus £20 a monththe balance after n months, a real-world linear pattern modelled by the nth term 20n + 50
What students will be able to do
Students will generate the terms of a sequence from a term-to-term rule or a position-to-term (nth term) rule, recognise geometric sequences plus square and triangular numbers, find the nth term of an arithmetic sequence, and use the nth term rule to check whether a number is a term or to model a real-world linear pattern.
- I can generate the terms of a sequence from a term-to-term rule.
- I can generate the terms of a sequence from its nth term (position-to-term) rule.
- I can recognise and continue a geometric sequence by multiplying by its common ratio.
- I can recognise square and triangular numbers and continue the pattern.
- I can find the nth term of an arithmetic sequence from its first term and common difference.
- I can use the nth term rule to check whether a number is a term in the sequence, or find which term has a given value.
- I can use a sequence to model a real-world linear pattern.
Standards this unit teaches
- KS3 Maths: AlgebraUK National Curriculum (England)Algebra
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Algebra" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "generate terms of a sequence from either a term-to-term or a position-to-term rule"; "recognise arithmetic sequences and find the nth term"; "recognise geometric sequences and appreciate other sequences that arise".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 5 coordinate plane & patterns teaching unitplotting and graphing 2 related number patterns, the bridge into treating a sequence as points that can be graphed
- Grade 8 linear functions & slope teaching unitthe same-age unit connecting a constant rate of change (slope) to the constant common difference that drives every arithmetic sequence here
- Variable in the glossary
Words to teach and display
- Sequence
- an ordered list of numbers or shapes that follows a rule
- Term
- 1 number in a sequence; the 1st term, 2nd term, and so on
- Term-to-term rule
- a rule that gives the next term from the term before it, such as 'add 4'
- nth term
- a rule (position-to-term) that gives ANY term directly from its position number n, without needing the terms before it
- Common difference
- the fixed amount added (or subtracted) between consecutive terms of an arithmetic sequence
- Geometric sequence
- a sequence formed by multiplying each term by the same non-zero number, called the common ratio
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Generating terms from a rule
ConcreteA term-to-term rule tells you how to get from 1 term to the next, such as 'start at 5, add 4 each time'. A position-to-term (nth term) rule instead gives you ANY term directly, by substituting its position number straight into a formula.
Starting at 5 and adding 4 repeatedly gives 5, 9, 13, 17, 21, ... Each new term is exactly 4 more than the one before it, the common difference.
A geometric sequence multiplies by the same common ratio instead: 2, 6, 18, 54, ... multiplies by 3 each time. Its differences are not constant, so it is geometric rather than arithmetic.
Not every sequence goes up by the same fixed amount. Square numbers (1, 4, 9, 16, 25, ...) and triangular numbers (1, 3, 6, 10, 15, ...) both follow clear rules, n squared and n(n+1)/2, but the GAP between consecutive terms keeps changing, so neither is an arithmetic sequence.
A sequence starts at 5, with the term-to-term rule 'add 4 each time'. Write the first 5 terms.
- Start at 5.
- Add 4 repeatedly: 5, 9, 13, 17, 21.
Answer: 5, 9, 13, 17, 21.
- Write the first 4 terms of the sequence starting at 20 with the rule 'subtract 6 each time'.
- Continue the geometric sequence 3, 9, 27, ... for 2 more terms.
- Why are the square numbers NOT an arithmetic sequence?
2. Finding the nth term of an arithmetic sequence
PictorialEvery arithmetic sequence has an nth term formula, dn + (a - d), where d is the common difference and a is the first term. Once you have it, you can plug in ANY position number n and get that term straight away, no counting required.
For 5, 9, 13, 17, ..., the common difference is d = 9 - 5 = 4, and the first term is a = 5, so the nth term is 4n + (5 - 4) = 4n + 1. Checking: n = 1 gives 4(1) + 1 = 5, n = 2 gives 4(2) + 1 = 9, both match.
