2. Number patterns

# 2. Number patterns ## Prime numbers and prime factorisation ### Introduction Welcome to our lesson on prime numbers and prime factorisation! Numbers are the building blocks of mathematics, and just like other things, they have unique properties. Today, we're going to explore some special types of numbers called **prime numbers** and learn a powerful technique called **prime factorisation**. Understanding these concepts will help you break down numbers into their most basic components, which is incredibly useful for solving many mathematical problems, especially when working with fractions, HCF, and LCM. ### Key Concepts #### Factors A **factor** of a number is a number that divides it exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 evenly. #### Prime Numbers A **prime number** is a natural number greater than 1 that has exactly two distinct positive factors: 1 and itself. * The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... * **Important Notes:** * The number 1 is **not** a prime number because it only has one factor (itself). * The number 2 is the **only** even prime number. #### Composite Numbers A **composite number** is a natural number greater than 1 that is not prime. This means it has more than two factors. * Examples of composite numbers include 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 8, 9, 10, 12, etc. #### Prime Factorisation **Prime factorisation** is the process of expressing a composite number as a product of its prime factors. Every composite number can be written as a unique product of prime numbers. There are two main methods for prime factorisation: ##### 1. Factor Tree Method In this method, you break down a number into any two factors. If a factor is prime, you circle it. If it's composite, you continue to break it down until all branches end in prime numbers. ##### 2. Division Method (Ladder Method) In this method, you repeatedly divide the number by the smallest possible prime number until the quotient is 1. The prime numbers used for division are the prime factors. When writing prime factorisation, we often use **index notation** (powers) to simplify the expression. For example, the prime factors of 12 are 2, 2, and 3. So, \[ 12 = 2 imes 2 imes 3 = 2^2 imes 3 \] #### Applications of Prime Factorisation Prime factorisation is a powerful tool used to find: * **Highest Common Factor (HCF):** The largest number that divides two or more numbers exactly. * **Lowest Common Multiple (LCM):** The smallest number that is a multiple of two or more numbers. ### Vocabulary * **Factor:** A number that divides another number exactly without a remainder. * **Prime Number:** A natural number greater than 1 that has exactly two factors: 1 and itself. * **Composite Number:** A natural number greater than 1 that has more than two factors (not prime). * **Prime Factorisation:** The process of expressing a composite number as a product of its prime factors. * **Index Notation:** A way of writing repeated multiplication using powers (e.g., \( 2^3 \) instead of \( 2 imes 2 imes 2 \)). * **Highest Common Factor (HCF):** The largest number that is a factor of two or more given numbers. * **Lowest Common Multiple (LCM):** The smallest number that is a multiple of two or more given numbers. ### Examples #### Example 1: Identifying Prime and Composite Numbers Is 7 a prime or composite number? What about 9? * **7:** Its factors are 1 and 7. Since it has exactly two factors, 7 is a **prime number**. * **9:** Its factors are 1, 3, and 9. Since it has more than two factors, 9 is a **composite number**. #### Example 2: Prime Factorisation of 60 using the Factor Tree Method ``` 60 / \ 2 30 / \ 2 15 / \ 3 5 ``` All the end branches (2, 2, 3, 5) are prime numbers. So, \[ 60 = 2 imes 2 imes 3 imes 5 = 2^2 imes 3 imes 5 \] #### Example 3: Prime Factorisation of 84 using the Division Method ``` 2 | 84 2 | 42 3 | 21 7 | 7 | 1 ``` So, \[ 84 = 2 imes 2 imes 3 imes 7 = 2^2 imes 3 imes 7 \] #### Example 4: Finding HCF using Prime Factorisation Find the HCF of 36 and 48. 