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: Find the differences between consecutive terms. Given sequence: -233, -199, -153, -119. Calculate the first differences: • D_1 = -199 - (-233) = -199 + 233 = 34 • D_2 = -153 - (-199) = -153 + 199 = 46 • D_3 = -119 - (-153) = -119 + 153 = 34 Step 2: Identify the pattern in the differences. The first differences are 34, 46, 34. This indicates an alternating pattern where the differences are 34, 46, 34, 46, . Step 3: Find the next two terms. Following the pattern, the next difference (D_4) will be 46. T_5 = T_4 + D_4 = -119 + 46 = -73 The subsequent difference (D_5) will be 34. T_6 = T_5 + D_5 = -73 + 34 = -39 The next two terms are -73, -39. Step 4: Find the general formula for T_n. The sequence of first differences D_k can be expressed as D_k = A + B(-1)^k. Using D_1 = 34 and D_2 = 46: For k=1: A - B = 34 For k=2: A + B = 46 Adding these equations: 2A = 80 A = 40. Subtracting the first from the second: 2B = 12 B = 6. So, the formula for the differences is D_k = 40 + 6(-1)^k. The general term T_n is given by T_n = T_1 + _k=1^n-1 D_k for n > 1. T_n = -233 + _k=1^n-1 (40 + 6(-1)^k) T_n = -233 + _k=1^n-1 40 + 6 _k=1^n-1 (-1)^k The first sum is 40(n-1). The second sum is 6 × ( (-1)^n-1-12 ). T_n = -233 + 40(n-1) + 3((-1)^n-1-1) T_n = -233 + 40n - 40 + 3(-1)^n-1 - 3 T_n = 40n - 276 + 3(-1)^n-1 The formula for T_n is T_n = 40n - 276 + 3(-1)^n-1. What's next?