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 for an arithmetic progression. Let a be the first term and d be the common difference. The n-th term is given by T_n = a + (n-1)d. The sum of the first n terms is given by S_n = (n)/(2)(2a + (n-1)d). Step 2: Formulate equations from the given information. From the statement "The eleventh term of an arithmetic progression is four times its second term": T_11 = 4 × T_2 a + (11-1)d = 4(a + (2-1)d) a + 10d = 4(a + d) (1) From the statement "The sum of the first seven terms of the same progression is 175": S_7 = 175 (7)/(2)(2a + (7-1)d) = 175 (7)/(2)(2a + 6d) = 175 (2) Step 3: Simplify Equation (1). a + 10d = 4a + 4d 10d - 4d = 4a - a 6d = 3a a = 2d (3) Step 4: Simplify Equation (2). Multiply both sides by (2)/(7): 2a + 6d = 175 × (2)/(7) 2a + 6d = 25 × 2 2a + 6d = 50 Divide by 2: a + 3d = 25 (4) Step 5: Substitute Equation (3) into Equation (4) to solve for d. Substitute a = 2d into a + 3d = 25: 2d + 3d = 25 5d = 25 d = (25)/(5) d = 5 Step 6: Substitute the value of d back into Equation (3) to solve for a. a = 2d a = 2 × 5 a = 10 The first term is 10 and the common difference is 5. The final answer is First term = 10, Common difference = 5.