Rising: 5, 8, 11, 14, …
Difference 3, so 3n: 3, 6, 9, 12
Each term is 2 more than 3n
nth term: 3n + 2
Check: n = 1 gives 3 + 2 = 5
Sequences and the nth term Why is the nth term of a linear sequence the equation of a line?
Foundation and Higher · AQA · 45 minutes
Students decide whether 100 is in a sequence, then plot the terms as dots and see that the nth term is the equation of the line through them. Slides to present, a vote that shows who's stuck, practice and homework.
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The dots are 5, 8, 11, 14, … above positions 1, 2, 3, 4, …, all on one straight line. Move the level line y = k. When does it meet that line exactly on a dot? So is 100 a term?
They cross at (2.67, 10)
The idea in one minute
In a linear sequence the terms go up (or down) by the same amount each time: the common difference. Plot each term above its position, as the point (n, term), and the dots lie on a straight line. The difference is the line's gradient, and the term before the first, the zero term, is where it meets the y-axis. So the nth term is dn + c, like y = mx + c. A number is a term when n comes out a whole number.
The lesson goes picture first. Students try to decide whether 100 is in 5, 8, 11, 14, … before anyone explains, then plot terms as dots, move the line to fit them and read off the rule. Only then do they find nth terms of rising and falling sequences, and test numbers by solving dn + c = N. The core is for Foundation and Higher; the stretch meets a quadratic sequence.
By the end, students can
- find any term from an nth term such as 4n + 3
- find the nth term of a rising or falling linear sequence
- explain the nth term as the line through the dots (n, term)
- decide whether a number is in a sequence, and which term it is
Key words
- term
- position
- common difference
- nth term
- zero term
- linear sequence
Before this lesson
Students should substitute into an expression such as 3n + 2, know the times tables and plot points. Gradient is met here: up for each 1 across.
Next: Straight-line graphs.
The lesson, step by step
45 minutes in total. The stretch is optional.
5 min 1 · Hook · students think
Is 100 in it?
Students decide whether 100 and 101 are in 5, 8, 11, 14, … before anyone explains.
Say“Here is a sequence: 5, 8, 11, 14, and so on. Is 100 in it? Is 101? Decide without writing out every term.”
Ask- How much does it go up each time?
- Which numbers that size are definitely in it?
Watch forSaying 100 is in it because it is near a term, or because 99 is in the 3 times table: keep it, the next step settles it.
For the teacher
Give two minutes of silent work, then vote. Some will list terms; some will notice every term is 2 more than a multiple of 3.
Don't explain yet. The question is the point: is there a rule that gives any term at once, and tells us whether a number can be one?
Take a vote with Ask the class: 100 yes or no, 101 yes or no. Gather one or two ways of deciding.
Shorter: Ask about 100 only; skip the vote (−2 min).
12 min 2 · Explain · you explain
Dots on a line, then the rule
Plot terms as dots: the difference is the gradient, the zero term is c. Then rules and testing a number.
Say“Put each term above its position. Watch what shape the dots make.”
Ask- The dots go up 2 for each 1 across: that is the line's gradient, its steepness. What would it be for 5, 8, 11, 14?
- If the sequence went back one step before the first term, what would it be?
Watch forWriting n + 2 for 1, 3, 5, 7: going up 2 each time means 2n, the 2 times table, then adjust.
Falling: 7, 5, 3, 1, …
Difference −2: gradient −2
Zero term: 7 + 2 = 9
nth term: 9 − 2n
Is it a term?
3n + 2 = 101, so 3n = 99 and n = 33: yes, the 33rd term
3n + 2 = 100, so 3n = 98 and n = 32.66…: no
100 falls between the 32nd term, 98, and the 33rd, 101
For the teacher
First slide: plot 1, 3, 5, 7 as (1, 1), (2, 3), (3, 5), (4, 7). The dots lie on a straight line. Each step right the line goes up 2. How far a line goes up for each 1 across is its gradient (its steepness; the straight-line-graphs lesson takes it further): the common difference is the gradient.
Second slide: start with c = 0, the line y = 2x: the 2 times table, every dot 1 too high. Move c to −1 and the line goes through every dot. At n = 0 the line is at −1: the zero term, the term before the first. So the nth term is 2n − 1.
Then without a graph: 5, 8, 11, 14 is 3n + 2 (3n is 3, 6, 9, 12; add 2), checked with n = 1. A falling sequence has a negative difference: 7, 5, 3, 1 is 9 − 2n. Last, a number is a term when dn + c = N gives a whole n. This answers the hook: 101 is the 33rd term; 100 is not a term.
Vote before moving c (Ask the class): what must c be for the line to go through every dot?
Shorter: Show the line through the dots without moving c; skip the falling sequence slide (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin, which is almost the same.”
Ask- What is the difference? So which times table do you start from?
Watch forStopping at the times table (5n for 6, 11, 16, …): check with n = 1.
A rising sequence
4, 7, 10, 13, …
Difference 3; 3n gives 3, 6, 9: add 1
nth term: 3n + 1
Then theirs
6, 11, 16, 21, …
A falling sequence
15, 12, 9, 6, …
Difference −3; zero term 15 + 3 = 18
nth term: 18 − 3n
Then theirs
20, 16, 12, 8, …
Is it a term?
Is 61 in 4, 7, 10, 13, …?