1. Prime factorise 36: \[ 36 = 2 imes 2 imes 3 imes 3 = 2^2 imes 3^2 \] 2. Prime factorise 48: \[ 48 = 2 imes 2 imes 2 imes 2 imes 3 = 2^4 imes 3 \] 3. Identify common prime factors and their lowest powers: * Common prime factor 2: Lowest power is \( 2^2 \) * Common prime factor 3: Lowest power is \( 3^1 \) 4. Multiply these lowest powers: \[ HCF(36, 48) = 2^2 imes 3 = 4 imes 3 = 12 \] #### Example 5: Finding LCM using Prime Factorisation Find the LCM of 36 and 48. 1. Prime factorise 36: \[ 36 = 2^2 imes 3^2 \] 2. Prime factorise 48: \[ 48 = 2^4 imes 3 \] 3. Identify all prime factors (common and uncommon) and their highest powers: * Prime factor 2: Highest power is \( 2^4 \) * Prime factor 3: Highest power is \( 3^2 \) 4. Multiply these highest powers: \[ LCM(36, 48) = 2^4 imes 3^2 = 16 imes 9 = 144 \] ### Exercises 1. List all factors of 20. 2. From the following list, identify all prime numbers: 1, 2, 9, 11, 15, 19, 21, 29. 3. Perform the prime factorisation for the following numbers, expressing your answer in index notation: a) 72 b) 100 c) 126 4. Using prime factorisation, find the HCF of 24 and 40. 5. Using prime factorisation, find the LCM of 15 and 25. ### Assignment 1. Find the prime factorisation of 150. Use both the factor tree method and the division method to verify your answer. 2. A number is written as \( 2^3 imes 3^2 imes 5 \). What is the number? 3. Determine whether the following statements are true or false. Explain your reasoning. a) All odd numbers are prime numbers. b) The product of two prime numbers is always a composite number. 4. Using prime factorisation, find the HCF and LCM of 80, 120, and 180. 5. Three bells ring at intervals of 12 minutes, 15 minutes, and 20 minutes respectively. If they all ring together at 9:00 AM, at what time will they next ring together? ### Summary Today, we've learned about the fundamental concepts of prime and composite numbers. We discovered that a **prime number** has exactly two factors (1 and itself), while a **composite number** has more than two factors. We mastered **prime factorisation**, which is expressing a number as a product of its prime factors, using both the factor tree and division methods. Finally, we saw how prime factorisation is a crucial tool for finding the **HCF** (Highest Common Factor) and **LCM** (Lowest Common Multiple) of numbers. These skills are essential for many areas of mathematics and will be very useful in your future studies! ## HCF (HCD) and LCM ### Introduction Welcome to this lesson on Number Patterns! Today, we will explore two important concepts: the Highest Common Factor (HCF), also known as the Highest Common Divisor (HCD), and the Lowest Common Multiple (LCM). Understanding HCF and LCM is very useful in mathematics, especially when working with fractions, solving problems involving cycles, and simplifying expressions. We'll learn how to find them using different methods and see how they apply to real-world situations. ### Key Concepts #### Understanding Factors and Multiples ##### Factors * **Definition:** A factor of a number is a number that divides it exactly without leaving a remainder. * **Example:** The factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 evenly. ##### Multiples * **Definition:** A multiple of a number is the result of multiplying that number by an integer. Essentially, multiples are the numbers you get when you count by a certain number. * **Example:** The multiples of 3 are 3, 6, 9, 12, 15, ... (obtained by \( 3 imes 1 \), \( 3 imes 2 \), \( 3 imes 3 \), etc.). #### Highest Common Factor (HCF) / Highest Common Divisor (HCD) ##### Definition * The HCF (or HCD) of two or more numbers is the largest number that is a factor of all of them. ##### How to Find HCF ###### Method 1: Listing Factors 1. List all the factors for each of the given numbers. 2. Identify the factors that are common to all the numbers. 3. The largest among these common factors is the HCF. * **Example:** Find the HCF of 12 and 18. * Factors of 12: 1, 2, 3, 4, 6, 12 * Factors of 18: 1, 2, 3, 6, 9, 18 * Common factors: 1, 2, 3, 6 * The highest common factor is 6. So, HCF(12, 18) = 6. ###### Method 2: Prime Factorization 1. Write each number as a product of its prime factors (prime factorization). 