Sequences Pattern. Grade 7 Mathematics: Sequences - Generating Number Patterns and Finding the nth Term Subtopic Navigator Introduction to Sequences What are Number Sequences? Arithmetic Sequences Finding the nth Term Formula Using the nth Term Formula Generating Number Patterns Cumulative Exercises Conclusion Lesson Objectives Understand the concept of number sequences and patterns Identify and generate arithmetic sequences Find the common difference in arithmetic sequences Derive and use the formula for the nth term of arithmetic sequences Generate number patterns using given rules Apply sequence knowledge to solve real-world problems Introduction to Sequences Have you ever noticed patterns in numbers? Like counting by 2s (2, 4, 6, 8...), or the days of the month? These ordered lists of numbers are called sequences. A sequence is a set of numbers arranged in a specific order according to a rule. Understanding sequences helps us predict future numbers, solve problems, and see mathematical patterns in the world around us. What are Number Sequences? A sequence is an ordered list of numbers. Each number in a sequence is called a term. Terms are usually written in order, separated by commas. We use specific notation to talk about different terms in a sequence. Sequence Notation: • First term: T₁ or a₁ • Second term: T₂ or a₂ • Third term: T₃ or a₃ • nth term: Tₙ or aₙ (general term we want to find) • The three dots (...) mean "and so on" or "continues" Example 1: Identifying Terms For the sequence: 5, 8, 11, 14, 17, ... Solution: T₁ = 5 (first term) T₂ = 8 (second term) T₃ = 11 (third term) T₄ = 14 (fourth term) T₅ = 17 (fifth term) Practice Problems For the sequence 3, 7, 11, 15, 19, ... find T₁, T₃, and T₅ Write the first five terms of a sequence where you start at 10 and add 3 each time What does Tₙ represent in a sequence? In the sequence 100, 95, 90, 85, 80, ... what is T₄? Create a sequence starting at 2 and multiplying by 2 each time Arithmetic Sequences An arithmetic sequence (also called a linear sequence) is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference and is usually represented by the letter d. Key Principle: To find the common difference (d): [latex]d = T_2 - T_1 = T_3 - T_2 = T_4 - T_3 = ...[/latex] If d is positive, the sequence increases. If d is negative, the sequence decreases. Example 2: Finding Common Difference Find the common difference in: 12, 9, 6, 3, 0, ... Solution: d = T₂ - T₁ = 9 - 12 = -3 d = T₃ - T₂ = 6 - 9 = -3 d = T₄ - T₃ = 3 - 6 = -3 Common difference = -3 (sequence decreases by 3 each time) Example 3: Generating Arithmetic Sequence Generate the first 5 terms of an arithmetic sequence with first term 7 and common difference 4. Solution: T₁ = 7 T₂ = 7 + 4 = 11 T₃ = 11 + 4 = 15 T₄ = 15 + 4 = 19 T₅ = 19 + 4 = 23 Sequence: 7, 11, 15, 19, 23, ... Practice Problems Find the common difference in: 4, 10, 16, 22, 28, ... Is 2, 5, 8, 11, 14, ... an arithmetic sequence? Why or why not? Generate the first 6 terms of an arithmetic sequence with T₁ = 20 and d = -2 Find the missing terms: 8, ___, 18, ___, 28, 33 What is the common difference if T₁ = 5 and T₄ = 14? Finding the nth Term Formula The most powerful tool in working with sequences is finding the nth term formula. This formula allows us to find ANY term in the sequence without listing all the previous terms! Formula for nth term of Arithmetic Sequence: [latex]T_n = a + (n - 1)d[/latex] Where: • Tₙ = nth term (what we want to find) • a = first term (T₁) • n = position of the term in the sequence • d = common difference Example 4: Deriving the nth Term Find the nth term formula for: 6, 11, 16, 21, 26, ... Solution: Step 1: Identify a = 6 (first term) Step 2: Find d = 11 - 6 = 5 Step 3: Apply formula: Tₙ = 6 + (n - 1)×5 Step 4: Simplify: Tₙ = 6 + 5n - 5 = 5n + 1 Formula: Tₙ = 5n + 1 Example 5: nth Term with Negative Difference Find the nth term for: 25, 21, 17, 13, 9, ... Solution: a = 25, d = 21 - 25 = -4 Tₙ = 25 + (n - 1)×(-4) = 25 - 4n + 4 = 29 - 4n Formula: Tₙ = 29 - 4n Practice Problems Find the nth term of: 3, 8, 13, 18, 23, ... Find the nth term of: 50, 46, 42, 38, 34, ... What is the nth term formula for: 7, 10, 13, 16, 19, ...? Derive the nth term for: 100, 97, 94, 91, 88, ... If a sequence starts at 15 and decreases by 3 each time, what is its nth term? Using the nth Term Formula Once we have the nth term formula, we can use it to find specific terms or check if a number is in the sequence. Example 6: Finding Specific Terms For the sequence with nth term Tₙ = 4n + 3: a) Find the first 3 terms b) Find the 10th term c) Find the 20th term Solution: a) T₁ = 4(1) + 3 = 7 T₂ = 4(2) + 3 = 11 T₃ = 4(3) + 3 = 15 First 3 terms: 7, 11, 15 b) T₁₀ = 4(10) + 3 = 43 c) T₂₀ = 4(20) + 3 = 83 Example 7: Checking if a Number is in the Sequence Is 47 in the sequence with nth term Tₙ = 5n - 3? Solution: Set Tₙ = 47 and solve for n: 5n - 3 = 47 5n = 50 n = 10 Since n = 10 is a whole number, 47 is the 10th term in the sequence. Practice Problems For Tₙ = 3n + 2, find T₅ and T₁₂ For Tₙ = 7n - 4, find the first 4 terms Is 100 in the sequence Tₙ = 4n + 8? If yes, which term? For Tₙ = 50 - 2n, find T₁₅ What is the 25th term of the sequence with nth term Tₙ = 2n + 10? Generating Number Patterns We can create various number patterns by following different rules. While arithmetic sequences add or subtract a constant number, other patterns might involve multiplication, squares, or more complex rules. Type of Pattern Example Sequence Rule Next 2 Terms Arithmetic (Add constant) 5, 9, 13, 17, 21, ... Add 4 each time 25, 29 Geometric (Multiply constant) 3, 6, 12, 24, 48, ... Multiply by 2 each time 96, 192 Square Numbers 1, 4, 9, 16, 25, ... n² (1², 2², 3², ...) 36, 49 Triangular Numbers 1, 3, 6, 10, 15, ... Add increasing numbers: +2, +3, +4, ... 21, 28 Cube Numbers 1, 8, 27, 64, 125, ... n³ (1³, 2³, 3³, ...) 216, 343 Example 8: Square Number Pattern Continue the square number pattern: 1, 4, 9, 16, 25, ... Solution: These are squares of natural numbers: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25 Next terms: 6² = 36, 7² = 49 Sequence continues: 36, 49, 64, 81, 100, ... Example 9: Real-World Pattern A taxi charges ₦300 for the first kilometer and ₦150 for each additional kilometer. Find the fare pattern for 1km, 2km, 3km, 4km, ... Solution: 1km: ₦300 2km: ₦300 + ₦150 = ₦450 3km: ₦300 + 2×₦150 = ₦600 4km: ₦300 + 3×₦150 = ₦750 Pattern: 300, 450, 600, 750, 900, ... This is arithmetic with a = 300, d = 150 Tₙ = 300 + (n-1)×150 = 150n + 150 Practice Problems Continue the pattern: 2, 4, 8, 16, 32, ... (next 3 terms) What type of pattern is: 1, 8, 27, 64, 125, ...? Generate the first 6 triangular numbers A plant grows 5cm each week. If it starts at 10cm, how tall after 6 weeks? Identify the pattern: 3, 6, 9, 12, 15, ... and find the next 2 terms Cumulative Exercises Solve these problems that combine all the concepts learned in this lesson. Find the next three terms: 4, 9, 14, 19, 24, ___, ___, ___ Identify the common difference: 15, 12, 9, 6, 3, ... Is this sequence arithmetic? 