3n + 1 = 61, so 3n = 60 and n = 20
Yes: the 20th term
Then theirs
Is 99 in 6, 11, 16, 21, …?
For the teacher
Work the first example aloud, writing each line. Then students solve its twin while you watch who checks with n = 1.
Do the same for each pair. The second pair catches students who lose the minus; the third, those who decide without solving for n.
Open the answers slide once most have finished their twins.
Shorter: Do the first and third pairs only (−3 min).
15 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- How can you check an nth term without me?
Watch forWriting the term-to-term rule with n (n + 5) for the nth term.
Show your working on paper. Check each nth term with n = 1 and n = 2. Write negative numbers with a minus sign. The stretch is optional.
The question that shows who's stuck
- What is the nth term of 4, 9, 14, 19, …?
In your Classroom, as they answer (an example) - A9
- B5
- C11
- D4
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- The nth term of a sequence is 4n + 3. What is the 10th term?
- The first three terms of 2n + 5 are ___, ___ and ___
- 3, 10, 17, 24, … has the nth term ___n − ___
- 30, 26, 22, 18, … has the nth term ___ − ___n
- Is 75 a term of the sequence 4, 7, 10, 13, …?
- Which term of 6, 11, 16, 21, … is 96? (Give its position, n.)
- The terms of a sequence are plotted as points (n, term). They lie on the line y = 4x − 3. What is the sequence?
- One sequence has nth term 4n + 1, another 50 − 3n. They have the same term at position n = ___, where both are ___
- The nth term is 3n + 2. Is the 20th term double the 10th term?
- Stretch: 2, 5, 10, 17, 26, … goes up 3, 5, 7, 9, so it is not linear. Compare it with n²: its nth term is n² + ___, and its 10th term is ___
Support · For question 3: the terms go up 7 each time, so start from 7n: 7, 14, 21, 28. How far is each term from that? For question 5: write the nth term first, then make it equal to the number and solve. A term needs a whole number n.
For the teacher
Everyone chooses before anyone explains. Then ask each group to put n = 1, 2, 3 into the rule they chose and compare with 4, 9, 14: a rule that goes up in 1s, or one that starts at the wrong number, shows itself at once.
Questions 1–2 build fluency; 3–4 change the look; 5–7 ask for reasons; 8 is exam-style; 9 asks about doubling. The stretch is a quadratic sequence (Higher).
Students show their working on paper or the board; the quiz keeps their final answers.
If many chose n + 5, put n = 1, 2, 3 into it together (6, 7, 8) and ask what the difference of n + 5 is, before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: why is a quadratic sequence not on a straight line? (+10 min).
5 min 5 · Exit · students show
Is 100 in this one?
One test of a number and one sentence about the graph show who has the whole idea.
Say“The nth term is 6n − 4. Is 100 a term? Show how you know, then say what the 6 is on a graph of the terms.”
Watch forRounding n to a whole number and saying yes.
Show your working; one sentence for the second part.
- The nth term of a sequence is 6n − 4. Is 100 a term? Show how you know. Then say in one sentence what the 6 means on a graph of the terms as points (n, term).
For the teacher
Students solve 6n − 4 = 100 and decide.
Then they say in one sentence what the 6 means on a graph of the dots (n, term).
The answers show who tests a number by solving, and who links the difference to the gradient: start the next lesson with whichever few did. Set the homework.
Longer: Plot the first four terms of 6n − 4 and draw the line through them (+5 min).
One to one: let the student plot the first few terms as dots and say what the line through them has to do with the rule; have them check every nth term with n = 1 and n = 2.
Homework: rules and terms
Show your working. Check every nth term with n = 1 and n = 2. Question 6 is for Higher.
- Write the first four terms of the sequence with nth term 5n − 2.
- Find the nth term of 9, 13, 17, 21, …
- Find the nth term of 40, 33, 26, 19, …
- Is 200 a term of 9, 13, 17, 21, …? Show how you know.
- Plot the first four terms of 2n + 1 as points (n, term). Draw the line through them. What is its gradient, and where does it meet the y-axis?
- (Higher) Find the nth term of 3, 9, 19, 33, 51, …
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching sequences and the nth term: common questions
How do you teach the nth term so that it makes sense?
Plot the terms as points (position, term). A linear sequence lies on a straight line: the common difference is how far it goes up each step, its gradient, and the term before the first is where it meets the y-axis. The nth term dn + c is that line's equation, with n for x.
Why do students write n + 3 for a sequence going up in 3s?
They write the term-to-term rule, add 3, with an n in it. Put n = 1, 2, 3 into n + 3: it gives 4, 5, 6, going up in 1s. Going up 3 each time is the 3 times table, 3n, moved up or down to fit the first term.
How do you know whether a number is in a sequence?
Write the nth term, make it equal to the number and solve for n. A whole number n means it is a term, and n says which one; otherwise it falls between two terms. Listing terms works for small numbers but not for large ones.
Is this nth term lesson suitable for Foundation GCSE?
Yes. The core finds and uses the nth term of rising and falling linear sequences, for students who can substitute into an expression. The stretch, a quadratic sequence compared with n², is Higher.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: A23 (terms from a term-to-term or position-to-term rule), A24 (arithmetic progressions; quadratic sequences) and A25 (the nth term of linear sequences; of quadratic sequences, Higher).
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