2. Identify all prime factors that are common to all numbers. 3. Multiply these common prime factors, each raised to the *lowest* power it appears in any of the factorizations. * **Example:** Find the HCF of 12 and 18. * Prime factorization of 12: \( 12 = 2^2 imes 3 \) * Prime factorization of 18: \( 18 = 2 imes 3^2 \) * Common prime factors are 2 and 3. * Lowest power of 2 is \( 2^1 \) (from 18). * Lowest power of 3 is \( 3^1 \) (from 12). * HCF = \( 2^1 imes 3^1 = 2 imes 3 = 6 \). #### Lowest Common Multiple (LCM) ##### Definition * The LCM of two or more numbers is the smallest positive number that is a multiple of all of them. ##### How to Find LCM ###### Method 1: Listing Multiples 1. List the multiples for each number. 2. Continue listing until you find the first multiple that appears in all lists. 3. This smallest common multiple is the LCM. * **Example:** Find the LCM of 4 and 6. * Multiples of 4: 4, 8, 12, 16, 20, 24, ... * Multiples of 6: 6, 12, 18, 24, 30, ... * Common multiples: 12, 24, ... * The lowest common multiple is 12. So, LCM(4, 6) = 12. ###### Method 2: Prime Factorization 1. Write each number as a product of its prime factors. 2. Identify all unique prime factors from all the numbers. 3. Multiply these prime factors, each raised to the *highest* power it appears in any of the factorizations. * **Example:** Find the LCM of 4 and 6. * Prime factorization of 4: \( 4 = 2^2 \) * Prime factorization of 6: \( 6 = 2 imes 3 \) * Unique prime factors are 2 and 3. * Highest power of 2 is \( 2^2 \) (from 4). * Highest power of 3 is \( 3^1 \) (from 6). * LCM = \( 2^2 imes 3^1 = 4 imes 3 = 12 \). ###### Method 3: Using HCF (for two numbers only) * For any two numbers, A and B, the product of their HCF and LCM is equal to the product of the numbers themselves. \[ A imes B = \text{HCF}(A, B) imes \text{LCM}(A, B) \] * This formula can be rearranged to find the LCM if you know the HCF (or vice-versa): \[ \text{LCM}(A, B) = \frac{A imes B}{\text{HCF}(A, B)} \] * **Example:** Find the LCM of 12 and 18, given that their HCF is 6. * \( \text{LCM}(12, 18) = \frac{12 imes 18}{6} = \frac{216}{6} = 36 \). ### Vocabulary * **Factor:** A number that divides another number exactly without leaving a remainder. * **Multiple:** The result of multiplying a number by an integer. Multiples are what you get when you multiply a number by 1, 2, 3, and so on. * **Prime Number:** A natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). * **Composite Number:** A natural number greater than 1 that is not prime (it has factors other than 1 and itself, e.g., 4, 6, 8, 9, 10). * **Prime Factorization:** The process of expressing a composite number as a product of its prime factors. * **HCF (Highest Common Factor) / HCD (Highest Common Divisor):** The largest factor that two or more numbers share. * **LCM (Lowest Common Multiple):** The smallest positive number that is a multiple of two or more numbers. ### Examples #### Example 1: Finding HCF of three numbers * **Question:** Find the HCF of 24, 36, and 48. * **Solution (using Prime Factorization):** * Prime factorization of 24: \( 24 = 2^3 \times 3 \) * Prime factorization of 36: \( 36 = 2^2 \times 3^2 \) * Prime factorization of 48: \( 48 = 2^4 \times 3 \) * Common prime factors are 2 and 3. * The lowest power of 2 across all numbers is \( 2^2 \). * The lowest power of 3 across all numbers is \( 3^1 \). * HCF \( = 2^2 \times 3^1 = 4 \times 3 = 12 \). #### Example 2: Finding LCM of three numbers * **Question:** Find the LCM of 15, 20, and 25. * **Solution (using Prime Factorization):** * Prime factorization of 15: \( 15 = 3 \times 5 \) * Prime factorization of 20: \( 20 = 2^2 \times 5 \) * Prime factorization of 25: \( 25 = 5^2 \) * Unique prime factors involved are 2, 3, and 5. * The highest power of 2 is \( 2^2 \). * The highest power of 3 is \( 3^1 \). * The highest power of 5 is \( 5^2 \). * LCM \( = 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 12 \times 25 = 300 \). #### Example 3: Real-world problem involving HCF * **Question:** A rectangular field is 180 meters long and 135 meters wide. It is to be divided into square plots of the same size. What is the largest