Academic Plug > Mathematics Notes > NUMBER PATTERNS Part 1 (First Guide)

NUMBER PATTERNS Part 1 (First Guide)

September 6, 2024
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Category: Mathematics Notes

In this page you will learn about NUMBER PATTERNS Part 1 (First Guide), a Mathematics Paper 1 topic that is also a recommended learning topic for grade 10 students. Study or download the notes with questions and answers.

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About NUMBER PATTERNS

In these notes you are going to be able to learn about NUMBER PATTERNS Part 1 (First Guide) as you go through notes and exercises to help you practice calculating and solving relevant questions. Make sure to go through all the notes, questions and answers and also complete the Mastery Quizzes so that you can deepen your understanding and master this topic.

Note that “NUMBER PATTERNS Part 1 (First Guide)” is a sub topic and you can also access more grade 10 notes, questions and answers under the main topic “Numbers And Number Patterns” which is one of the recommended learning topics in the South African Grade 10 Mathematics Curriculum.

NUMBER PATTERNS Part 1 (First Guide )

You’ll notice that patterns appear in nearly every area of mathematics! They frequently assist us in understanding concepts, and most importantly, they help us develop critical thinking skills.

Mastery in mathematics is often associated with the ability to: explore, explain, identify, and extend patterns; make and test conjectures (essentially guessing how something functions); confirm or refute these guesses; create general rules or principles; and recognize connections both within mathematical topics and to real-world situations.

Number patterns, also known as sequences, are enjoyable to work with, and we’ll explore many different examples. It’s encouraged that you attempt solving them independently as much as you can. Nevertheless, different methods will also be provided to guide you. Embrace the challenge and have fun!

Promoting awareness of typical patterns!

Example 1

Take into account the sequences below.

1. 2 ; 4 ; 6 ; 8 ; …

2. 3 ; 6 ; 9 ; 12 ; …

3. 2 ; 4 ; 8 ; 16 ; …

4. 1 ; 4 ; 9 ; 16 ; …

These are series of numbers, where each number is called a TERM. We represent terms using the symbol T, as shown below: T1 = 2 indicates that the first term is 2 T2 = 4 indicates that the second term is 4

For each scenario, the following determine that:

(a) what the next term will be,
(b) what the tenth term is,
(c) what the 100th term is, and
(d) what the nth term is for any given position ‘n’.

How have things been going? Can you see how important it is to think critically? It allows you to build skills for analysis, prediction, and generalization. Now, let’s explore some potential strategies…

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Can you see how important it is to think critically? It enables you to build skills in analysis, prediction, and generalization. Now, let’s explore some potential strategies…

There were multiple methods to determine the next number in the sequence. You could say: “We simply add 2 each time,” which works, but recognizing that this represents the set of even numbers is also valid.

You could also think of it this way

2n represents the GENERAL TERM for even numbers, or we can refer to it as the STANDARD FORM of an even number, as an even number is always equal to 2 multiplied by a natural number.

Consider a sequence of lockers labeled 1, 2, 3, and so on. Now, let’s insert the even numbers into these lockers:

n	1	2	3	4	5	10	100	n
2n	2	4	6	8	10	20	200	2n

The term associated with a locker is determined by its number. For example, if the locker number is

8, the term is 16.

If the locker number is 14, then the term would be 28.

Generally, for a locker number

$n$

, the term

$T_n$

is given by

$T_n = 2n$

The concluding GENERALISATIONS (‘rules’) for the examples presented in (1) to (4) in Example 1 are:

Example 2

Next, consider the following four number sequences:

(5) 1 ; 3 ; 5 ; 7 ; . . . HINT: compare with (1)
(6) 2 ; 5 ; 8 ; 11; . . . HINT: compare with (2)
(7) 1 ; 2 ; 4 ; 8 ; . . . HINT: compare with (3)
(8) 1 ; 8 ; 27 ; 64 ; . . . HINT: compare with (4)

Examine the sequences by comparing (5) with (1), (6) with (2), (7) with (3), and (8) with (4).

Did you see that…

– the values in (5) were each exactly 1 less than those in (1)?
– the values in (6) were each exactly 1 less than those in (2)?
– the sequence in (7) began 1 term earlier, starting with 1 (= ) rather than , but otherwise followed the same pattern as (3)?
– in (8), we have cubes, while in (4) we had squares?

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Example 3

Lets work with triangles made of matchsticks:

Number of triangles: (1) 1 ; (2) 2 ; (3) 3

Number of matchsticks: (1) 3 ; (2) 5 ; (3) 7

Exercises:

Determine a formula that connects the count of triangles to the number of matchsticks used.
Describe how this formula works.
Verify the formula with two additional examples (for 4 and 5 triangles).
For the given formula: 4.1 How many matchsticks are needed to create 50 triangles? 4.2 How many triangles can be formed with 99 matchsticks?

Answers

The number of matchsticks needed to form $n$ triangles can be expressed as $2n + 1$
Explanation: To create the first “A,” we require 3 matchsticks. Subsequently, each additional triangle will need 2 matchsticks.
Total number of matchsticks = 3 + (number of additional triangles) × 2 For n triangles in total:

Number of matchsticks = 3 + (n -1) x 2
= 3 + 2n-2
= 2n + 1

You observed that the sequence 3, 5, 7, and so on consists of odd numbers, each of which is one greater than an even number. The formula for the nth term is Tn = 2n + 1. To calculate the total number of matchsticks, use the formula: 2 multiplied by the number of triangles plus 1.

