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Grade 10 Module 1 Page 1
Contents Module 1 Algebriac Expressions Module 2 Exponents Module 3 Number Patterns Module 4 Equations & Inequalities Module 5 Trigonometry Module 6 Functions Module 7 Euclidean Geometry Module 8 Analytical Geometry Module 9 Financial Mathematics Module 10 Statistics Module 11 Measurement Module 12 Probability
MODULE 3
NUMBER PATTERNS
Grade 10 Module 3 Page 53
LINEAR NUMBER PATTERNS
3; 5; 7; 9;11;..........
Consider the number pattern:
The first term is 1T 3
The second term is 2T 5
The third term is 3T 7
The fourth term is 4T 9
and so forth
Grade 10 Module 3 Page 53
The pattern is formed by adding 2 to each new term. We say that the constant difference between the terms is 2. A number pattern with a constant difference is called a linear number pattern.
Grade 10 Module 3 Page 53
Now consider the rule . We can use this rule to generate terms as follows:
T 2 1n n
1T 2(1) 1 3
2T 2(2) 1 5
3T 2(3) 1 7
4T 2(4) 1 9
Grade 10 Module 3 Page 53
You will have probably noticed that this rule is the general or nth term of the number pattern This general rule will enable one to determine terms such as the 100th term or 1000th term.
100T 2(100) 1 201
3; 5; 7; 9;11;..........
For example:
1000T 2(1000) 1 2001
Grade 10 Module 3 Page 53
But the question arises: How do we get the general rule from the number pattern ? 3; 5; 7; 9;11;..........
The general rule for any linear number pattern takes the form , so let’s explore this a little further. We can determine the first few terms using this rule.
T 2 1n n
Tn bn c
Grade 10 Module 3 Page 53
Tn bn c
1T (1)b c b c
2T (2) 2b c b c
3T (3) 3b c b c
4T (4) 4b c b c
b c 2b c 3b c 4b c
b b b
The constant difference is b and the first term is b c
Grade 10 Module 3 Page 54
b c 2b c 3b c 4b c
b b b
b
b c
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2b 3b c
2 3c
3 2c
1c
Tn bn c
T 2 1n n
Grade 10 Module 3 Page 54
EXAMPLE 1
6 ;10 ;14 ;18 ;..............
Consider the number pattern:
(a) Determine the nth term of the pattern.
Grade 10 Module 3 Page 54
b
b c
4b 6b c
4 6c
6 4c
2c
Tn bn c
T 4 2n n
Grade 10 Module 3 Page 54
(b) Determine the 1000th term.
T 4 2n n
1000T 4(1000) 2 4002
(c) Which term of the number pattern equals 202?
T 4 2n n
202 4 2n
200 4n
50 n
50T 202
Grade 10 Module 3 Page 54
EXAMPLE 2 Chains of squares can be built with matchsticks as follows:
Grade 10 Module 3 Page 54
(a) Use matches and create a chain of 4 squares. How many matches were used?
13 matches were used.
(b) Create a chain of 5 squares. How many matches were used?
16 matches were used.
Grade 10 Module 3 Page 55
(c) Determine a conjecture (rule) for calculating the number of matches in a chain of n squares.
4; 7;10;13; .........
The number pattern is:
Grade 10 Module 3 Page 55
b
b c
3b 4b c
3 4c
4 3c
1c
Tn bn c
T 3 1n n
Grade 10 Module 3 Page 55
(d) Now determine how many matches will be needed to build a chain of 100 squares.
T 3 1n n
100T 3(100) 1 301
Grade 10 Module 3 Page 56
ALTERNATIVE METHODS Alternative method 1
3; 5; 7; 9;11;..........Consider the number pattern:
b
1Tc b
Subtract b from the first term to get c:
3 2c
1c
1T
T 2 1n n
Grade 10 Module 3 Page 56
Examples
5;12;19; 26; .........(a) Consider the number pattern:
b
1Tc b
Subtract b from the first term to get c:
5 7c
2c
1T
T 7 2n n
2 2 2
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8; 6; 4; 2; .........
(b) Consider the number pattern:
b
1Tc b
Subtract b from the first term to get c:
8 ( 2)c
10c
1T
T 2 10n n
Grade 10 Module 3 Page 56
Alternative method 2
3; 5; 7; 9;11;..........Consider the number pattern:
We have already determined previously that . Therefore we know that . What you now do is write down the multiples of and compare these multiples to the original pattern.
T 2n n c
2b
2b
Grade 10 Module 3 Page 56
In the second row, you work out what must be added to or subtracted from each multiple so as to obtain the original pattern.
Multiples of 2
What to do to get original number
Original pattern
1
2 4 6 8 10 12
3 5 7 9 11 13
1 1 1 1 1
Grade 10 Module 3 Page 56,57
The constant number being added in the second row represents the value of c.
T 2 1n n [ 2 ; 1]b c
Grade 10 Module 3 Page 57
Examples
5; 7; 9;11; .........
(a) Consider the number pattern:
2b
Grade 10 Module 3 Page 57
Multiples of 2
What to do to get original number
Original pattern
3
2 4 6 8 10 12
5 7 9 11 13 15
3 3 3 3 3
2b 3c
T 2 3n n
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4; 9;14;19; .........
(b) Consider the number pattern:
5b
Grade 10 Module 3 Page 57
Multiples of 5
What to do to get original number
Original pattern
1
5 10 15 20 25 30
4 9 14 19 24 29
1 1 1 1 1
5b 1c
T 5 1n n