Focus Revision Mathematics Mathayom 1-3 samplebook by Pelangi Publishing Thailand

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Pelangi Publishing (Thailand) Co., Ltd ISBN 978-616-541-354-1 ขอมูลทางบรรณานุกรมของหอสมุดแหงชาติ Focus revision mathematics.-- Bangkok : Pelangi Publishing (Thailand), 2021. 272 p.-- (Focus Revision). 1. Mathematics -- Study and teaching (Secondary). I. Title. 510.7 ISBN 978-616-541-354-1 พิมพครั้งแรก เดือนธันวาคม 2564 ราคา 295 บาท จัดทําโดย 1213/364 ซอยลาดพร้าว 94 ถนนลาดพร้าว แขวงพลับพลา เขตวังทองหลาง กรุงเทพฯ 10310 โทรศัพท 0-2935-6368-9 โทรสาร 0-2934-8160 # 0 พิมพที่ บริษัท สยามไทเกอร์ อินแทคน์ จำ�กัด 106 ซอยเสนานิคม 1 ซอย 42 แยก 11 ถนนเสนานิคม 1 แขวงลาดพร้าว เขตลาดพร้าว 10230 โทรศัพท์ 0-2116-4944

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Contents Mathayom 1 Chapter 1

1 10

Chapter 7 Linear Equations

53 59

Chapter 2 Fractions

12 20

Chapter 8 Linear Equations in Two Variables

60 64

Chapter 3 Decimals

22 26

Chapter 9 Geometrical Constructions

65 73

Chapter 4 Indices

28 36

Chapter 10 Solid Geometry

75 80

Chapter 5 Exponential Notation

38 43

Chapter 11 Statistics

83 91

Chapter 6 Ratios, Proportions and Percentages

44 51

Assessment Paper 1

97 102

Chapter 7 Parallel Lines and Angles

141 146

Chapter 2 Real Numbers

103 110

Chapter 8 Transformations

148 156

Chapter 3 Polynomials

113 118

Chapter 9 Congruence

159 162

Chapter 10 Mean and Data Presentation

164 172

Chapter 11 Geometrical Constructions

175 179

Assessment Paper 2

181

Chapter 7 Similarity

229 234

Chapter 2 Quadratic Equations in One Variable 193

Chapter 8 Trigonometric Ratios

Mastery Practice

198

235 241

Chapter 3 Systems of Linear Equations

199 204

Chapter 9 Circles

243 251

Chapter 4 Quadratic Functions

205 212

Chapter 10 Box Plots

253 257

Chapter 5 Factorization of Polynomials

213 220

Chapter 11 Probability

259 264

Assessment Paper 3

266

Number Sequences and Integers Mastery Practice Mastery Practice Mastery Practice

Mastery Practice Mastery Practice Mastery Practice

Mastery Practice Mastery Practice

Mathayom 2 Chapter 1

Exponential Notation Mastery Practice Mastery Practice Mastery Practice

Chapter 4 Solving Quadratic Polynomial Equations Mastery Practice

Chapter 5 Pythagoras’ Theorem Mastery Practice

Chapter 6 Surface Areas and Volumes of Prisms and Cylinders Mastery Practice

120 126 127 131

Mastery Practice

Mastery Practice Mastery Practice

Mastery Practice

133 139

Mathayom 3 Chapter 1

Linear Inequalities Mastery Practice

Mastery Practice

Chapter 6 Surface Areas and Volumes

of Pyramids, Cones and Spheres Mastery Practice

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184 191

Mastery Practice Mastery Practice

Mastery Practice

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Special features in This Book Summarizes and links the key concepts in a chapter.

Lists important words related to concepts in a chapter together with the glossary in Thai.

Provides questions that facilitate the development of concepts and practice for mastery at the end of the chapter.

Enables students to learn to reason, solve problems and present well-explained solutions.

Extends students’ learning experiences through the internet.

Provides short notes that highlight important information for students.

Enables students to revise the key learning outcomes of a chapter.

Provides questions that challenge students’ way of thinking.

Provides direct access to the answers of the practices by scanning the QR codes given.

