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Chapter 13: Sequences 13.1 – Term-to-term rule Introduction Sequences are all around you….

Ø Door numbers follow a sequence

v A sequence is a set of numbers or objects that form a pattern.

v Each number in a sequence is called a term.

Example of a sequence:

The first term is 3 and the third term is 15.

The dots … show that the sequence goes on forever. (to infinity)

v The terms of a sequence are related by a rule.

Term – to – term rule

A term-to-term rule tells you how to find the next term form the

previous term.

Example 1: Describe the rule for each of the following sequences in words. a) 3, 9, 15, 21, 27, …

Start at 3 and add 6.

b) 25,21,17,13, …

Start at 25 and subtract 4.

3, 9, 15, 21, 27, …

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c) 2, 4, 8, 16, 32, …

Start at 2 and multiply by 2.

Example 2: Find the value of each unknown letter. i) 5, 10, 15, a, b, …

This sequence starts from 5 and adds 5.

Answer: a is 20 and b is 25.

ii) 6, 10, 14, 18, c, d, …

The sequence starts at 6 and adds 4.

Answer: c is 22 and d is 26.

Example 3: Generate the first five terms of the sequence described by these term – to

– term rules:

1st term(s) Term – to – term rule Sequence

5

add 3 5, 8, 11, 14, 17, …

4

subtract 2 4, 2, 0, −2, −4, …

3

double

50000

divide by 10

1, 1

add the two previous terms

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13.2 – Position-to-term rule

A position – to – term rule tells you how to find a term from its position

in the sequence.

Example 1: Write down the first five terms of these sequences: a) Each term is the term number, multiplied by 5, plus 1.

Term Number: 1 2 3 4 5

Sequence 6 11 16 21 26

Working: 1st term: 1 x 5 = 5 + 1 = 6

2nd term: 2 x 5 = 10 + 1 = 11

3rd term: 3 x 5 = 15 + 1 = 16

4th term: 4 x 5 = 20 + 1 = 21

5th term: 5 x 5 = 25 + 1 = 26

What is the term – to – term rule?

Start from 6 and add 5. b) Each term is the term number, multiplied by 2, plus 5

Term Number 1 2 3 4 5

Sequence 7 9 11 13 15

Working: 1st term: 1 x 2 = 2 + 5 = 7

2nd term: 2 x 2 = 4 + 5 = 9

3rd term: 3 x 2 = 6 + 5 = 11

4th term: 4 x 2 = 8 + 5 = 13

5th term: 5 x 2 = 10 + 5 = 15

What is the term – to – term rule?

Start from 7 and add 2.

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Are you noticing a relationship between term – to – term rules and position – to term

rules?

13.3 – Sequences in diagrams Sequences often occur in diagrams. Here is a pattern made with matches. a) Draw the next two patterns: b) Complete the following table:

Pattern 1 2 3 4 5

No. of matches

3

6 9 12 15

c) How many matches are being added on each time? 3 matches each time.

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d) Write down the term – to term rule and position – to – term rule. Term – to – term rule Position – to – term rule Start from 3 and add 3 each time

Multiply the term number by 3

e) How many matches are needed for:

a) the 6th pattern?

6 × 3 = 18

b) the 10th pattern?

10 × 3 = 30

c) the 100th pattern?

100 × 3 = 300 Example 2: Here is a sequence of tile patterns.

i) Draw the next two patterns.

You can generate any term of a sequence if you know the position – to – term rule.

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ii) Complete the following table:

Pattern 1 2 3 4 5

No. of edges

4

7 10

iii) Write down the term – to term rule and position – to – term rule. Term – to – term rule Position – to – term rule

iv) How many edges are there in:

a) the 6th pattern?

b) the 10th pattern?

c) the 100th pattern?