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Conjectures
1.
2.
3.
4.
5.
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
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 = 12
2 odd numbers  22  4
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.
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 + 24
25 + 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:
, Twhere
T1  3; T2  5 etc.
n
 It is useful to determine a general rule in order to determine the
next or any other term of the sequence. Tn  Tn1  2
 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; ………..
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:
Tn  Tn1  2
T1  2(1)  1  3
T2  2(2)  1  5
T3  2(3)  1  7
 We refer to the constant adding of 2 as a common difference
between terms
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.
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; ………….
T1  0  d  a
T2  1 d  a
.
.
Tn  (n  1)d  a
This sequence is called an Arithmetic sequence
Worked Example
 Given:
4; 9; 14; 19;…….
Note there is a common difference (d) of 5 between successive
terms. T2  T1  9  4  5
d  5
Pattern: Tn  5  n  1
e.g.
T2  5  2  1  10  1  9
 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:
Tn  (n  1)d  a
=8(n -1)  7
 8n -1
Test Your Knowledge
Question 1
Determine the general term of the sequence:
1; 5; 9; 13; 17;……. ………
Answer
A Tn = (n-1)5-1
C Tn = (n-1)4-1
B 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; ……..
8 4
 2
4 2
 In this sequence the ratio between two consecutive terms are
constant. This type of sequence are referred to as a Geometric
Sequence
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:
r
The general term is
given by: T  ar n 1
Tn
Tn 1
e.g. r 
n
T4 16

2
T3 8
1
= 6( ) n 1
3
Test Your Knowledge
1.
Determine the common ratio and the general term of:
6; 2;
2
;......
3
Solution
2
6; 2; ;..........
3
2
2 1 1
3
r    
2 3 2 3
T2 2 1
and:
 
T1 6 3
5. Use graphs to represent
number patterns
 If we again look at the example
1; 5; 9; 13; 17; 21; …….
T1  1
T2  5
T3  9
.
.Tn  4( n  1)  1
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.
This graph if plotted
shows a geometric
sequence.
It represents
exponential decrease
and the common
1
ratio r = 2
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