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
a) Step 1: Analyze the pattern for the number of dots. Pattern 1: 1 dot (1^2) Pattern 2: 4 dots (2^2) Pattern 3: 9 dots (3^2) This sequence is n^2. For pattern 4, the number of dots is 4^2 = 16. This is already given in the table. Step 2: Analyze the pattern for the number of crosses. Pattern 1: 5 crosses Pattern 2: 8 crosses Pattern 3: 11 crosses This is an arithmetic progression with a common difference d = 8-5 = 3. The formula for the n-th term is a_n = a_1 + (n-1)d. Here a_1 = 5 and d = 3. So, a_n = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2. For pattern 4, the number of crosses is 3(4) + 2 = 12 + 2 = 14. Step 3: Analyze the pattern for the sum of crosses and dots. Pattern 1: 1+5 = 6 Pattern 2: 4+8 = 12 Pattern 3: 9+11 = 20 For pattern 4, the sum is 16+14 = 30. The completed table is: | pattern | 1 | 2 | 3 | 4 | | :---------------------- | :-- | :-- | :-- | :-- | | number of dots | 1 | 4 | 9 | 16 | | number of crosses | 5 | 8 | 11 | 14 | | sum of crosses and dots | 6 | 12 | 20 | 30 | b) i) Step 1: Use the n-th term formula for the number of crosses derived in part (a), which is 3n+2. Step 2: Substitute n=45 into the formula. Number of crosses = 3(45) + 2 Number of crosses = 135 + 2 Number of crosses = 137 The number of crosses needed for pattern 45 is 137. ii) Step 1: Analyze the sequence for the sum of crosses and dots: 6, 12, 20, ... Find the first differences: 12-6=6, 20-12=8. Find the second differences: 8-6=2. Since the second difference is constant, the n-th term is a quadratic sequence of the form An^2 + Bn + C. Step 2: Determine the coefficients A, B, and C. For a quadratic sequence, 2A = second difference, so 2A = 2 A = 1. The first term of the first differences is 3A+B, so 3(1)+B = 6 3+B=6 B=3. The first term of the sequence is A+B+C, so 1+3+C = 6 4+C=6 C=2. Step 3: Write the n-th term. The n-th term for the sum of crosses and dots is n^2 + 3n + 2. The n-th term for the sum of crosses and dots is n^2 + 3n + 2. That's 2 down. 3 left today — send the next one.