Number Sequences
Linear sequences and the nth term
In this note
- 2.7Continue a sequence and use a term-to-term rule.
- 2.7Find and use the nth term \(an+b\) of a linear sequence, including decreasing ones.
- 2.7Decide whether a number is a term; find the first term above or below a value; compare two sequences.
Formula bank
Two ways to describe a sequence
A term-to-term rule says how to get the NEXT term (5, 9, 13, 17: add 4). The nth term is a formula using the position \(n\) (\(4n+1\)): it gives the 100th term at once. Term-to-term rules can be more than 'add': 'multiply by 2 and add 3' from 1 gives 1, 5, 13, 29.
Try it yourself
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Answer 27
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Answer 47
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Answer 1, 3, 5
Finding the nth term
In a linear sequence the terms change by the same amount \(d\). The \(d\)-times table goes up in the same steps, so the sequence is \(dn\) shifted by a constant.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 4n | 4 | 8 | 12 | 16 |
| sequence | 5 | 9 | 13 | 17 |
| difference | +1 | +1 | +1 | +1 |
General methodHow to find the nth term of any linear sequence
- Find the common difference \(d\) (2nd term − 1st term). Negative if the sequence goes down.
- Find the zero term \(c\): first term − \(d\) (the term one step BEFORE the first).
- nth term \(=dn+c\) (write \(29-6n\) rather than \(-6n+29\)).
- Check with the 2nd term.
Example Find the nth term of 23, 17, 11, 5, …
\(d=-6\); zero term \(23+6=29\).
- \(-6n+29\)
- check n = 2: 29 − 12 = 17 ✓
Answer \(29-6n\)
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Answer \(3n+8\)
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Answer \(19-3n\)
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Answer \(6n-4\)
Using the nth term
General methodHow to use an nth term
- A given term: substitute n (50th term of 4n + 1 = 201).
- Is it a term? Solve nth term = number; n must be a positive whole number.
- First term above (or below) a value: solve the inequality, then round n to the next whole number in the right direction.
- Two sequences: write an inequality with both nth terms at the same n.
Example Is 99 a term of \(4n+1\)?
Solve 4n + 1 = 99.
- \(4n=98\)
- \(n=24.5\)
Answer No: n is not a whole number
Example First term of \(4n+1\) greater than 500?
Solve, round n up, substitute back.
- \(4n+1>500\Rightarrow n>124.75\)
- \(n=125\)
Answer 501
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Answer Yes, the 33rd
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Answer 8
Ways they test it
Type 1 Next term and nth term
"7, 12, 17, 22, … (a) next term (b) nth term." (3 marks)
Spot it a list of terms.
- Common difference.
- dn + (first − d).
For example 7, 12, 17, 22
- d = 5
- zero term 2
Answer 27; \(5n+2\)
Type 2 Decreasing sequence
"Find the nth term of 31, 26, 21, 16." (2 marks)
Spot it terms going down.
- d negative; zero term = first term − d (bigger).
For example 31, 26, 21, 16
- d = −5
- zero term 36
Answer \(36-5n\)
Type 3 Is it a term?
"Explain why 150 is not a term of 4n + 3."
Spot it "explain", "show that … is (not) a term".
- Solve: n not whole.
- Or: all terms are odd.
Type 4 First term past a value
"Find the first negative term."
Spot it "first term greater / less than".
- Inequality.
- Round n the right way; give the TERM.
Type 5 Two sequences
"Smallest n for which 6n + 1 is greater than 100 − 3n." (3 marks)
Spot it two nth terms.
- Inequality.
- Next whole number.
Mistakes to avoid
- ✗ 5, 9, 13 → \(n+4\)✓ \(4n+1\)The difference goes in front of \(n\).
- ✗ 31, 26, 21 → \(31-5n\)✓ \(36-5n\)Use the zero term.
- ✗ First term > 500: n = 124✓ n = 125Round up.
- ✗ Giving n when the TERM is asked✓ substitute backRead the question.
Summary
- Linear: constant difference \(d\); nth term \(dn+c\) with \(c\) = first term − \(d\).
- Is it a term? Solve; n must be a positive whole number.
- First term past a value: inequality, round n, then give the term.
If you see …
- terms going downd is negative.
- "is … a term?"solve for n.
Practice
Practice: easy, medium, hard
Easy
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- d = 4, zero term 1
Answer \(4n+1\)
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- \(4(50)+1\)
Answer 201
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- −3
Answer 4
Medium
Guide me
What is it asking? the nth term
What do we know? decreasing by 5
Plan d = −5, zero term 36
Show working
- d = −5
- 36 − 5n
Answer \(36-5n\)
Guide me
What is it asking? n
What do we know? \(19-3n=-35\)
Plan solve
Show working
- \(3n=54\)
Answer 18
Guide me
What is it asking? n
What do we know? \(4n+3=203\)
Plan solve
Show working
- \(4n=200\)
Answer 50
Hard
Guide me
What is it asking? a term
What do we know? \(36-5n<0\)
Plan solve, round up, substitute
Show working
- \(n>7.2\Rightarrow n=8\)
- \(36-40=-4\)
Answer −4
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(36-5n\) | B1 | ||
| n = 8 | M1 | ||
| −4 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? n
What do we know? two nth terms
Plan \(6n+1>100-3n\)
Show working
- \(9n>99\Rightarrow n>11\)
- next whole number: 12
Answer 12
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| inequality | M1 | ||
| n > 11 | M1 | ||
| 12 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a term
What do we know? \(6n-4>200\)
Plan solve, round up, substitute
Show working
- \(n>34\Rightarrow n=35\)
- \(6(35)-4\)
Answer 206
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(6n-4\) | B1 | ||
| n = 35 | M1 | ||
| 206 | A1 | ||
| Total: 3 marks | |||
Can I do this?
- I can continue a sequence and use a term-to-term rule.
- I can find the nth term of an increasing or decreasing linear sequence.
- I can decide whether a number is a term and find terms past a value.
- I can compare terms of two sequences.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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