Sequences: – Generating number patterns and finding the nth term.

SEQUENCES Generating number patterns and finding the nth term. Grade 7 Mathematics: Sequences – Generating Number Patterns and Finding the nth Term Subtopic Navigator Introduction Understanding Sequences Generating Number Patterns Arithmetic Sequences Finding the nth Term Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Define and recognize sequences in mathematics. Generate number patterns based on rules. Identify arithmetic sequences and their properties. Find the nth term of a sequence. Lesson Introduction A sequence is an ordered list of numbers that follow a particular rule or pattern. Each number in a sequence is called a term. Sequences appear frequently in everyday life, such as patterns of odd numbers, even numbers, or multiples. In mathematics, we often want to generate these patterns and find a formula to identify any term in the sequence, known as the nth term. Understanding Sequences A sequence is written as a list of terms separated by commas, e.g., [latex]2, 4, 6, 8, 10, dots[/latex]. The position of each term is important. The first term is usually called [latex]a_1[/latex], the second term [latex]a_2[/latex], and so on. Example 1: Write down the first five terms of the sequence of odd numbers. Solution: The odd numbers are [latex]1, 3, 5, 7, 9[/latex]. Example 2: Identify the 4th term of the sequence [latex]10, 20, 30, 40, 50, dots[/latex]. Solution: The terms are increasing by 10 each time. The 4th term is [latex]40[/latex]. Exercises (Understanding Sequences) Write down the first six terms of the even numbers. Find the 5th term of the sequence [latex]7, 14, 21, 28, dots[/latex]. Generating Number Patterns Number patterns are generated by applying a consistent rule. The rule could be "add 3 each time," "multiply by 2," or "subtract 4 each time." Recognizing the rule is key to extending the sequence. Example 3: Generate the next three terms of the sequence [latex]5, 10, 15, 20, dots[/latex]. Solution: Rule: Add 5 each time. Next three terms: [latex]25, 30, 35[/latex]. Example 4: Generate the next three terms of [latex]2, 4, 8, 16, dots[/latex]. Solution: Rule: Multiply by 2. Next three terms: [latex]32, 64, 128[/latex]. Exercises (Generating Number Patterns) Find the next three terms of [latex]3, 6, 9, 12, dots[/latex]. Find the next three terms of [latex]1, 2, 4, 8, dots[/latex]. Arithmetic Sequences An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This difference is called the common difference, [latex]d[/latex]. The general form of an arithmetic sequence is: [latex]a, a + d, a + 2d, a + 3d, dots[/latex] Example 5: Find the common difference in [latex]4, 7, 10, 13, dots[/latex]. Solution: Each term increases by 3, so [latex]d = 3[/latex]. Example 6: Write down the next three terms of [latex]12, 9, 6, 3, dots[/latex]. Solution: Here, [latex]d = -3[/latex]. Next three terms: [latex]0, -3, -6[/latex]. Exercises (Arithmetic Sequences) Find the common difference of [latex]15, 20, 25, 30, dots[/latex]. Write the next three terms of [latex]50, 45, 40, 35, dots[/latex]. Finding the nth Term The nth term of an arithmetic sequence can be found using the formula: [latex]a_n = a + (n-1)d[/latex], where: [latex]a[/latex] is the first term, [latex]d[/latex] is the common difference, [latex]n[/latex] is the position of the term. Example 7: Find the nth term of [latex]2, 5, 8, 11, dots[/latex]. Solution: First term [latex]a = 2[/latex], common difference [latex]d = 3[/latex]. Formula: [latex]a_n = 2 + (n-1)3 = 3n - 1[/latex]. Example 8: Find the 10th term of [latex]7, 10, 13, 16, dots[/latex]. Solution: [latex]a = 7[/latex], [latex]d = 3[/latex]. [latex]a_{10} = 7 + (10-1) times 3 = 34[/latex]. Exercises (Finding the nth Term) Find the nth term of [latex]4, 9, 14, 19, dots[/latex]. Find the 15th term of [latex]3, 7, 11, 15, dots[/latex]. Applications and Mixed Problems Example 9: A staircase has steps arranged so that the first step has 2 tiles, the second has 4, the third has 6, and so on. How many tiles are on the 20th step? Solution: Sequence: [latex]2, 4, 6, 8, dots[/latex], [latex]a = 2[/latex], [latex]d = 2[/latex]. [latex]a_{20} = 2 + (20-1)times 2 = 40[/latex] tiles. Example 10: The number of seats in each row of a hall increases by 3 compared to the previous row. If the first row has 25 seats, how many seats are in the 12th row? Solution: [latex]a = 25[/latex], [latex]d = 3[/latex]. [latex]a_{12} = 25 + (12-1)times 3 = 25 + 33 = 58[/latex] seats. Exercises (Applications) A flowerbed is arranged with 5 flowers in the first row, 8 in the second, 11 in the third, and so on. How many flowers are in the 15th row? A pile of bricks is stacked so that the bottom layer has 50 bricks, the second has 47, the third has 44, and so on. How many bricks are in the 8th layer? Cumulative Exercises Write down the first six terms of the sequence of multiples of 9. Find the 12th term of [latex]5, 10, 15, dots[/latex]. Determine the nth term of [latex]6, 11, 16, 21, dots[/latex]. Write the next four terms of [latex]100, 90, 80, 70, dots[/latex]. Find the 20th term of [latex]7, 14, 21, dots[/latex]. Identify the rule that generates [latex]1, 4, 9, 16, dots[/latex]. Is [latex]150[/latex] a term in the sequence [latex]5, 10, 15, dots[/latex]? Explain. Find the common difference in [latex]2, 9, 16, 23, dots[/latex]. Write the nth term of [latex]3, 8, 13, 18, dots[/latex]. A savings plan starts with ₦500 in the first month and increases by ₦200 each month. How much will be saved in the 10th month? Conclusion/Recap In this lesson, we explored sequences, focusing on generating number patterns, arithmetic sequences, and finding the nth term. These skills are vital in identifying patterns in mathematics and applying them to real-world problems, such as arrangements, savings plans, and building designs. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c