Points out conceptual misunderstandings of students and explains the correct principles to help students to grasp the underlying mathematical concepts.

Contains exam-oriented questions including some from O-NET Paper (2019) to challenge students and help them develop higher-order thinking skills.

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PTE

CHA

Number Sequences and Integers

Number patterns and sequences

Number sequences A list of numbers that follows a particular pattern

Number patterns An arrangement of numbers that follows a particular rule

Rules for multiplication and division of two integers (+) × (+) = (+) (+) ÷ (+) = (+) (–) × (–) = (+) (–) ÷ (–) = (+) (+) ÷ (–) = (–) (+) × (–) = (–) (–) × (+) = (–) (–) ÷ (+) = (–)

Mathayom 1

Numbers

Integers

Positive integers 1, 2, 3 Zero 0 Negative integers -3, -2, -1

Arithmetic operations involving integers Addition Subtraction

Multiplication Division

Solving problems involving integers

Glossary Integer จำ�นวนเต็ม

Number sequence ลำ�ดับจำ�นวน

A whole number (not a fraction) that can be positive, negative, or zero

A list of numbers that follows a particular pattern or rule

Negative integer จำ�นวนเต็มลบ

Positive integer จำ�นวนเต็มบวก

A whole number with a negative sign Negative number จำ�นวนลบ

A number that is less than zero Number pattern แบบรูปของจำ�นวน

The pattern in which a list of numbers that follows a particular rule

A whole number with a positive sign or without any sign Positive number จำ�นวนบวก

A number that is greater than zero Term พจน์

An element in a sequence

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Number Patterns and Sequences

1.1

A number sequence is a list of numbers that follows a particular pattern or rule. The numbers in a sequence are called terms. Number pattern is the arrangement of numbers that follows a particular rule through the operation of addition, subtraction, multiplication or division. Mathayom 1

Examples of sequences

Number pattern types

4, 8, 12, 16, 20, ... +4 +4

33, 31, 29, 27, 25, ... –2

–2

9, 16, 25, 36, 49, ... 3

27, 64, 125, 216, 343, ... 3

Even numbers

Each term is obtained by adding 4 to the previous term.

Odd numbers

Each term is obtained by subtracting 2 from the previous term.

Square numbers

Each term is the square of the number.

Cube numbers

Each term is the cube of the number.

Triangular numbers

Each term is obtained by adding one more each time.

Fibonacci numbers

Each term is obtained by adding the two previous numbers.

5, 7, 10, 14, 19, ... +2 +3

Pattern

1, 1, 2, 3, 5, 8, ... 1+1 1+2 2+3 3+5

A number sequence can be extended by determining its pattern beforehand.

Example 1

Example 2

Describe each of the following number sequences. Find the 5th term and 6th term of the number sequences.

Find the missing term in each of the following number sequences.

(a) 217, 229, 241, 253, ...

(b) 406, 391, 376,

(b) 512, 729, 1,000, 1,331, ...

(a) 217, 229, 241, 253, 265, 277, ... +12

+12

Odd numbers. The 5th term is 265 and the 6th term is 277. (b) 512, 729, 1,000, 1,331, 1,728, 2,197, ... 83

103

113

, 346, ...

Solution

Solution +12

, 55, 66, 78, ...

(a) 28, 36,

123

133

(a) 28, 36, 45, 55, 66, 78, ... +8

+10 +11 +12

The missing term is 45. (b) 406, 391, 376, 361, 346, ... –15

–15

The missing term is 361.

Cube numbers. The 5th term is 1,728 and the 6th term is 2,197.

Mathematics Mathayom 1

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To find the value of any term in a number sequence, first find the rule or pattern behind the sequence.

In a number sequence, if the difference between one term and the next is a constant, we can ﬁnd nth term with the formula:

4, 10, 16, 22, 28, ... +6

(a) The sequence above has a difference of +6 between the consecutive terms. (b) The pattern continues by adding 6 each time. 1st term, T1 = 4 2nd term, T2 = T1 + 6 = 4 + 6 3rd term, T3 = T2 + 6 = 4 + 6 + 6 = 4 + (2 × 6) 4th term, T4 = T3 + 6 = 4 + (3 × 6) nth term,Tn = 4 + [(n – 1) × 6]

Tn = a1 + (n – 1) d Where, a1 is the ﬁrst term.

d is the common difference between the terms.

