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: Identify the input and output pairs from the flow diagram. The given input-output pairs are: • Input (n) = 5, Output (T_n) = 17 • Input (n) = 10, Output (T_n) = 32 • Input (n) = 15, Output (T_n) = 47 • Input (n) = 1, Output (T_n) = 5 Step 2: Determine the rule for the sequence. Assume the rule is linear, of the form T_n = an + b. Calculate the change in output for a given change in input: For inputs 5 and 10: Change in input: 10 - 5 = 5 Change in output: 32 - 17 = 15 The coefficient a is change in outputchange in input = (15)/(5) = 3. So, the rule is T_n = 3n + b. Step 3: Find the value of b. Substitute one of the input-output pairs into the rule T_n = 3n + b. Using n=1 and T_n=5: 5 = 3(1) + b 5 = 3 + b b = 5 - 3 b = 2 The rule is T_n = 3n + 2. Step 4: Verify the rule with other pairs. For n=5: T_5 = 3(5) + 2 = 15 + 2 = 17. (Matches) For n=10: T_10 = 3(10) + 2 = 30 + 2 = 32. (Matches) For n=15: T_15 = 3(15) + 2 = 45 + 2 = 47. (Matches) The rule is consistent for all given pairs. The rule for the flow diagram is: T_n = 3n + 2 That's 2 down. 3 left today — send the next one.