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 extended patterns: 2. Extend the following pattern. a) Pattern: 1, 2, 3, 5, 7, 11, ___, ___, ___ The tip indicates that prime numbers are used. The pattern starts with 1, followed by the sequence of prime numbers. Step 1: Identify the prime numbers after 11. The prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. The prime numbers after 11 are 13, 17, and 19. The extended pattern is: 1, 2, 3, 5, 7, 11, 13, 17, 19. b) Pattern: 1, 3, 7, 15, ___, ___, ___ Rule: multiply by 2 and add 1. Step 1: Apply the rule to the last given number, 15. 15 × 2 + 1 = 30 + 1 = 31 Step 2: Apply the rule to the new number, 31. 31 × 2 + 1 = 62 + 1 = 63 Step 3: Apply the rule to the next new number, 63. 63 × 2 + 1 = 126 + 1 = 127 The extended pattern is: 1, 3, 7, 15, 31, 63, 127. c) Pattern: 100, 52, 28, ___, ___, ___ Rule: divide by 2 and add 2. Step 1: Apply the rule to the last given number, 28. 28 ÷ 2 + 2 = 14 + 2 = 16 Step 2: Apply the rule to the new number, 16. 16 ÷ 2 + 2 = 8 + 2 = 10 Step 3: Apply the rule to the next new number, 10. 10 ÷ 2 + 2 = 5 + 2 = 7 The extended pattern is: 100, 52, 28, 16, 10, 7. 3. Create two of your own number patterns and ask your friend to extend it. a) Pattern: 5, 10, 15, 20, ___, ___, ___ Rule: Add 5 to the previous number. b) Pattern: 1, 4, 9, 16, ___, ___, ___ Rule: The n-th term in the sequence is n^2. 3 done, 2 left today. You're making progress.