Mathayom 1

n is the term number.

(c) The number pattern for this sequence is: Tn = 4 + [(n – 1) × 6] (d) For example, the 50th term is found by setting n = 50 in the formula above. T50 = 4 + [(50 – 1) × 6] = 298

Example 3

Example 4

Find the nth term and 20th term of the following number sequence. 1,120, 1,095, 1,070, 1,045, 1,020, ....

Find 9th term of the following number sequence. 6, 18, 54, 162, 486, ....

Solution

1,120, 1,095, 1,070, 1,045, 1,020, ...

6, 18, 54, 162, 486, ...

–25

a1 = 1,120, d = -25 Tn = 1,120 + (n – 1)(-25) = 1,120 – 25n + 25 = 1,145 – 25n T20 = 1,145 – 25(20) = 645

Example 5 Write the first 5 terms of a sequence if its number pattern is Tn = 4n – 1.

Solution T1 = 4(1) – 1 = 3 T2 = 4(2) – 1 = 7 T3 = 4(3) – 1 = 11 T4 = 4(4) – 1 = 15 T5 = 4(5) – 1 = 19 The sequence is 3, 7, 11, 15, 19, ...

×3

T1 = 6 T2 = T1 × 3 = 6 × 3 T3 = T2 × 3 = 6 × 3 × 3 = 6 × 32 T4 = T3 × 3 = 6 × 32 × 3 = 6 × 33 Tn = 6 × 3(n – 1) T9 = 6 × 3(9 – 1) = 6 × 38 = 39,366

Example 6 What is the 12th term of the sequence if given by the formula (a) 3n + 6? (b) 9n – 11? (c) 15n + 7?

Solution (a) T12 = 3(12) + 6 = 42 (b) T12 = 9(12) – 11 = 97 (c) T12 = 15(12) + 7 = 187

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Integers

1.2

An integer is a whole number that has a positive sign (+) or a negative sign (–), including zero.

Mathayom 1

A positive integer is a whole number with a positive sign or without any sign. Examples: +5, 106 A negative integer is a whole number with a negative sign. Examples: -2, -139 ..., –3, –2, –1, 0, 1, 2, 3, ...

 Negative integers

Example 7 (a) Write -67 in words. (b) State the integers from the list. 0.12, 25,000, 7 3 , –16.2, 0, + 11 , -136 5 3

Solution (a) Negative sixty-seven (b) The integers are 25,000, 0 and -136.

 Positive integers

The value is decreasing

Zero

The value is increasing

–3 –2 –1 0 1 2 3 Negative integers Positive integers

Fractions and decimals are not integers. Zero is neither positive nor negative.

Horizontal number line

Integers can be represented on a horizontal number line or a vertical number line. Hence, we can compare their values. (a) On a number line, the positive integers are to the right of zero and the negative integers are to the left of zero.

The value is increasing

(b) An integer is always greater than the integers to its left and smaller than the integers to its right. Example: –1 is greater than –3 but is smaller than 1.

The value is decreasing

3 2 1 0 –1 –2 –3

Vertical number line

Example 8

Example 9

State the values of p, q and r in the following number lines. (a) (b)

–24

–12

(a) Which integer is smaller, –16 or 12? (b) Which integer is greater, –3 or –9?

Solution (a) -16 (b) –3

Solution (a) p = -9, q = -4, r = 11 (b) p = 0, q = 12, r = 24

Mathematics Mathayom 1

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Integers can be arranged in ascending or descending order.

Example 11

Write the given integers in ascending order.

Arrange the integers in descending order.

-35, 11, -65, -5, 0, 365, 18, -653, 152, -21

-158, 201, -170, 302, -285, 112, -303, -113

Identify the greatest negative integer and the smallest positive integer.

Determine the greatest integer and the smallest integer.

