MODULE 3 NUMBER PATTERNS - All Copy Publishers · Grade 10 Module 1 Page 1 Contents Module 1...

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

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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

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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

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EXAMPLE 2 Chains of squares can be built with matchsticks as follows:

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(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.

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(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

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(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

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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

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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

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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