possible length of the side of each square plot? * **Solution:** To find the largest possible side length of a square that can fit exactly into both the length and width, we need to find the HCF of 180 and 135. * Prime factorization of 180: \( 180 = 2^2 \times 3^2 \times 5 \) * Prime factorization of 135: \( 135 = 3^3 \times 5 \) * Common prime factors are 3 and 5. * Lowest power of 3 is \( 3^2 \). * Lowest power of 5 is \( 5^1 \). * HCF \( = 3^2 \times 5 = 9 \times 5 = 45 \). * **Answer:** The largest possible length of the side of each square plot is 45 meters. #### Example 4: Real-world problem involving LCM * **Question:** Two bells ring at intervals of 15 minutes and 20 minutes respectively. If they both ring together at 10:00 AM, when will they next ring together? * **Solution:** We need to find the smallest time interval after which both bells will ring together again. This means finding the LCM of 15 and 20. * Prime factorization of 15: \( 15 = 3 \times 5 \) * Prime factorization of 20: \( 20 = 2^2 \times 5 \) * Unique prime factors are 2, 3, and 5. * Highest power of 2 is \( 2^2 \). * Highest power of 3 is \( 3^1 \). * Highest power of 5 is \( 5^1 \). * LCM \( = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 \). * The bells will ring together again after 60 minutes, which is 1 hour. * **Answer:** They will next ring together at 11:00 AM (10:00 AM + 1 hour). ### Exercises 1. Find the HCF of 72 and 108. 2. Find the LCM of 16 and 24. 3. Find both the HCF and LCM of 30, 45, and 60. 4. Three different types of fruit are sold in bags. Apples are sold in bags of 12, oranges in bags of 18, and bananas in bags of 24. A caterer wants to buy an equal minimum number of each type of fruit. What is the minimum number of each fruit type he must buy? 5. A gardener has 60 red roses, 84 yellow roses, and 108 white roses. She wants to arrange them in rows, with each row containing only one color and having the same number of roses. What is the greatest number of roses she can put in each row? ### Assignment 1. Determine the HCF of 90, 120, and 150. 2. Determine the LCM of 8, 12, and 18. 3. Two cyclists start riding from the same point, in the same direction, at the same time. One completes a lap in 30 minutes, and the other in 45 minutes. After how many minutes will they both be at the starting point again? 4. A school is organizing a fundraising event. They have 144 lollipops and 108 chocolates. They want to make identical gift bags, with no items left over. What is the greatest number of gift bags they can make? How many lollipops and chocolates will be in each bag? 5. If the HCF of two numbers is 8 and their product is 384, find their LCM using the formula \( A \times B = \text{HCF}(A, B) \times \text{LCM}(A, B) \). ### Summary In this lesson, we have learned about HCF (Highest Common Factor) and LCM (Lowest Common Multiple). The HCF is the largest number that divides two or more numbers exactly, while the LCM is the smallest positive number that is a multiple of two or more numbers. We explored different methods to find HCF and LCM, including listing factors/multiples and using prime factorization. We also saw how these concepts are applied in various real-life scenarios, from arranging items into equal groups to synchronizing events. ## Square roots and cube roots ### Introduction Welcome to this lesson on square roots and cube roots! These are fundamental concepts in mathematics that help us understand the inverse operations of squaring and cubing numbers. By the end of this lesson, you will be able to identify, calculate, and apply square roots and cube roots. ### Key Concepts #### Understanding Square Roots * The square root of a number is a value that, when multiplied by itself, gives the original number. * The symbol for square root is \( \sqrt{} \). * Every positive number has two square roots: a positive one (called the principal square root) and a negative one. For example, \[ 3 \times 3 = 9 \text{ and } (-3) \times (-3) = 9 \] So, the square roots of 9 are 3 and -3. However, the symbol \( \sqrt{9} \) usually refers to the principal (positive) square root, which is 3. #### Perfect Squares * A perfect square is a number that is the square of an integer. * Examples: \( 1 \ (1^2) \), \( 4 \ (2^2) \), \( 9 \ (3^2) \), \( 16 \ (4^2) \), etc. #### Finding Square Roots * **Prime Factorization Method:** * Break down the number into its prime factors. * Group identical prime factors in pairs. * Take one factor from each pair and multiply them together. * Example: To find \( \sqrt{36} \) \[ 36 = 2 \times 2 \times 3 \times 3 \] Group in pairs: \( (2 \times 2) \times (3 \times 3) \) Take one from each pair: \( 2 \times 3 = 6 \) So, \( \sqrt{36} = 6 \). * **Calculator Method:** * Simply use the square root key (\( \sqrt{} \)) on your calculator. #### Understanding Cube Roots * The cube root of a number is a value that, when multiplied by itself three times, gives the original number. * The symbol for cube root is \( \sqrt[3]{} \). * Unlike square roots, every real number (positive or negative) has exactly one real cube root. * Example: \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \). * Example: \( \sqrt[3]{-27} = -3 \) because \( (-3) \times (-3) \times (-3) = -27 \). #### Perfect Cubes * A perfect cube is a number that is the cube of an integer. * Examples: \( 1 \ (1^3) \), \( 8 \ (2^3) \), \( 27 \ (3^3) \), \( 64 \ (4^3) \), etc. #### Finding Cube Roots * **Prime Factorization Method:** * Break down the number into its prime factors. * Group identical prime factors in threes. * Take one factor from each group and multiply them together. * Example: To find \( \sqrt[3]{64} \) \[ 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \] Group in threes: \( (2 \times 2 \times 2) \times (2 \times 2 \times 2) \) Take one from each group: \( 2 \times 2 = 4 \) So, \( \sqrt[3]{64} = 4 \). * **Calculator Method:** * Use the cube root key (often \( \sqrt[3]{} \) or a function like \( x^{1/3} \)) on your calculator. ### Vocabulary * **Square Root:** A number that when multiplied by itself gives the original number. (Symbol: \( \sqrt{} \)) * **Principal Square Root:** The positive square root of a number. * **Perfect Square:** An integer that is the square of another integer. * **Cube Root:** A number that when multiplied by itself three times gives the original number. (Symbol: \( \sqrt[3]{} \)) * **Perfect Cube:** An integer that is the cube of another integer. * **Radical:** The symbol \( \sqrt{} \) used to indicate a root. * **Radicand:** The number inside the radical symbol. ### Examples 1. **Find the principal square root of 49.** * Solution: We need a number that, when multiplied by itself, equals 49. Since \( 7 \times 7 = 49 \), the principal square root of 49 is 7. * \( \sqrt{49} = 7 \) 2. **Calculate \( \sqrt{144} \) using prime factorization.** * Solution: \[ 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \] Group in pairs: \( (2 \times 2) \times (2 \times 2) \times (3 \times 3) \) Take one from each pair: \( 2 \times 2 \times 3 = 12 \) So, \( \sqrt{144} = 12 \). 3. **Find the cube root of 125.** * Solution: We need a number that, when multiplied by itself three times, equals 125. Since \( 5 \times 5 \times 5 = 125 \), the cube root of 125 is 5. * \( \sqrt[3]{125} = 5 \) 4. **Calculate \( \sqrt[3]{216} \) using prime factorization.** * Solution: \[ 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \] Group in threes: \( (2 \times 2 \times 2) \times (3 \times 3 \times 3) \) Take one from each group: \( 2 \times 3 = 6 \) So, \( \sqrt[3]{216} = 6 \). ### Exercises 1. Find the principal square root of the following numbers: * a) 25 * b) 81 * c) 169 * d) 400 2. Use prime factorization to find the square root of: * a) 64 * b) 225 3. Find the cube root of the following numbers: * a) 27 * b) 1000 * c) 343 4. Use prime factorization to find the cube root of: * a) 729 * b) 1728 ### Assignment 1. Determine if the following numbers are perfect squares, perfect cubes, both, or neither: * a) 1 * b) 64 * c) 100 * d) 216 * e) 50 2. A square garden has an area of 289 square meters. What is the length of one side of the garden? 