Solution

-653, -65, -35, -21, -5, 0, 11, 18, 152, 365

302, 201, 112, -113, -158, -170, -285, -303

The greatest negative integer is -5 and the smallest positive integer is 11.

The greatest integer is 302 and the smallest integer is -303.

Mathayom 1

Example 10

Uses of positive and negative numbers A positive number is a number that is greater than zero. Examples: +6, 15 A negative number is a number that is less than zero. It is always with a negative sign (–). Examples: -6, -124 The positive and negative numbers are used in real life situations. Here are some examples: Positive numbers

Negative numbers

Weight gain

Weight loss

Profit

Loss

Above sea level

Below sea level

Situation

(a) An increase in value or a decrease in value (b) Values greater than zero or less than zero

Examples A weight gain of 3 kg ⇒ +3 kg A weight loss of 2 kg ⇒ -2 kg 20% discount for sales ⇒ -20% Food prices rose by 5% ⇒ +5% 265 m above sea level ⇒ +265 m 150 m below sea level ⇒ -150 m

Positive and negative numbers can be used to represent opposite directions. For example, if 5 km to the North is written as +5 km, then 24 km to the South can be written as -24 km.

Example 12 Use a positive number or a negative number to represent each situation. (a) A loss of 2,500 Baht (b) A 2°C increase in temperature (c) A submarine is 245 m below sea level (d) 800 m to right if 20 m to the left is written as -20 m.

Solution (a) -2,500 Baht

(b) +2°C

(d) +800 m

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Addition and Subtraction of Integers

1.3

Addition of integers Adding a positive integer Example: -3 + (+9) = -3 + 9 =6 +9

Mathayom 1

–3 –2 –1 0 1 2 3 4 5 6

Adding a negative integer Example: 5 + (-8) = 5 – 8 = -3 –8 –3 –2 –1 0

✓ When two same signs appear next to each other, replace with a positive sign. ✓ Move to the right 9 steps.

✓ When two different signs appear next to each other, replace with a negative sign. ✓ Move to the left 8 steps.

Example 13

Example 14

-15 + (-4) + (+2) =

8 + (+3) + (-4) =

Solution

-15 + (-4) + (+2) = -15 – 4 + 2 = -19 + 2 = -17

8 + (+3) + (-4) = 8 + 3 - 4 = 11 - 4 =7

–4

–4 6

–20 –19 –18 –17 –16 –15 –14

Properties of addition Commutative property of addition a + b = b + a

25 + 13 = 13 + 25

Associative property of addition

(a + b) + c = a + (b + c)

(8 + 16) + 7 = 8 + (16 + 7)

Identity property of addition

a+0=a a + (-a) = 0

9+0=9 6 + (-6) = 0

Example 15

Integers with like signs are integers with the same sign.

Fill in the blanks. (a) 56 + 24 = 24 + (b) (-34 + 15) + 43 = -34 + (

+ 43)

Integers with unlike signs integers with the different signs.

are

Solution (a) 56 + 24 = 24 + 56 (b) (-34 + 15) + 43 = -34 + ( 15 + 43)

Mathematics Mathayom 1

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Simplify 3 + (-5). ✗ 3 + (-5) = –8

✓ 3 + (-5) = –2

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Subtraction of integers Subtracting a positive integer Example: -4 - (+5) = -4 - 5 = -9 –5 –9 –8 –7 –6 –5 –4

Subtracting a negative integer Example: -3 – (-9) = -3 + 9 =6

Mathayom 1

✓ When two different signs appear next to each other, replace them with a negative sign. ✓ Move to the left 5 steps.

✓ When two same signs appear next to each other, replace them with a positive sign. ✓ Move to the right 9 steps.

+9 –3 –2 –1 0 1 2 3 4 5 6

Example 16

Example 17

-2 - (+1) - (-3) =

10 - (-3) - (-2) =

Solution

-2 – (+1) – (-3) = -2 – 1 + 3

= 13 + 2

= 15

–1 –4

–3

10 – (-3) – (-2) = 10 + 3 + 2

= -3 + 3

–2

–1

+3 0

+2 12

Multiplication and Division of Integers

1.4

Multiplication of integers Multiplication is the process of repeated addition of integers.