Because the nth term is a genuine formula, it plots as a straight line when the term number n is on 1 axis and the term's value is on the other, the same constant-rate-of-change idea as a linear function's slope.
Find the nth term of the sequence 5, 9, 13, 17, ..., then find the 20th term.
- Common difference: d = 9 - 5 = 4.
- nth term = dn + (a - d) = 4n + (5 - 4) = 4n + 1.
- 20th term: substitute n = 20: 4(20) + 1 = 80 + 1 = 81.
Answer: nth term = 4n + 1; the 20th term is 81.
- An arithmetic sequence has first term 2 and common difference 6. Find the nth term.
- Why does the coefficient of n always equal the common difference, not the first term?
3. Using the nth term rule to solve problems
AbstractThe nth term rule works both ways. Given a term number, substitute it in to find the term's value. Given a value, set the nth term formula equal to it and solve for n, if n comes out as a positive whole number, that value IS a term (and n tells you which one).
The same idea models real growing patterns: a fixed starting amount plus a constant rate per step is exactly an nth term formula, whether that is a market stall's daily target, a phone plan's monthly cost, or a matchstick pattern's stick count.
A market stall charges a £10 pitch fee plus £3 per hour. Write the nth term for the total cost after n hours, then find the cost after 6 hours. Then check: is £34 ever the exact total cost?
- Total cost = 3n + 10 (common difference £3, and n = 0 would give the fixed fee alone, so the constant is 10).
- After 6 hours: 3(6) + 10 = 18 + 10 = 28.
- Is 34 a possible total? Solve 3n + 10 = 34: n = 24 / 3 = 8, a positive whole number, so yes, £34 is the cost after exactly 8 hours.
Answer: Total cost = 3n + 10; after 6 hours it is £28, and £34 occurs after exactly 8 hours.
- A phone plan costs a £15 sign-up fee plus £25 a month. Write the nth term for the total cost after n months.
- Is 100 a term in the sequence with nth term 6n + 5? Explain how you know without listing every term.
Common misconceptions and how to address them
MisconceptionThe nth term of 5, 9, 13, 17, ... is 5n, using the first term as the coefficient of n.
Why it happens: Students assume the coefficient of n is simply the first term, rather than the amount the sequence actually increases by each step.
How to address it: The coefficient of n is always the COMMON DIFFERENCE (how much the sequence goes up by each time), never the first term. Here d = 4, so the nth term starts '4n...', then the constant is chosen so it matches when n = 1: 4(1) + 1 = 5. The nth term is 4n + 1, not 5n.
MisconceptionTo find the 8th term of the sequence with nth term 3n + 2, some students calculate 3 x 8 = 24 and stop, forgetting to add the 2.
Why it happens: They apply only the multiplication part of the formula and forget the constant term also applies to every single term, not just some of them.
How to address it: Substitute n into the WHOLE expression, every term: 3(8) + 2 = 24 + 2 = 26. The constant is part of the formula for every position, not an extra step you can skip.
MisconceptionSquare numbers must be an arithmetic sequence, because the early gaps (3, 5, 7, ...) look like they follow a simple pattern.
Why it happens: Students notice the gaps look orderly and assume 'orderly' means 'constant', without checking whether the gap ITSELF stays the same.
How to address it: Check at least 3 gaps before deciding a sequence is arithmetic. Square numbers 1, 4, 9, 16, 25 have gaps 3, 5, 7, 9, increasing by 2 each time, not constant, so square numbers are NOT arithmetic and do not have a single 'add a fixed number' rule.
MisconceptionTo check whether a number is a term in a sequence, students list terms one by one, and if they do not spot it quickly, conclude it is not a term.
Why it happens: Listing terms is unreliable for large values or for terms far down the sequence, and gives up too early without any proof.
