Linear sequences.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Linear sequences

Algebra

Lesson 5 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Algebra

Lesson 5 of 7

Answer each one, then check.

1

Write down the next term: 3, 7, 11, 15, ...

2

What is the difference between the terms of 20, 17, 14, ...?

3

Solve 3n + 2 = 50.

4

Last lesson: work out 2n + 1 when n = 10.

5

Last lesson: make n the subject of T = 4n − 1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Algebra

Lesson 5 of 7

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

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Learning Objectives

Algebra

Lesson 5 of 7

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.

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Sequences

Algebra

Lesson 5 of 7

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.

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Where the nth Term Comes From

Algebra

Lesson 5 of 7

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding the nth Term

Algebra

Lesson 5 of 7

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.

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Finding the nth Term

Algebra

Lesson 5 of 7

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

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A Decreasing Sequence

Algebra

Lesson 5 of 7

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

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Is It in the Sequence?

Algebra

Lesson 5 of 7

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).

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A Matchstick Pattern

Algebra

Lesson 5 of 7

4, 7, 10, ...: each new square needs 3 more matchsticks.

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.

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Using the Pattern

Algebra

Lesson 5 of 7

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

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Key Terms

Algebra

Lesson 5 of 7

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.

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Algebra

Lesson 5 of 7

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.

WHAT 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).

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Can I...?

Algebra

Lesson 5 of 7

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.

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Summary & Exam Focus

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.

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