Example 18

Example 19

-2 × (+6) =

-3 × (-8) =

Solution

-2 × (+6) = -(2 × 6)

-3 × (-8) = +(3 × 8)

= -12

= 24

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Example 20

Example 21

-4 × (+6) × (-4) =

-12 × (-3) × (-5) =

Solution

4 × (+6) x (-4) = (4 × 6) × (-4) = 24 × (-4) = -(24 × 4) = -96

-12 × (-3) × (-5) = +(12 × 3) × (-5) = 36 × (-5) = -(36 × 5) = -180

Mathayom 1

Properties of multiplication Commutative property of a×b=b×a multiplication

(–7) × 10 = 10 × (–7)

Associative property of multiplication

(a × b) × c = a × (b × c)

(3 × 15) × 9 = 3 × (15 × 9)

Distributive property of multiplication

a × (b + c) = (a × b) + (a × c) a × (b – c) = (a × b) – (a × c)

9 × (25 + 18) = (9 × 25) + (9 × 18) 2 × (8 – 3) = (2 × 8) – (2 × 3)

Identity property of multiplication

a×0=0 a×1=a

7×0=0 12 × 1 = 12

Division of integers Division is the process of equal sharing or equal gathering. Dividing integers is the opposite operation of multiplication.

Example 22

Example 23

12 ÷ (-3) =

-28 ÷ (-2) =

Solution

12 ÷ (-3) = -(12 ÷ 3) = -4

-28 ÷ (-2) = +(28 ÷ 2) = 14

✓ When a negative integer divided by a positive integer or vice versa, the quotient is always negative. ✓ When a negative integer divided by a negative integer or a positive integer multiply by a positive integer, the quotient is always positive.

Example 24

Example 25

-100 ÷ (-2) ÷ (+5) =

-320 ÷ (-8) ÷ (-2) =

Solution

-100 ÷ (-2) ÷ (+5) = +(100 ÷ 2) ÷ (+5) = 50 ÷ (+5) = +(50 ÷ 5) = 10

-320 ÷ (-8) ÷ (-2) = +(320 ÷ 8) ÷ (-2) = 40 ÷ (-2) = -(40 ÷ 2) = -20

Mathematics Mathayom 1

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1.5

Combined Operations of Integers

Combined operations are also called mixed operations. (a) Work out the calculations within the brackets first. (b) Then, perform the multiplication or division, working from left to right. (c) Lastly, perform the addition or subtraction, working from left to right.

–9 + (11 – 8) =

Mathayom 1

Example 26 Let’s learn more about integers.

Solution –9 + (11 – 8) = –9 + 3 = -6

Example 27

Example 28

–108 ÷ (–6) × (–7) =

–18 + (9 – 34) ÷ (–5) =

Solution

–108 ÷ (–6) × (–7) = 18 × (–7) = -126

–18 + (9 – 34) ÷ (–5) = -18 + (–25) ÷ (–5) = –18 + 5 = -13

Solving problems involving integers

Example 29

Daily Application

Example 30

Mount Everest is the world‛s highest mountain. It is 8,850 m above sea level. The Dead Sea is the world's lowest point of land at 423 m below sea level. Find the difference between these two points.

In an experiment, the initial temperature of a solution is 14°C. The temperature dropped by 23°C when cooled. Then, the temperature is increased by 5°C when heated. Find the final temperature of the solution.

Solution

8,850 – (-423) = 9,273 m

14 – 23 + 5 = -4°C

The difference between these two points is 9,273 m.

The final temperature of the solution is -4°C.

Understand number sequences. Recognize and explain number patterns. Specify or give examples and compare integers. Arrange integers in order. Apply integers in daily life situations. Solve problems involving addition, subtraction, multiplication and division of integers. Apply the knowledge of integers to solve problems.

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Mastery Practice 1.1

9. A number sequence is given below.

Number Patterns and Sequences

7, 15, 23, 32, 39, 47, 55, …

1. Find the missing terms in the number sequences. (a) 1, 9,

, 729, 6,561,

Mathayom 1

(b) 452, 226, 113, (c) 4, 9,

, 28.25,

, 25,

Can you spot the mistake and give correct answer for the sequence?

