GCSE · Maths · AQA · Spec 8300 · Foundation
Triangular, square, cube number and arithmetic sequences
Some numbers secretly belong to two number families at once — and one belongs to three. Let's catch them.
Mind the gap
Which family is each sequence?
Pick a sequence, then pick its family. Work out the gaps between neighbouring terms first.
Still to sort
Arithmetic (0)
Same gap every time — up or down.
Where the line is: Rising steadily isn't enough. 1, 3, 6, 10 rises steadily, but its gaps are +2, +3, +4, so it is not arithmetic. The gap must be exactly the same every single time.
Square numbers (0)
Each term is a whole number times itself. Odd-number gaps are a clue, not the proof.
Where the line is: Square gaps go up by 2 each time (+3, +5, +7). Triangular gaps go up by 1 (+2, +3, +4). That's a quick way to tell the two apart — but 2, 5, 10, 17 has odd gaps too and isn't square, so check the terms are n × n.
Triangular numbers (0)
Each term is on the list 1, 3, 6, 10, 15, 21, … — and the gap grows by one each time.
Where the line is: Growing gaps are a clue, not proof. 2, 4, 7, 11 has gaps +2, +3, +4, but none of its terms is triangular. Check the terms are really on the triangular list.
Cube numbers (0)
Each term is a whole number times itself three times.
Where the line is: The gaps grow fast (+7, +19, +37), so don't rely on them — test a term directly: is it n × n × n?
None of these (0)
It fails every test above.
Look at what happens BETWEEN the terms, not just at the terms themselves.
Predict, then check
These two sequences start the same way, then split apart.
Sequence A: 1, 3, 5, 7, … and Sequence B: 1, 3, 6, 10, … What is the next term of each?
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Every number family leaves a fingerprint. Learn to read it.
What you need to know
- Square numbers: 1, 4, 9, 16, 25, 36, … — each is a whole number times itself (n × n).
- Cube numbers: 1, 8, 27, 64, 125, … — each is a whole number times itself, times itself again (n × n × n).
- Triangular numbers: 1, 3, 6, 10, 15, 21, … — keep adding the next counting number: +2, +3, +4, …
- An arithmetic sequence goes up or down by the same amount every time, like 5, 8, 11, 14, … (always +3).
- Write the gaps between neighbouring terms underneath a sequence for a clue to its family — then check the terms really belong to that family.
- A number can be in more than one family: 1 is in all three, 36 is square and triangular, 64 is square and cube.
The big picture
Square numbers come from multiplying a whole number by itself (4 × 4 = 16), and cube numbers from multiplying it by itself three times (3 × 3 × 3 = 27). Triangular numbers build up by adding the next counting number each time: 1, then 1 + 2, then 1 + 2 + 3, and so on. An arithmetic sequence adds (or takes away) the same amount every time. The gaps between neighbouring terms give you a quick clue to the family — but always check the terms themselves too, and remember that one number can belong to more than one family.
Key points
Worked example
Problem
An arithmetic sequence starts at 50, and each term is 7 less than the one before. (a) Work out the 6th term. (b) Which one of the first five terms is both a square number and a triangular number?
⚠ Watch out
Treating 'squared' as 'times two' and 'cubed' as 'times three' — so the 4th square number becomes 8 instead of 16, and the 3rd cube number becomes 9 instead of 27.
Memory hook
Mind the gap: same gap → arithmetic. Gap growing by 1, or odd-number gaps? Suspect triangular or square — then check the terms really are on that list. And 'square' means a number times ITSELF — never times 2.
Check yourself
Without writing anything down: is 100 a square number, a cube number, both, or neither? What would you multiply to prove it?
Flashcards
(6)What is a square number?
What is a cube number?
How do you build the triangular numbers?
How can you tell a sequence is arithmetic?
What do the gaps between square numbers look like?
Name a number that is both square and cube, and one that is both square and triangular.
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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