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.
ScanSolve AI Answer
Here are the solutions for the matchstick pattern questions: 3.1 Investigate and demonstrate the physical structure and then complete the table below. Step 1: Analyze the pattern for the number of matchsticks. Pattern 1 (1 triangle): 3 matchsticks Pattern 2 (2 triangles): 5 matchsticks Pattern 3 (3 triangles): 7 matchsticks The number of matchsticks increases by 2 for each additional triangle. This is an arithmetic progression. Step 2: Determine the "Repeated addition" and "The rule" based on the examples. The "Repeated addition" shows 1 plus 2 added n times, where n is the pattern number. "The rule" simplifies this to 1 + (2 × n). Step 3: Complete the table using these observations. For Pattern number 4: Number of matchsticks: 7 + 2 = 9. Repeated addition: 1+2+2+2+2. The rule: 1+(2 × 4). For Pattern number 10: Number of matchsticks: 1 + (2 × 10) = 1 + 20 = 21. Repeated addition: 1+2+2+2+2+2+2+2+2+2+2. The rule: 1+(2 × 10). The completed table is: |c|c|c|c|c|c| Pattern number & 1 & 2 & 3 & 4 & 10 \\ Number of matchsticks & 3 & 5 & 7 & 9 & 21 \\ Repeated addition & 1+2 & 1+2+2 & 1+2+2+2 & 1+2+2+2+2 & 1+2+2+2+2+2+2+2+2+2+2 \\ The rule & 1+(2 × 1) & 1+(2 × 2) & 1+(2 × 3) & 1+(2 × 4) & 1+(2 × 10) \\ 3.2 Draw the fourth stage in the space provided below. The fourth stage consists of 4 triangles joined in a row, sharing sides. It uses 9 matchsticks. ` /\ / \ /____\ /\ /\ / \ / \ /____\/____\ /\ /\ /\ / \ / \ / \ /____\/____\/____\ /\ /\ /\ /\ / \ / \ / \ / \ /____\/____\/____\/____\ ` (Note: Due to text-based limitations, this is a simplified representation. In a drawing, the matchsticks would be distinct lines forming the triangles.) 3.3 Use the general rule to find the number of matchsticks you will use to build a pattern with 20 triangles. Step 1: Identify the general rule from the table. The general rule for the number of matchsticks (N) for n triangles is: N = 1 + (2 × n) Step 2: Substitute n=20 into the rule. N = 1 + (2 × 20) N = 1 + 40 N = 41 The number of matchsticks needed for a pattern with 20 triangles is 41. Last free one today — make it count tomorrow, or type /upgrade for unlimited.