3. A cubic box has a volume of 512 cubic centimeters. What is the length of one side of the box? 4. Simplify: * a) \( \sqrt{196} + \sqrt[3]{1} \) * b) \( \sqrt[3]{64} \times \sqrt{25} \) ### Summary In this lesson, we explored square roots and cube roots. * A **square root** is the inverse operation of squaring a number. \( \sqrt{a} = b \) means \( b^2 = a \). * A **cube root** is the inverse operation of cubing a number. \( \sqrt[3]{a} = b \) means \( b^3 = a \). * We learned to identify **perfect squares** and **perfect cubes**. * We practiced finding square roots and cube roots using both **prime factorization** and **calculators**. * Remember that the principal square root is always positive, while a cube root can be positive or negative depending on the original number. ## Elementary sequences ### Introduction Welcome to the exciting world of number patterns! Have you ever noticed how things in nature or everyday life often follow a certain order? Think about the petals of a flower, the scales on a pineapple, or even the way numbers are organised on a calendar. In mathematics, we often encounter lists of numbers that follow a specific rule or pattern. These ordered lists are called **sequences**. In this lesson, we will explore what sequences are, how to identify the patterns within them, and how to predict what comes next. Understanding sequences is a fundamental skill that helps us in various areas of mathematics and problem-solving. ### Key Concepts #### What is a Sequence? A **sequence** is an ordered list of numbers. Each number in the sequence is called a **term**. The order of the terms is very important, as it helps us understand the pattern. * **Example 1:** \[ 2, 4, 6, 8, \dots \] * This is a sequence where each number is obtained by adding 2 to the previous one. * **Example 2:** \[ 1, 3, 5, 7, \dots \] * This sequence consists of odd numbers, where each number is obtained by adding 2 to the previous one. * **Example 3:** \[ 1, 4, 9, 16, \dots \] * This sequence consists of perfect squares: \( 1^2, 2^2, 3^2, 4^2 \), and so on. #### Terms of a Sequence Each number in a sequence is called a **term**. We often use a special notation to refer to specific terms: * The first term is denoted by \( T_1 \) (or \( a_1 \)) * The second term is denoted by \( T_2 \) (or \( a_2 \)) * The third term is denoted by \( T_3 \) (or \( a_3 \)) * And so on... * The term in the \( n \)-th position is called the **nth term** or **general term**, denoted by \( T_n \) (or \( a_n \)). For the sequence \[ 2, 4, 6, 8, \dots \]: * \( T_1 = 2 \) * \( T_2 = 4 \) * \( T_3 = 6 \) * \( T_4 = 8 \) #### Identifying Number Patterns The key to working with sequences is to identify the rule or pattern that connects the terms. Here are some common types of patterns you might encounter: 1. **Adding or Subtracting a Constant Number:** * Example: \[ 5, 10, 15, 20, \dots \] (Adding 5 each time) * Example: \[ 20, 18, 16, 14, \dots \] (Subtracting 2 each time) 2. **Multiplying or Dividing by a Constant Number:** * Example: \[ 3, 6, 12, 24, \dots \] (Multiplying by 2 each time) * Example: \[ 81, 27, 9, 3, \dots \] (Dividing by 3 each time) 3. **Other Patterns:** Some sequences might involve squares, cubes, or more complex rules relating the term number to the term value. #### The General Term (nth Term) The **general term**, or **nth term**, is a formula that allows you to find any term in the sequence if you know its position \( n \). It's a powerful tool because it means you don't have to list all the terms to find a specific one, like the 100th term. * For the sequence \[ 2, 4, 6, 8, \dots \], the general term is \( T_n = 2n \). * To find the 5th term, substitute \( n=5 \): \( T_5 = 2(5) = 10 \). * For the sequence \[ 1, 3, 5, 7, \dots \], the general term is \( T_n = 2n - 1 \). * To find the 4th term, substitute \( n=4 \): \( T_4 = 2(4) - 1 = 8 - 1 = 7 \). **How to find the general term (for simple linear patterns):** 1. Find the common difference (the number added or subtracted between terms). Let's call this \( d \). 2. The general term will often look like \( T_n = dn + c \), where \( c \) is a constant. 