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 sequence. We are given the input and output values for problem 5: Input (n): 1, 2, 3, 4 Output (T_n): 7, 9, 11, 13 First, find the common difference between consecutive output values: 9 - 7 = 2 11 - 9 = 2 13 - 11 = 2 The common difference is 2. This indicates a linear rule of the form T_n = an + b, where a is the common difference. So, a = 2. The rule is T_n = 2n + b. Step 2: Find the value of b. Substitute one of the input-output pairs into the rule. Using n=1 and T_1=7: 7 = 2(1) + b 7 = 2 + b b = 7 - 2 b = 5 The rule for the sequence is T_n = 2n + 5. Step 3: Calculate the missing output values using the rule T_n = 2n + 5. For Input n=10: T_10 = 2(10) + 5 = 20 + 5 = 25 For Input n=100: T_100 = 2(100) + 5 = 200 + 5 = 205 Step 4: Complete the table and list the terms. The rule is: T_n = 2n + 5 The completed table is: Input | 1 | 2 | 3 | 4 | 10 | 100 ---|---|---|---|---|---|--- Output | 7 | 9 | 11 | 13 | 25 | 205 The terms are: T_1 = 7 T_2 = 9 T_3 = 11 T_4 = 13 T_10 = 25 T_100 = 205 3 done, 2 left today. You're making progress.