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 to determine the type of sequence. The given terms are: T_1 = 0 T_2 = 3 T_3 = 8 T_4 = 15 First differences: T_2 - T_1 = 3 - 0 = 3 T_3 - T_2 = 8 - 3 = 5 T_4 - T_3 = 15 - 8 = 7 Second differences: 5 - 3 = 2 7 - 5 = 2 Since the second differences are constant, the sequence is quadratic, meaning the general term T_n can be expressed in the form T_n = An^2 + Bn + C. Step 2: Determine the coefficients A, B, and C. The second difference is equal to 2A. 2A = 2 A = 1 Now substitute A=1 into the general formula: T_n = n^2 + Bn + C. Use the first two terms to form a system of equations: For n=1: T_1 = 1^2 + B(1) + C = 0 1 + B + C = 0 (1) For n=2: T_2 = 2^2 + B(2) + C = 3 4 + 2B + C = 3 (2) Step 3: Solve the system of equations for B and C. From equation (1), B + C = -1. From equation (2), 2B + C = -1. Subtract equation (1) from equation (2): (2B + C) - (B + C) = -1 - (-1) B = 0 Substitute B=0 into equation (1): 1 + 0 + C = 0 C = -1 Step 4: Write the formula for T_n. Substitute A=1, B=0, and C=-1 into T_n = An^2 + Bn + C: T_n = 1n^2 + 0n - 1 T_n = n^2 - 1 Step 5: Calculate T_10 using the derived formula. Substitute n=10 into the formula for T_n: T_10 = (10)^2 - 1 T_10 = 100 - 1 T_10 = 99 The formula for T_n is n^2 - 1. The value of T_10 is 99. Send me the next one 📸