1, 3, 6, 10, 15, ... Explain why or why not Find the nth term of: 8, 13, 18, 23, 28, ... Using Tₙ = 3n + 7, find T₁, T₄, and T₁₀ Find the 12th term of: 5, 8, 11, 14, 17, ... Generate first 5 terms of arithmetic sequence with a = 12, d = -3 Find which term equals 35 in: 3, 7, 11, 15, 19, ... Write the sequence of square numbers from 1² to 8² A staircase has steps that increase in height: 10cm, 13cm, 16cm, 19cm,... Find the height of the 10th step Find nth term: 30, 27, 24, 21, 18, ... Is 50 in the sequence with nth term Tₙ = 4n + 6? If yes, which term? Find missing terms: 6, ___, 16, ___, 26, 31 A savings plan starts with ₦100 and adds ₦50 each week. How much after 8 weeks? For the pattern 2, 6, 18, 54, ... what type of pattern is it and what are the next 2 terms? Show/Hide Answers Problem: Find the next three terms: 4, 9, 14, 19, 24, ___, ___, ___ Answer: Common difference = 5, so next terms: 29, 34, 39 Problem: Identify the common difference: 15, 12, 9, 6, 3, ... Answer: d = 12 - 15 = -3 (or 9 - 12 = -3, etc.) Problem: Is this sequence arithmetic? 1, 3, 6, 10, 15, ... Answer: No. Differences: 3-1=2, 6-3=3, 10-6=4, 15-10=5. Differences are not constant. Problem: Find the nth term of: 8, 13, 18, 23, 28, ... Answer: a=8, d=5 → Tₙ = 8 + (n-1)×5 = 8 + 5n - 5 = 5n + 3 Problem: Using Tₙ = 3n + 7, find T₁, T₄, and T₁₀ Answer: T₁ = 3(1)+7=10, T₄=3(4)+7=19, T₁₀=3(10)+7=37 Problem: Find the 12th term of: 5, 8, 11, 14, 17, ... Answer: a=5, d=3 → T₁₂ = 5 + (12-1)×3 = 5 + 33 = 38 Problem: Generate first 5 terms of arithmetic sequence with a = 12, d = -3 Answer: 12, 9, 6, 3, 0 Problem: Find which term equals 35 in: 3, 7, 11, 15, 19, ... Answer: Sequence: Tₙ=4n-1. Solve 4n-1=35 → 4n=36 → n=9. 35 is 9th term. Problem: Write the sequence of square numbers from 1² to 8² Answer: 1, 4, 9, 16, 25, 36, 49, 64 Problem: A staircase has steps: 10cm, 13cm, 16cm, 19cm,... Find height of 10th step Answer: a=10, d=3 → T₁₀ = 10 + (10-1)×3 = 10 + 27 = 37cm Problem: Find nth term: 30, 27, 24, 21, 18, ... Answer: a=30, d=-3 → Tₙ=30+(n-1)×(-3)=30-3n+3=33-3n Problem: Is 50 in the sequence with nth term Tₙ = 4n + 6? Answer: Solve 4n+6=50 → 4n=44 → n=11. Yes, it's the 11th term. Problem: Find missing terms: 6, ___, 16, ___, 26, 31 Answer: Arithmetic with d=5 (31-26=5). Sequence: 6, 11, 16, 21, 26, 31 Problem: Savings: ₦100 start, add ₦50 each week. Amount after 8 weeks? Answer: Sequence: 100, 150, 200, 250,... Tₙ=100+50(n-1)=50n+50. T₈=50×8+50=₦450 Problem: Pattern: 2, 6, 18, 54, ... Type and next 2 terms Answer: Geometric (multiply by 3). Next terms: 54×3=162, 162×3=486 Conclusion/Recap In this lesson, we've explored the fascinating world of number sequences. We learned that a sequence is an ordered list of numbers following a pattern, and arithmetic sequences have a constant difference between terms. The most important skill we developed was finding and using the nth term formula (Tₙ = a + (n-1)d), which allows us to find any term without listing all previous ones. We also explored different types of patterns including arithmetic, geometric, square numbers, and triangular numbers. These skills are not just mathematical exercises - they help us predict, analyze, and solve problems in everyday life, from calculating savings to understanding growth patterns. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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