3. For 4 triangles: Number of matchsticks = 9 and 2(4)+ 1 = 9 ✓
For 5 triangles: Number of matchsticks = 11 and 2(5) + 1 = 11✓

4.1 Number of matchsticks for 50 triangles = 2(50) + 1 = 101

4.2 Number of matchsticks = 99 means

2n + 1 = 99
2n = 98
n = 49
the number of triangles = 49

Example 4

Now lets find the initial three terms of each sequence given by the general term:

Compare these sequences with the ones from before.!

Answers

(9) 3 ; 5 ; 7 ; …

(10) 4 ; 7 ; 10 ; …

(11) 6 ; 12 ; 24 ; …

(12) 2 ; 8 ; 18 ; …

Comparisons

(9) The terms are each one greater than the set of even numbers in (1).

(10) The terms are all 1 more than the multiples of 3 in (2).

(11) All the terms are 3 multiplied by a power of 2, as shown in (3).

(12) The terms are each twice the squares, as shown in (4).

It’s important to be familiar with the fundamental sets of numbers, such as multiples of 2, 3, 4, 5, and so on.

The initial powers of 2, 3, 4, and 5.
Squares of numbers
Cubes

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Two-digit numbers, three-digit numbers, and so on…

The number is 57, with 5 as the tens digit and 7 as the units digit.

Can you express the NUMBER using its individual DIGITS? Remember how you did it in Grade 2…

57 = 5 x 10 + 7
32 = 3 x 10 + 2
86 = 8x 10 + 6

Exercises

If the tens place is represented by $x$ and the units place by $y$ , what is the value of the number?
What if you exchanged the digits?
How about three-digit numbers?
456=…..+…..+…..
If the digit in the hundreds place is $a$ , the digit in the tens place is $b$ , and the digit in the units place is $c$ , then the number can be expressed as…

Answers

49 = 4 x 10 + 9
If the digits are $x$ and $y$ , the number can be expressed as $10x + y$ .
The number with the digits exchanged would be expressed as 10y + x.
456 = 4 x 100 + 5 x 10 + 6
A three-digit number can be expressed as $100 a + 10 b + c$ , where a $b$ , and $c$ are the digits of the number.

Consistent differences of ‘1st’ and ‘2nd’ between terms and their consequences.

Example 1

Consider the following sequence: 2, 5, 8, 11, 14, …

The term number, n : 1 2 3 4 5

2 5 8 11 14

1st differences: 3 3 3 3 ← (these are constant)

This is the general term which represents a linear function!

The consistent difference of 3 between each term indicates that this sequence is associated with multiples of 3.

Example 2

Consider the following sequence: 0 ; 3 ; 8 ; 15 ; 24 ; …

The term number, n: 1 2 3 4 5

0 3 8 15 24

1st differences: 3 5 7 9 ← (Different values show that the function is not linear)

2nd differences: 2 2 2 ← (these are constant)

The numbers in this sequence can be described as being ‘one less than the perfect squares’, such as (1, 4, 9, 16, …).

With the general term is a quadratic function

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What if the third differences remain consistent?

Example 3

Consider the following sequence: 1 ; 8; 27 ; 64 ; …

The term number, n: 1 2 3 4 5

1 8 27 64 125

1st differences: 7 19 37 61 ← (Not constant and not linear)

2nd differences: 12 18 24 ← (Not constant and not quadratic)

3rd differences: 6 6 ← (Not constant and not quadratic)

Thus we should know that is a CUBIC FUNCTION!
This offers a glimpse into the fascinating work that lies ahead with patterns! Exploring number patterns can be both enjoyable and engaging, giving us an opportunity to develop a strong grasp of various mathematical concepts.

Ask Plug Bot “Explain with examples what Fibonacci sequence is”

There two patterns that are important for you to know:

Pascal’s triangle (covered in 3. Algebraic Expressions)
Fibonacci sequence (Plug Bot can explain that for you very precisely )

Practice Exercises

1. Determine the next two terms in each of the following sequences:
1.1 1 ; 4 ; 9 ; 16 ; . .
1.2 2 ; 2 ; 4 ; 6 ; 10 ; ..
1.3 1 ; 3 ; 6 ; 10 ; ..
1.4 6 ; 12 ; 24 ; …

2. Consider the following sequence: 4, 9, 14, 19, and so on…
2.1

2.2

2.3

2.4 What term in the sequence is equal to 114?

3.1 1 ; 4 ; 7 ; 10 ; 13 ; . . .

3.2 2 ; 6 ; 10 ; 14 ; 18 ; …

3.3 5 ; 15 ; 25 ; 35 ; 45 ; . . .

3.4 4 ; 13 ; 22 ; 31 ; 40 ; ..

4. Consider the following: ; ;

4.1 Confirm whether these statements are accurate?

4.2 Write the next two items in the ‘sequence/pattern’ and verify these as well.

4.3 Describe what you observe and create a rule based on that.

4.4 Express your rule from section 4.3 in algebraic form.

4.5 Demonstrate that it is true in all cases.

4.6 Apply the rule to solve the following: 92 x 94 = … -1.

5.1 What is the units digit of the product of the first 100 prime numbers?

5.2 Identify the three smallest two-digit numbers that have an odd number of factors.

Answers:

1.1 25 ; 36

1.2 16 ; 26

1.3 15 ; 21

1.4 48 ; 96

2.1 = 5n -1

2.2 Linear; the initial difference remains consistent

2.3 249

2.4 The 23rd term

3.1 = 3n - 2

3.2 = 4n - 2

3.3 = 10n - 5

3.4 = 9n - 5

4.1 Yes

4.2

4.3 The result of multiplying two numbers that have a difference of 2 is equal to one less than the square of their average (the value halfway between them).

4.4

4.5

4.6

5.1 = 0

5.2 16 ; 25 ; 36

See Answer Calculations/Explanations