Number Sequences
Quadratic, cubic and combined sequences
In this note
- 2.7Use second differences to recognise a quadratic sequence and find its nth term \(an^{2}+bn+c\) .
- 2.7Find the nth term of simple cubic sequences such as \(n^{3}+1\) and \(n^{3}-n\).
- 2.7Find unknown coefficients when the form is given; spot combinations like \(n^{2}+2^{n}\).
Before you start: linear sequences and special sequences.
Formula bank
Second differences
Take differences of the terms (1st differences), then differences of those (2nd differences). For \(n^{2}\): 1, 4, 9, 16 → 3, 5, 7 → 2, 2. For \(an^{2}\) the second difference is 2a. So a constant 2nd difference means quadratic, and \(a=\) half of it.
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| term | 6 | 15 | 28 | 45 | 66 |
| 1st diff | 9 | 13 | 17 | 21 | |
| 2nd diff | 4 | 4 | 4 |
General methodHow to find the nth term of any quadratic sequence
- Work out the 1st and 2nd differences. Constant 2nd difference → quadratic.
- \(a=\) 2nd difference ÷ 2.
- Subtract \(an^{2}\) from each TERM (not from the differences).
- What is left is linear: find its nth term \(bn+c\) as usual.
- nth term \(=an^{2}+bn+c\). Check with another term.
Example Find the nth term of 6, 15, 28, 45, 66.
2nd difference 4 → \(2n^{2}\); subtract it and finish the linear part.
- \(2n^{2}\): 2, 8, 18, 32, 50
- terms − 2n²: 4, 7, 10, 13, 16 = 3n + 1
- check n = 5: 50 + 15 + 1 = 66 ✓
Answer \(2n^{2}+3n+1\)
Try it yourself
Show answer
Answer \(n^{2}+2n\)
Show answer
Answer \(2n^{2}-1\)
Hint
Show answer
Answer \(11-n^{2}\)
Cubic and combined sequences
General methodHow to find the nth term of a cubic or combined sequence
- Compare with the fastest-growing family: \(n^{3}\) (1, 8, 27, 64) or \(2^{n}\) (2, 4, 8, 16).
- Subtract it from each term.
- Recognise what is left (a constant, n, n², …).
| Sequence | Compare with | Left over | nth term |
|---|---|---|---|
| 2, 9, 28, 65 | \(n^{3}\) | +1 | \(n^{3}+1\) |
| 0, 6, 24, 60 | \(n^{3}\) | \(-n\) | \(n^{3}-n\) |
| 5, 7, 11, 19 | \(2^{n}\) | +3 | \(2^{n}+3\) |
Example Find the nth term of 3, 8, 17, 32, 57.
Subtract \(2^{n}\): 1, 4, 9, 16, 25.
- left over \(n^{2}\)
Answer \(2^{n}+n^{2}\)
When the form is given
General methodHow to find a and b when the nth term is \(an^{2}+bn\)
- Substitute each given position n and its term: one equation each.
- Simplify, then solve the simultaneous equations.
Example nth term \(an^{2}+bn\); 2nd term 14, 5th term 65. Find a and b.
Substitute n = 2 and n = 5.
- \(4a+2b=14\Rightarrow2a+b=7\)
- \(25a+5b=65\Rightarrow5a+b=13\)
- subtract: \(3a=6\)
Answer \(a=2,\ b=3\)
Ways they test it
Type 1 Quadratic nth term
"Find the nth term of 3, 11, 25, 45, 71." (3 marks)
Spot it 2nd differences constant.
- \(a\) from the 2nd difference (M1).
- Subtract \(an^{2}\), linear leftover (M1, A1).
For example 3, 11, 25, 45, 71
- 2nd diff 6 → 3n²
- leftover 0, −1, −2, −3, −4 = 1 − n
Answer \(3n^{2}-n+1\)
Type 2 Form given
"The nth term is \(an^{2}+bn\). The 2nd term is 14 and the 5th is 65."
Spot it letters in a given nth term.
- Substitute.
- Simultaneous equations.
Type 3 Cubic or combination
"Find the nth term of 3, 8, 17, 32, 57."
Spot it neither linear nor quadratic.
- Compare with \(n^{3}\) or \(2^{n}\).
Type 4 A table of sequences
"Complete the table" with linear, quadratic and geometric rows (0580 P4)
Spot it several sequences.
- Identify each type first.
Mistakes to avoid
- ✗ 2nd difference 6 → \(6n^{2}\)✓ \(3n^{2}\)The 2nd difference is 2a.
- ✗ Leftover found from the 1st differences✓ subtract \(an^{2}\) from the TERMSThe leftover is a sequence of terms.
- ✗ Never checking✓ check n = 3 or 4One slip changes every term.
Summary
- Constant 2nd difference → quadratic; \(a=\tfrac12\) of it.
- Subtract \(an^{2}\) from the terms; the leftover is linear.
- Cubics and combinations: compare with \(n^{3}\) or \(2^{n}\).
- Form given: substitute and solve simultaneous equations.
If you see …
- 1st differences not constantfind the 2nd differences.
- 2nd differences not constanttry n³ or 2ⁿ.
Practice
Practice: easy, medium, hard
Easy
Show working
- 2nd diff 2 → n²
- leftover 4, 8, 12, … = 4n
Answer \(n^{2}+4n\)
Show answer
Answer 3
Show working
- cubes + 1
Answer \(n^{3}+1\)
Medium
Guide me
What is it asking? nth term
What do we know? 2nd diff 2 → n²
Plan subtract n², linear leftover
Show working
- leftover 2, −1, −4, −7, −10 = 5 − 3n
Answer \(n^{2}-3n+5\)
Guide me
What is it asking? nth term
What do we know? 2nd diff 8
Plan a = 4
Show working
- leftover −2, −4, −6, −8 = −2n
Answer \(4n^{2}-2n\)
Guide me
What is it asking? nth term
What do we know? compare with cubes 1, 8, 27, 64
Plan leftover −1, −2, −3, −4
Show working
- −n
Answer \(n^{3}-n\)
Hard
Guide me
What is it asking? nth term
What do we know? 2nd diff 6
Plan a = 3; subtract 3n²
Show working
- leftover 0, −1, −2, −3, −4 = 1 − n
Answer \(3n^{2}-n+1\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(3n^{2}\) | M1 | ||
| leftover | M1 | ||
| \(3n^{2}-n+1\) | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a
What do we know? two equations
Plan substitute n = 2 and 5
Show working
- \(2a+b=7\), \(5a+b=13\)
- \(3a=6\)
Answer 2
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| both equations | M1 | ||
| a = 2 | A1 | ||
| b = 3 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? nth term
What do we know? not quadratic (2nd diffs 4, 6, 10)
Plan subtract \(2^{n}\)
Show working
- leftover 1, 4, 9, 16, 25
Answer \(2^{n}+n^{2}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(2^{n}\) seen | M1 | ||
| \(2^{n}+n^{2}\) | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can use 1st and 2nd differences to identify a sequence.
- I can find the nth term of a quadratic sequence.
- I can find the nth term of simple cubic and combined sequences.
- I can find coefficients when the form is given.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.
Your progress on this note is saved when JavaScript is on.