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's the solution for question 1: a) To complete the table, we need to observe the patterns for the number of shaded, unshaded, and total triangles. Step 1: Analyze the pattern for "Number of shaded triangles". Pattern 1: 2 Pattern 2: 4 Pattern 3: 6 This is an arithmetic progression where the number of shaded triangles is 2 × Pattern number. For Pattern 4: 2 × 4 = 8. Step 2: Analyze the pattern for "Number of unshaded triangles". Pattern 1: 0 Pattern 2: 2 Pattern 3: 4 Pattern 4: 6 This is an arithmetic progression where the number of unshaded triangles is 2 × (Pattern number - 1). Step 3: Analyze the pattern for "Total number of triangles". Pattern 1: 2 Pattern 2: 6 Pattern 3: 10 This is an arithmetic progression with a common difference of 4. For Pattern 4: 10 + 4 = 14. Alternatively, Total number of triangles = Number of shaded triangles + Number of unshaded triangles. For Pattern 4: 8 + 6 = 14. The completed table is: |l|c|c|c|c| Pattern number & 1 & 2 & 3 & 4 \\ Number of shaded triangles & 2 & 4 & 6 & 8 \\ Number of unshaded triangles & 0 & 2 & 4 & 6 \\ Total number of triangles & 2 & 6 & 10 & 14 \\ b) i) Find the number of unshaded triangles in pattern 6. From the pattern observed in Step 2, the number of unshaded triangles is 2 × (Pattern number - 1). For Pattern 6: 2 × (6 - 1) = 2 × 5 = 10. The number of unshaded triangles in pattern 6 is 10. ii) Find the nth term, for the total number of triangles in pattern n. From the table, the sequence for the total number of triangles is 2, 6, 10, 14, ... This is an arithmetic progression with the first term a = 2 and common difference d = 6 - 2 = 4. The formula for the nth term of an arithmetic progression is T_n = a + (n-1)d. Step 1: Substitute the values of a and d into the formula. T_n = 2 + (n-1)4 Step 2: Simplify the expression. T_n = 2 + 4n - 4 T_n = 4n - 2 The nth term for the total number of triangles in pattern n is 4n - 2. Send me the next one 📸