This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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3 and Exercise 12.4. Exercise 12.3: Matchstick Houses Step 1: Describe the pattern (Question 1). The pattern shows houses made of matchsticks. Each subsequent diagram is formed by adding one more house to the right of the previous one, sharing a vertical matchstick with the house to its left. Step 2: Make the next diagram (Question 2). To make the next diagram (Diagram 4), you must add one more house to the right of Diagram 3, sharing a vertical matchstick. Diagram 4 will consist of four houses joined together in a row. Step 3: Count matchsticks and fill the table (Question 3). • Diagram 1: 5 matchsticks • Diagram 2: 5 (for 1st house) + 4 (for 2nd house) = 9 matchsticks • Diagram 3: 9 (for 2 houses) + 4 (for 3rd house) = 13 matchsticks • Diagram 4: 13 (for 3 houses) + 4 (for 4th house) = 17 matchsticks | Diagram number | 1 | 2 | 3 | 4 | 5 | 7 | 12 | 101 | |---|---|---|---|---|---|---|---|---| | Number of matchsticks | 5 | 9 | 13 | 17 | | | | | Step 4: Find the rule (Question 4). The number of matchsticks increases by 4 for each additional house. This is an arithmetic sequence with a common difference of 4. Let n be the diagram number. The rule is: Number of matchsticks = 4n + 1 Diagram number → Multiply by 4 and add 1 → number of matches. Step 5: Test the rule for diagrams 2 and 3 (Question 5). • For Diagram 2 (n=2): Number of matchsticks = 4(2) + 1 = 8 + 1 = 9. This matches the count. • For Diagram 3 (n=3): Number of matchsticks = 4(3) + 1 = 12 + 1 = 13. This matches the count. Step 6: Fill in the table for 5, 7, and 12 houses (Question 6). • For Diagram 5 (n=5): Number of matchsticks = 4(5) + 1 = 20 + 1 = 21. • For Diagram 7 (n=7): Number of matchsticks = 4(7) + 1 = 28 + 1 = 29. • For Diagram 12 (n=12): Number of matchsticks = 4(12) + 1 = 48 + 1 = 49. Updated table: | Diagram number | 1 | 2 | 3 | 4 | 5 | 7 | 12 | 101 | |---|---|---|---|---|---|---|---|---| | Number of matchsticks | 5 | 9 | 13 | 17 | 21 | 29 | 49 | | Step 7: Calculate houses for 101 matches (Question 7). We use the rule: Number of matchsticks = 4n + 1. Given 101 matchsticks: 101 = 4n + 1 Subtract 1 from both sides: 100 = 4n Divide by 4: n = (100)/(4) n = 25 So, 25 houses are joined together. Final updated table: | Diagram number | 1 | 2 | 3 | 4 | 5 | 7 | 12 | 25 | |---|---|---|---|---|---|---|---|---| | Number of matchsticks | 5 | 9 | 13 | 17 | 21 | 29 | 49 | 101 | --- Exercise 12.4: Number Sequences For each sequence, the next three numbers are provided, along with the rule. 1. 3; 6; 9; 12; 15; 18; 21 Rule: Add 3 to the previous number. 2. 0; 2; 4; 6; 8; 10; 12 Rule: Add 2 to the previous number. 3. 78; 67; 62; 51; 46; 35 Rule: Alternately subtract 11, then subtract 5 from the previous number. (78 - 11 = 67, 67 - 5 = 62, 62 - 11 = 51, 51 - 5 = 46, 46 - 11 = 35)