, … ,…

, 49, …

2. What are the next three terms in the number sequences? (a) 2, 12, 32, 62, … (b) 34, 23, 12, 1, -10, … 3. Find the nth term for each sequence. (a) 6, 11, 16, 21, 26, … (b) 14, 28, 42, 56, 70, … (c) 5, 2, –1, –4, –7, … 4. A number sequence has nth term 4n + 3.

1.2

Integers

10. Write each integer in words. (a) -52

(b) +678

(d) -423

11. Write each integer in numerals. (a) Negative sixty-five (b) Positive seven hundred and fifty-two (c) Negative ninety-four 12. Determine the positions of integers on the given number line.

(a) Write the first 5 terms of the sequence. (b) Anong wrote that 74 is a term in this sequence. Explain why her answer is wrong. 5. (a) What is the 8th term of the sequence 5n + 2? (b) What is the 100th term of the s?equence 7n - 4? 6. If the 9th term in a number sequence is T9 = -4 and the common difference is d = -2, (a) write a rule that can find the value of any term in the sequence. (b) find the 15th term and 95th term in the sequence. 7. Calculate the sum of the 40th term and 100th term of the sequence 8, 3, -2, -7, -12, ... 8. A sequence of numbers is given below. 2, 11, x, 29, y, 47, 56 Find the value of y - x.

Mathematics Mathayom 1

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-7, 14, -21, -14, -35

–15

–6 –3

13. Identify which integer is greater. (a) -14, 15

(b) -96, 0

(d) -1, 11

14. Arrange the following integers in ascending order. (a) -66, 58, 0, 14, -99, 21 (b) -7, -22, 14, 24, -145, 105 15. Arrange the following descending order.

integers

sequences

(a) -1, 0, -11, 107, -124, 11 (b) 564, -554, -447, 52, 541 16. Complete integers. (a) -2, -8, (b) -10,

the

following , -128,

, 8,

, -2,048, ... , 26, 35, ...

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(a) 7°C below the melting point (b) A 2% rise in price (c) Move up 100 stairs (d) A loss of 350 Baht

1.3

Addition and Subtraction of Integers

18. 101 – (-182) = 19. 932 + (-459) = 20. 39 - (-16) – 25 = 21. 365 + 123 – (-15) = 22. -82 - 109 + (-127) =

1.4

32. In a test, students are required to answer 60 questions. 5 marks will be given for each correct answer, 2 marks will be deducted for each wrong answer and 1 mark will be deducted for each question that was not answered. Somsak answered 42 questions correctly and 11 questions wrongly. (a) Calculate the total marks obtained by Somsak. (a) The passing mark for the test is 200 marks. Did Somsak pass the test?

Mathayom 1

17. Use an integer to represent each situation.

33. The temperature of a cold storage room drops constantly by 20°C in 5 hours. Find the drop in temperature each hour. 34. The eagle is ﬂying 85 m in the sky. The fish is swimming 164 m vertically below the eagle. How deep is the fish swimming?

Multiplication and Division of Integers

23. -14 × 9 = 24. -8 × (-17) = 25. 28 ÷ (-4) = 26. -342 ÷ (-38) =

1.5

The diagram shows the positions of participants in a performance on a drawing sketch.

Combined Operations of Integers

27. –11 × 28 – 36 = 28. –58 + (7 – 28) ÷ (–7) = 29. –124 ÷ (–4) × (–9) = 30. The temperature of a city was -2°C in the morning. If the temperature dropped 1°C, what is the temperature of the city now? 31. An elevator started at the third floor and went up 23 floors. It then came down 11 floors and went back up 9 floors. At what floor was it stopped?

6th 5th 4th 3rd 2nd 1st Given the number of participants positioned in each row is according to a speciﬁc pattern. (a) State the pattern of the sequence above. (b) How many participants are there in the 6th row? (c) If there are 60 persons volunteer to take part in the performance, is that enough? Give your reason.

Answers

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