How to address it: Solve the nth term formula equal to the target number algebraically for n. If n comes out as a positive whole number, the value IS a term (and n tells you exactly which one); if not, it can never be a term, no matter how far you listed.
Guided practice (with answers)
1. Write the first 4 terms of the sequence starting at 3 with the rule 'add 5 each time'.
Answer: 3, 8, 13, 18.
2. The nth term of a sequence is 6n - 2. Write the first 3 terms.
Answer: 4, 10, 16, substituting n = 1, 2, 3.
3. Continue the geometric sequence 2, 6, 18, 54 for 2 more terms.
Answer: 162 and 486, because each term is multiplied by the common ratio 3.
4. Find the next 2 triangular numbers after 1, 3, 6, 10.
Answer: 15 and 21, because the 5th triangular number is (5 x 6)/2 = 15 and the 6th is (6 x 7)/2 = 21.
5. An arithmetic sequence has first term 7 and common difference 3. Find the nth term.
Answer: 3n + 4, because nth term = dn + (a - d) = 3n + (7 - 3) = 3n + 4.
6. The nth term of a sequence is 5n + 1. Find the 12th term.
Answer: 61, because 5(12) + 1 = 60 + 1 = 61.
7. Is 50 a term in the sequence with nth term 4n + 2?
Answer: Yes, because 4n + 2 = 50 gives n = 12, a positive whole number, so 50 is the 12th term.
Independent practice worksheets
Practise generating sequences, finding the nth term, and applying it to solve problems, with computed, never-wrong answer keys.
Differentiation
- Always write out the first 5 terms alongside their position numbers (1, 2, 3, 4, 5) in a 2-row table before trying to spot the nth term formula.
- For 'is X a term?' questions, encourage solving the equation step by step rather than guessing, and checking the answer is a positive whole number every time.
- Keep a worked 3-4-5-... style example of square numbers visible, with the gaps written underneath, to make 'not arithmetic' concrete.
- Use real contexts (a phone plan, a market stall) for every abstract nth term problem to keep the coefficient and constant meaningful.
- Investigate the nth term of the triangular numbers, n(n+1)/2, and use it to check whether a large number (e.g. 190) is a triangular number.
- Compare the growth of a geometric sequence with an arithmetic sequence that has a similar start, and explain why repeated multiplication eventually grows much faster than repeated addition.
- Find the nth term of a sequence that decreases (negative common difference), and use it to find when the sequence first becomes negative.
- Design a real-world scenario with 2 different linear pricing plans (e.g. 2 gyms) and use their nth terms to find the number of months after which 1 plan becomes cheaper than the other.
Assessment: exit ticket
A three-question exit ticket sampling generating terms, finding the nth term, and checking whether a number is a term.
1. Write the first 4 terms of the sequence starting at 10 with the rule 'subtract 3 each time'.
Answer: 10, 7, 4, 1.
2. Find the nth term of the sequence 2, 7, 12, 17, ...
Answer: 5n - 3, because d = 5 and the constant is 2 - 5 = -3.
3. The nth term of a sequence is 4n - 5. Is 40 a term? Explain.
Answer: No, because 4n - 5 = 40 gives n = 11.25, not a whole number, so 40 is not a term.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 generating terms (section 1), Lesson 2 finding the nth term (section 2), Lesson 3 applying the nth term rule plus the exit ticket (section 3).
- Geometric sequences are taught directly through their constant common ratio; square and triangular numbers are included as contrast cases that satisfy the curriculum's 'other sequences that arise' wording and show what does NOT have a constant difference or ratio.
- Language to keep repeating: the coefficient of n is the common difference, never the first term; substitute into the WHOLE formula; check at least 3 gaps before calling a sequence arithmetic; solve for n and check it is a positive whole number.
- The functionGraph figure in section 2 deliberately reuses the same linear-graphing engine already used for slope and linear functions elsewhere on the site, so students see an arithmetic sequence as simply a linear function restricted to whole-number inputs.
- Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.