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
Step 1: Determine the rule for the pattern. Observe the given input and output values: • Input 1 gives Output 5 • Input 2 gives Output 9 • Input 3 gives Output 13 Calculate the difference between consecutive output values: 9 - 5 = 4 13 - 9 = 4 The common difference is 4. This indicates a linear relationship of the form T_n = an + b, where a is the common difference. So, T_n = 4n + b. Using the first pair (Input 1, Output 5): 5 = 4(1) + b 5 = 4 + b b = 1 The rule is T_n = 4n + 1. Step 2: Complete the flow diagram. The rule for the flow diagram is 4n + 1. For the input value 20, calculate the output: T_20 = 4(20) + 1 = 80 + 1 = 81 The completed flow diagram is: Input values: 1 → 2 → 3 → 20 → Middle box: 4n+1 Output values: → 5 → 9 → 13 → 81 Step 3: Complete the table. Using the rule T_n = 4n + 1: For Stage 1: T_1 = 4(1) + 1 = 5 For Stage 2: T_2 = 4(2) + 1 = 9 For Stage 3: T_3 = 4(3) + 1 = 13 For Stage 4: T_4 = 4(4) + 1 = 17 For Stage 5: T_5 = 4(5) + 1 = 21 The completed table is: | Stage | 1 | 2 | 3 | 4 | 5 | |-----------------------|---|---|---|---|----| | Number of match sticks | 5 | 9 | 13| 17| 21 | The missing values are: • The rule in the flow diagram: 4n+1 • The output for input 20: 81 • The number of match sticks for Stage 5: 21 3 done, 2 left today. You're making progress.