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: Define the terms of the arithmetic progression. Let the first term be a and the common difference be d. The first three terms of the arithmetic progression are: t_1 = a t_2 = a + d t_3 = a + 2d Step 2: Use the information about the average of the first three terms to form an equation. The average of the first three terms is 56. (t_1 + t_2 + t_3)/(3) = 56 Substitute the expressions for t_1, t_2, t_3: (a + (a+d) + (a+2d))/(3) = 56 (3a + 3d)/(3) = 56 Divide by 3: a + d = 56 (*1) Step 3: Use the information about the sum of the first two terms to form another equation. The sum of the first two terms is 96. t_1 + t_2 = 96 Substitute the expressions for t_1, t_2: a + (a+d) = 96 2a + d = 96 (*2) Step 4: Solve the system of two linear equations. We have the system: 1) a + d = 56 2) 2a + d = 96 Subtract equation (1) from equation (2): (2a + d) - (a + d) = 96 - 56 2a + d - a - d = 40 a = 40 Step 5: Substitute the value of a into equation (*1) to find d. a + d = 56 40 + d = 56 d = 56 - 40 d = 16 The first term is a=40 and the common difference is d=16. The first term is 40 and the common difference is 16.