conjectures. 1. investigate number patterns to make and test conjectures 2. generalise relationships...

Conjectures

1. Investigate number patterns to make and test conjectures

2. Generalise relationships in number patterns3. Investigate and generalise number patterns where

there is a constant difference between consecutive terms

4. Investigate and generalise number patterns where there is a constant ratio between consecutive terms

5. Use graphs to represent number patterns

1. Investigate number patterns to make and test conjectures

A conjecture is a statement which, although evidence can be found to support it, has not been proved to be true or false.

Example:Conjecture: The sum of the first n odd numbers equals the square of n

1=1 1+3 = 4 1+3+5=9 1+3+5+7=16

The answers represent the square of:1 odd number =

2 odd numbersThe next step will be to find the general proof for this statement or to show that

it is incorrect by using a counter proof.

21

22 4

Test Your Knowledge

Look at the pattern below:1 + 2 = 3

4 + 5 +6 = 7+ 8 9 + 10 +11 + 12 = 13 + 14 + 15

Make a conjecture about the last number in the 5th row. Test your conjecture by writing the next two rows. What will the last number be at both ends of the 10th row?

Solutions

1. 35

16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 2425 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35

3. 100 and 120

2. Generalise relationships in number patterns

Number patterns are a regular occurrence in Mathematics. Ordered list of numbers are called a sequence and the numbers we

referred to as terms with symbol: , where It is useful to determine a general rule in order to determine the

next or any other term of the sequence.This can be in the form of a recursive pattern (each new term is

defined in relation to some terms which have been made previously) e.g.

3; 5; 7; 9; ………..

nT 1 23; 5 etc.T T

1 2n nT T

2. Generalise relationships in number patterns

We see that by adding 2 we can determine the next term and make the conjecture that the general term can be determined using the rule:

We refer to the constant adding of 2 as a common difference between terms

1 2n nT T

1

2

3

2(1) 1 3

2(2) 1 5

2(3) 1 7

T

T

T

3. Investigate and generalise number patterns where there is a constant difference between consecutive terms

Look at the following patterns of numbers:5; 10; 15; …………

We want to know what the next three numbers / terms of this sequence will be as well as the general term of the sequence.

Each sequence is a group of numbers that has two very important properties:

the terms are listed in a specific order and there is a rule which enables you to continue with

the sequence.

Either the rule is given or you have to determine it using the first three terms of the sequence.

Consider the sequence: a; a + d; a + 2d; a + 3d; ………….

This sequence is called an Arithmetic sequence

1

2

0

1

.

.

( 1)n

T d a

T d a

T n d a

3. Investigate and generalise number patterns where there is a constant

difference between consecutive terms

Worked Example Given:

4; 9; 14; 19;……. Note there is a common difference (d) of 5 between successive terms.

If there exists a first common difference between successive terms, it is called a linear pattern

If a = first term and is equal to 7 and d = common difference = 8, then:

2 1

n

2

9 4 5

5

Pattern: T 5 1

e.g. T 5 2 1 10 1 9

T T

d

n

( 1)

=8( -1) 7

8 -1

nT n d a

n

n

Test Your Knowledge

Question 1

Determine the general term of the sequence:1; 5; 9; 13; 17;……. ………

Answer

A Tn = (n-1)5-1 B Tn = (n-1)4+1C Tn = (n-1)4-1 D Tn = (n-1)5+1

4. Investigate and generalise number patterns where there is a constant ratio between consecutive terms

Another pattern to investigate:Consider: 2; 4; 8; 16; ……..

In this sequence the ratio between two consecutive terms are constant. This type of sequence are referred to as a Geometric Sequence

8 42

4 2

4. Investigate and generalise number patterns where there is a constant ratio between consecutive terms

Consecutive means that the numbers are next to each other in the sequence.In a geometric sequence r is called the constant ratio and:

1

4

3

16e.g. 2

8

n

n

TrT

TrT

The general term is given by: 1

11 = 6( )

3

nn

n

T ar

Test Your Knowledge

1. Determine the common ratio and the general term of:

6; 2; 2;......

3

Solution

2

1

26;2; ;..........

32

2 1 132 3 2 3T 2 1

and: T 6 3

r

5. Use graphs to represent number patterns

If we again look at the example1; 5; 9; 13; 17; 21; …….

This represents a linear discrete graphwhere the difference between consecutive terms are constant. If points are discrete it means they are

not joined with a line.

1

2

3

1

5

9

.

. 4( 1) 1n

T

T

T

T n

1

2

This graph if plotted shows a geometric sequence.

It represents exponential decrease and the common ratio r =

1

1

2

Test Your Knowledge

1. Plot the graph of the following sequence: 21; 18; 15; 12;………..

Solution

Bibliography

Examples from Oxford “Mathematics Plus” Grade 10