EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Linear sequences
Algebra · Lesson 5 of 7
Teacher copy - includes the notes for whoever is teaching from it.
Last Lesson and Before
Answer each one, then check.
1. Write down the next term: 3, 7, 11, 15, ...
19
2. What is the difference between the terms of 20, 17, 14, ...?
−3
3. Solve 3n + 2 = 50.
n = 16
4. Last lesson: work out 2n + 1 when n = 10.
21
5. Last lesson: make n the subject of T = 4n − 1.
n = (T + 1)/4
Learning Objectives
1. Continue a sequence using a term-to-term rule.
2. Generate terms of a sequence from its nth term.
3. Find the nth term of a linear (arithmetic) sequence.
4. Decide whether a number is a term of a sequence.
5. Solve problems with linear sequences and patterns.
Sequences
A sequence is a list of numbers that follows a rule.
▸ Term. Each number in a sequence. The 1st term, 2nd term and so on.
▸ Term-to-term rule. How to get from one term to the next: "add 3".
▸ Position-to-term rule (nth term). A formula for any term from its position n: the nth term 3n + 2 gives 5, 8, 11, ...
▸ Linear (arithmetic) sequence. One that goes up or down by the same amount - the common difference - every time.
Where the nth Term Comes From
The sequence 5, 8, 11, 14, ... goes up by 3, so compare it with the 3 times table.
|
Position n |
3n |
Term |
Term − 3n |
|---|---|---|---|
|
1 |
3 |
5 |
+2 |
|
2 |
6 |
8 |
+2 |
|
3 |
9 |
11 |
+2 |
|
4 |
12 |
14 |
+2 |
|
n |
3n |
3n + 2 |
+2 |
Finding the nth Term
Works for every linear sequence.
|
1 Difference Find the common difference, d. |
2 Times table Write down dn: the d times table. |
3 Compare How do you get from dn to the sequence? |
4 Adjust Add or subtract that number. |
5 Check Put n = 1 and n = 2 into your answer. |
Finding the nth Term
|
Find the nth term of 7, 11, 15, 19, ... |
1. The common difference is +4
so it starts 4n
2. The 4 times table
4, 8, 12, 16
3. Compare with the sequence
7 − 4 = 3: each term is 3 more
4. Write the nth term
4n + 3
5. Check n = 3
4 × 3 + 3 = 15
Answer: 4n + 3
A Decreasing Sequence
|
Find the nth term of 20, 17, 14, 11, ... |
1. The common difference is −3
so it starts −3n
2. The −3 times table
−3, −6, −9, −12
3. Compare with the sequence
20 − (−3) = 23: each term is 23 more
4. Write the nth term
−3n + 23, usually written 23 − 3n
5. Check n = 1
23 − 3 = 20
Answer: 23 − 3n
Is It in the Sequence?
|
Is 100 a term of the sequence with nth term 4n + 3? |
1. Set the nth term equal to 100
4n + 3 = 100
2. Solve
4n = 97, so n = 24.25
3. n must be a whole number (a position)
24.25 is not a whole number
Answer: No - 100 is not in the sequence (the 24th term is 99 and the 25th is 103).
A Matchstick Pattern
|
Each new square needs 3 more matchsticks, so the sequence 4, 7, 10, ... has a common difference of 3. Comparing with 3n (3, 6, 9) shows each pattern uses one more: the nth pattern needs 3n + 1 matchsticks - the extra 1 is the first stick on the left. |
4, 7, 10, ...: each new square needs 3 more matchsticks. |
Using the Pattern
|
Pattern n uses 3n + 1 matchsticks. (a) How many matchsticks does pattern 20 use? (b) What is the largest pattern that can be made with 100 matchsticks? |
1. (a) Substitute n = 20
3 × 20 + 1 = 61
2. (b) The number of matchsticks must be 100 or fewer
3n + 1 ≤ 100
3. Solve
3n ≤ 99, so n ≤ 33
4. Pattern 33 uses exactly 3 × 33 + 1 = 100
it fits
Answer: (a) 61 matchsticks (b) Pattern 33
Key Terms
|
Sequence A list of numbers or patterns that follows a rule. |
Term One number in a sequence. |
|
Term-to-term rule The rule that gets you from one term to the next. |
nth term A formula that gives any term from its position, n. |
|
Common difference The amount a linear sequence goes up or down by each time. |
Linear (arithmetic) sequence A sequence with a common difference. |
Your Task: Sequence Detectives
12 minutes
|
For each sequence, find the nth term and the 50th term. Then decide whether 150 is in it. (a) 6, 11, 16, 21, ... (b) 1, 4, 7, 10, ... (c) 40, 36, 32, 28, ... (d) −5, −3, −1, 1, ... 1. Find the difference. 2. Write the nth term and check it. 3. Solve nth term = 150. |
A good answer shows: (a) 5n + 1; 251; 150 is not in it (n = 29.8). (b) 3n − 2; 148; 150 is not in it (n = 50.67). (c) 44 − 4n; −156; 150 is not in it (it is decreasing from 40). (d) 2n − 7; 93; 150 is not in it (n = 78.5).