3. Substitute \( n=1 \) and \( T_1 \) into the formula to find \( c \). ### Vocabulary * **Sequence:** An ordered list of numbers that follows a specific pattern or rule. * **Term:** Each individual number in a sequence. * **Pattern:** The discernible rule or relationship that determines the order and values of the terms in a sequence. * **General Term (nth Term):** A formula, usually denoted as \( T_n \) or \( a_n \), that describes any term in a sequence based on its position \( n \). ### Examples **Example 1: Finding the next terms** Find the next three terms in the sequence \[ 7, 10, 13, 16, \dots \] * **Step 1: Identify the pattern.** Look at the difference between consecutive terms: * \( 10 - 7 = 3 \) * \( 13 - 10 = 3 \) * \( 16 - 13 = 3 \) The pattern is to add 3 to the previous term. * **Step 2: Calculate the next terms.** * Next term after 16: \( 16 + 3 = 19 \) * Next term after 19: \( 19 + 3 = 22 \) * Next term after 22: \( 22 + 3 = 25 \) * **Answer:** The next three terms are 19, 22, 25. **Example 2: Finding a specific term using the general term** Find the 7th term of the sequence whose general term is \( T_n = 5n - 2 \). * **Step 1: Use the given general term formula.** We want the 7th term, so we substitute \( n=7 \) into the formula. * **Step 2: Calculate.** * \( T_7 = 5(7) - 2 \) * \( T_7 = 35 - 2 \) * \( T_7 = 33 \) * **Answer:** The 7th term of the sequence is 33. **Example 3: Finding the general term (nth term) for an arithmetic sequence** Find the general term for the sequence \[ 3, 7, 11, 15, \dots \] * **Step 1: Find the common difference (d).** * \( 7 - 3 = 4 \) * \( 11 - 7 = 4 \) * \( 15 - 11 = 4 \) The common difference is \( d = 4 \). * **Step 2: Write the preliminary form of the general term.** Since the common difference is 4, the general term will be in the form \( T_n = 4n + c \). * **Step 3: Use the first term to find c.** We know \( T_1 = 3 \). Substitute \( n=1 \) and \( T_1 = 3 \) into the formula: * \( 3 = 4(1) + c \) * \( 3 = 4 + c \) * \( c = 3 - 4 \) * \( c = -1 \) * **Step 4: Write the complete general term.** * \( T_n = 4n - 1 \) * **Answer:** The general term for the sequence is \( T_n = 4n - 1 \). ### Exercises 1. Find the next three terms for each of the following sequences: a) \[ 4, 9, 14, 19, \dots \] b) \[ 50, 45, 40, 35, \dots \] c) \[ 2, 6, 18, 54, \dots \] d) \[ 1, 1, 2, 3, 5, \dots \] (Hint: This is a special sequence where each term is the sum of the two preceding ones after the first two) 2. For the sequence \[ 10, 17, 24, 31, \dots \]: a) Find the common difference. b) Find the 6th term. c) Find the general term \( T_n \). d) Use the general term to find the 12th term. 3. A sequence is given by the general term \( T_n = 3n + 5 \). a) Find the first 4 terms of the sequence. b) Find the 10th term of the sequence. 4. Find the general term (nth term) for each of these sequences: a) \[ 6, 8, 10, 12, \dots \] b) \[ 100, 90, 80, 70, \dots \] c) \[ 0, 3, 6, 9, \dots \] ### Assignment 1. List the first five terms of a sequence defined by \( T_n = n^2 - 1 \). 2. The cost of hiring a bicycle is \( \$5 \) for the first hour and \( \$2 \) for each additional hour. Create a sequence showing the cost for hiring the bicycle for 1, 2, 3, 4, and 5 hours. What is the general term for the cost (\( C_n \)) for \( n \) hours? 3. Consider the sequence \[ 1.5, 3.0, 4.5, 6.0, \dots \] a) What is the common difference? b) Write down the 7th term. c) Find the general term \( T_n \). 4. Identify the pattern and find the missing numbers: a) \[ 1, 8, 27, \underline{\quad}, 125, \underline{\quad}, \dots \] b) \[ 64, 32, \underline{\quad}, 8, 4, \underline{\quad}, \dots \] ### Summary In this lesson, we've learned about **sequences**, which are ordered lists of numbers following a specific **pattern**. Each number in a sequence is called a **term**. We use \( T_n \) to denote the **nth term**, which is also known as the **general term**. We explored how to: * Identify the pattern in a sequence (e.g., by finding a common difference or ratio). * Find subsequent terms in a sequence. * Determine the **general term** (nth term) for simple linear sequences, which allows us to find any term in the sequence without listing them all out. Understanding sequences is a key step in seeing the order and predictability in numbers, which is very useful in many mathematical concepts!