Number Sequences
Square, cube, triangular and geometric sequences
In this note
- 2.7Recognise square, cube and triangular numbers, and shifted or scaled versions.
- 2.7Recognise geometric sequences and write their nth term \(ar^{n-1}\).
- 2.7Find the nth term of fraction sequences and combinations such as \(n^{2}+n\) or \(\dfrac{3^{n}}{n+1}\).
Before you start: linear sequences.
Formula bank
The basic families
General methodHow to find the nth term of any non-linear sequence
- Check the differences: constant → linear (previous note). Not constant → look further.
- Each term × the same number? → geometric: \(ar^{n-1}\).
- Otherwise compare with \(n^{2}\) (1, 4, 9, 16) or \(n^{3}\) (1, 8, 27, 64) term by term: + or − a number? × a number? Shifted along (\((n+1)^{2}\))? + n?
- Fractions: find the tops and the bottoms separately.
- Check with n = 1 and n = 2.
| Sequence | Compare with n² | nth term |
|---|---|---|
| 3, 6, 11, 18 | +2 | \(n^{2}+2\) |
| 2, 8, 18, 32 | ×2 | \(2n^{2}\) |
| 4, 9, 16, 25 | shifted one place | \((n+1)^{2}\) |
| 2, 6, 12, 20 | +n | \(n^{2}+n\) |
Example Find the nth term of 4, 7, 12, 19, 28.
Compare with \(n^{2}\): 1, 4, 9, 16, 25 - each term is 3 more.
- \(n^{2}+3\)
- check n = 5: 28 ✓
Answer \(n^{2}+3\)
Try it yourself
Show answer
Answer \(n^{2}+5\)
Show answer
Answer \(3n^{2}\)
Show answer
Answer \(n^{3}-1\)
Geometric sequences
In a geometric sequence each term is the previous one × the same common ratio \(r\): \(a, ar, ar^{2}, \dots\). The first term has been multiplied 0 times, so the nth term is \(ar^{n-1}\).
General methodHow to find the nth term of a geometric sequence
- r = second term ÷ first term (check with the next pair).
- a = the first term.
- nth term \(=a\times r^{n-1}\). Only r is raised to the power.
Example Find the nth term of 2, 6, 18, 54.
a = 2, r = 3.
- \(2\times3^{n-1}\)
- check n = 4: \(2\times27=54\) ✓
Answer \(2\times3^{n-1}\)
Try it yourself
Show answer
Answer \(5^{n-1}\)
Show answer
Answer \(64\times0.5^{n-1}\)
Fraction sequences and combinations
Example Find the nth term of \(\tfrac25,\tfrac47,\tfrac69,\dots\)
Tops 2, 4, 6 and bottoms 5, 7, 9 are both linear. (Don't simplify 6/9 - it hides the pattern.)
- tops \(2n\); bottoms \(2n+3\)
Answer \(\dfrac{2n}{2n+3}\)
Try it yourself
Show answer
Answer \(\dfrac n{n+1}\)
Ways they test it
Type 1 Complete a table of sequences (0580 P4)
"Complete the table: next term and nth term for each sequence."
Spot it several sequences side by side.
- Linear, square-based, geometric: identify each.
- Check each nth term with n = 1.
Type 2 Shifted squares or cubes
"Find the nth term of 3, 12, 27, 48." (2 marks)
Spot it differences not constant.
- Compare with \(n^{2}\) / \(n^{3}\).
For example 3, 12, 27, 48
- \(n^{2}\): 1, 4, 9, 16 → ×3
Answer \(3n^{2}\)
Type 3 Geometric
"Find the nth term of 2, 6, 18, 54." (2 marks)
Spot it multiplying each time.
- \(ar^{n-1}\).
Type 4 Fractions / combinations
"Find the nth term of \(\tfrac32,\tfrac93,\tfrac{27}4,\tfrac{81}5\)."
Spot it fractions in a list.
- Tops and bottoms separately.
For example tops 3, 9, 27, 81; bottoms 2, 3, 4, 5
- \(3^{n}\) over \(n+1\)
Answer \(\dfrac{3^{n}}{n+1}\)
Mistakes to avoid
- ✗ 2, 6, 18 → \(6^{n-1}\)✓ \(2\times3^{n-1}\)Only r is raised to the power.
- ✗ 4, 9, 16 → \(n^{2}\)✓ \((n+1)^{2}\)Check n = 1.
- ✗ Treating 3, 12, 27 as linear✓ \(3n^{2}\)Check the differences first.
- ✗ 3, 6, 12 → \(3\times2^{n}\)✓ \(3\times2^{n-1}\)n = 1 must give 3.
Summary
- Know \(n^{2}\), \(n^{3}\), \(\tfrac{n(n+1)}2\).
- Compare term by term with a family: +, −, ×, or shifted.
- Geometric: \(ar^{n-1}\).
- Fractions: tops and bottoms separately.
If you see …
- differences not constantcompare with n² or n³.
- multiplying each timegeometric.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer \(n^{2}\)
Show answer
Answer 10
Show answer
Answer \(4^{n}\)
Medium
Guide me
What is it asking? nth term
What do we know? squares starting at 3²
Plan shifted along
Show working
- \(3^{2},4^{2},5^{2},\dots\)
Answer \((n+2)^{2}\)
Guide me
What is it asking? nth term
What do we know? compare with cubes
Plan 1, 8, 27 → +2
Show working
- +2
Answer \(n^{3}+2\)
Guide me
What is it asking? nth term
What do we know? halving
Plan a = 100, r = 0.5
Show working
- \(ar^{n-1}\)
Answer \(100\times0.5^{n-1}\)
Hard
Guide me
What is it asking? nth term
What do we know? fractions
Plan tops 3, 5, 7; bottoms 4, 7, 10
Show working
- tops 2n + 1
- bottoms 3n + 1
Answer \(\dfrac{2n+1}{3n+1}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| both parts | B2 | ||
| Total: 2 marks | |||
Guide me
What is it asking? nth term
What do we know? tops powers of 3
Plan bottoms n + 1
Show working
- \(3^{n}\) over \(n+1\)
Answer \(\dfrac{3^{n}}{n+1}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\dfrac{3^{n}}{n+1}\) | B2 | ||
| Total: 2 marks | |||
Guide me
What is it asking? nth term
What do we know? compare with 1, 4, 9, 16
Plan +1, +2, +3, +4
Show working
- +n
Answer \(n^{2}+n\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(n^{2}+n\) | B2 | ||
| Total: 2 marks | |||
Can I do this?
- I can recognise square, cube, triangular and geometric sequences.
- I can find the nth term of shifted or scaled squares and cubes.
- I can write the nth term of a geometric sequence.
- I can find the nth term of a fraction sequence or combination.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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