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
Here's how to find the common difference: Step 1: Write down the formula for the n-th term of an arithmetic progression (A.P.). The n-th term is given by a_n = a + (n-1)d, where a is the first term and d is the common difference. Step 2: Use the given information to form equations. "The sum of the 2nd and 5th terms of an arithmetic progression (A.P) is 42." The 2nd term is a_2 = a + (2-1)d = a + d. The 5th term is a_5 = a + (5-1)d = a + 4d. So, a_2 + a_5 = 42 (a + d) + (a + 4d) = 42 2a + 5d = 42 (Equation 1) "The difference between the 6th and 3rd terms is 12." The 6th term is a_6 = a + (6-1)d = a + 5d. The 3rd term is a_3 = a + (3-1)d = a + 2d. So, a_6 - a_3 = 12 (a + 5d) - (a + 2d) = 12 a + 5d - a - 2d = 12 3d = 12 (Equation 2) Step 3: Solve Equation 2 for d. 3d = 12 d = (12)/(3) d = 4 The common difference is 4. The final answer is 4