1. Grade 10 Mathematics
Conjectures

2. Table of Contents
Investigate number patterns to make and test conjectures
Generalise relationships in number patterns
Investigate and generalise number patterns where there is a constant difference between consecutive terms
Investigate and generalise number patterns where there is a constant ratio between consecutive terms
Use graphs to represent number patterns

3. 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
=16
The answers represent the square of:
1 odd number =
2 odd numbers
The next step will be to find the general proof for this statement or to show that it is incorrect by using a counter proof.

4. Test Your Knowledge Look at the pattern below: 1 + 2 = 3
= 7+ 8 =
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?

5. Solutions 35 = =
and 120

6. 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; ………..

7. 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

8. 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.

9. 3. Investigate and generalise number patterns where there is a constant difference between consecutive terms
Consider the sequence:
a; a + d; a + 2d; a + 3d; ………….
This sequence is called an Arithmetic sequence

10. 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:

11. Test Your Knowledge Question 1
Determine the general term of the sequence:
1; 5; 9; 13; 17;……. ………
Answer
A Tn = (n-1) B Tn = (n-1)4+1
C Tn = (n-1) D Tn = (n-1)5+1

12. 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

13. 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:
The general term is given by:

14. Test Your Knowledge
Determine the common ratio and the general term of:
6; 2;

15. Solution

16. 5. Use graphs to represent number patterns
If we again look at the example
1; 5; 9; 13; 17; 21; …….
This represents a linear discrete graph
where the difference between
consecutive terms are constant.
If points are discrete it means they are not joined with a line.

17. This graph if plotted shows a geometric sequence.1
This graph if plotted shows a geometric sequence.
It represents exponential decrease and the common ratio r =

18. Test Your Knowledge Plot the graph of the following sequence:
21; 18; 15; 12;………..

19. Solution

20. Bibliography
Examples from Oxford “Mathematics Plus”
Grade 10