Note: The trick answer is that 150 is in none of them - a good discussion of what "not a whole number" tells you.
Can I...?
☐ Continue a sequence.
☐ Describe a term-to-term rule.
☐ Work out terms from an nth term.
☐ Find the nth term of an increasing sequence.
☐ Find the nth term of a decreasing sequence.
☐ Decide whether a number is in a sequence.
☐ Find the nth term of a pattern.
☐ Use an nth term to solve a problem.
Summary
✓ A linear sequence has a common difference.
✓ nth term = (difference) × n, then adjust with a number.
✓ To test a number, set the nth term equal to it: a whole-number n means it is in the sequence.
✓ For patterns, the difference is how many more items each new pattern needs.
|
EXAM FOCUS Here are the first five terms of a sequence: 5, 9, 13, 17, 21. Find an expression for the nth term. (2 marks) "n + 4" is the term-to-term rule written wrongly, not an nth term. The nth term of a sequence going up by 4 always starts 4n. |
Exam Practice: Linear Sequences
Answer all questions. Show your working. Questions 1 to 3 are non-calculator. · 20 minutes
▸ Question 1 · 2 marks · Non-calculator. Here are the first five terms of a sequence: 5, 9, 13, 17, 21. Find an expression, in terms of n, for the nth term of the sequence.
▸ Question 2 · 2 marks · Non-calculator. Is 99 a term of the sequence 5, 9, 13, 17, 21, ...? You must give a reason for your answer.
▸ Question 3 · 2 marks · Non-calculator. Find an expression for the nth term of the sequence 30, 26, 22, 18, ...
▸ Question 4 · 3 marks · Calculator. The diagram shows the first three patterns in a sequence made from matchsticks. Pattern 1 uses 4 matchsticks, pattern 2 uses 7 and pattern…
▸ Question 5 · 3 marks · Calculator. Sequence A has nth term 3n + 5. Sequence B has nth term 50 − 2n. For which value of n are the nth terms of the two sequences equal, and…
Question 1 · 2 marks · Non-calculator
|
“Here are the first five terms of a sequence: 5, 9, 13, 17, 21. Find an expression, in terms of n, for the nth term of the sequence.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ 4n + 1. B2
▸ 4n + k, where k ≠ 1. B1
▸ Model answer. 4n + 1
Question 2 · 2 marks · Non-calculator
|
“Is 99 a term of the sequence 5, 9, 13, 17, 21, ...? You must give a reason for your answer.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ 4n + 1 = 99, or listing terms to 97 and 101. M1
▸ No, with a correct reason. C1
▸ Model answer. 4n + 1 = 99 gives n = 24.5, which is not a whole number, so 99 is not a term. (The terms go 97, 101.)
Question 3 · 2 marks · Non-calculator
|
“Find an expression for the nth term of the sequence 30, 26, 22, 18, ...” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 3 · mark scheme
2 marks available. Award a mark for each point made.
▸ 34 − 4n. B2
▸ −4n + k, where k ≠ 34. B1
▸ Model answer. 34 − 4n
Question 4 · 3 marks · Calculator
|
The diagram shows the first three patterns in a sequence made from matchsticks. Pattern 1 uses 4 matchsticks, pattern 2 uses 7 and pattern 3 uses 10. Jack has 150 matchsticks. What is the largest pattern number he can make? (3 marks) |
|
Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ 3n + 1, or the difference of 3 used. P1
▸ 3n + 1 = 150 or 3n + 1 ≤ 150, or 49.6… seen. P1
▸ 49. A1
▸ Model answer. The nth pattern uses 3n + 1 matchsticks. 3n + 1 ≤ 150, so 3n ≤ 149 and n ≤ 49.67. The largest pattern is pattern 49 (it uses 148 matchsticks).
Question 5 · 3 marks · Calculator
|
“Sequence A has nth term 3n + 5. Sequence B has nth term 50 − 2n. For which value of n are the nth terms of the two sequences equal, and what is that term?” |
HOW TO ANSWER IT Command word: Calculator. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ 3n + 5 = 50 − 2n. M1
▸ n = 9. A1
▸ 32. A1
▸ Model answer. 3n + 5 = 50 − 2n, so 5n = 45 and n = 9. The term is 3 × 9 + 5 = 32.