EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Linear sequences
Algebra · Lesson 5 of 7
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).
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. |