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
You're on a roll — let's find the first term and common difference for this arithmetic progression. Let the first term be a and the common difference be d. The formula for the n-th term is aₙ = a + (n-1)d. The formula for the sum of the first n terms is Sₙ = (n/2)[2a + (n-1)d]. Step 1: Use the first condition to form an equation. "The ninth term of an arithmetic progression is three times the third term." a₉ = a + (9-1)d = a + 8d a₃ = a + (3-1)d = a + 2d So, we have: a + 8d = 3(a + 2d) a + 8d = 3a + 6d Rearrange the terms: 8d - 6d = 3a - a 2d = 2a Divide by 2: d = a (Equation 1) Step 2: Use the second condition to form another equation. "The sum of the first 4 terms is 30." S₄ = (4/2)[2a + (4-1)d] S₄ = 2[2a + 3d] So, we have: 2(2a + 3d) = 30 Divide by 2: 2a + 3d = 15 (Equation 2) Step 3: Solve the system of equations. Substitute Equation 1 (d = a) into Equation 2: 2a + 3(a) = 15 2a + 3a = 15 5a = 15 Divide by 5: a = 3 Now, use Equation 1 to find d: d = a d = 3 The first term of the progression is 3 and the common difference is 3. Got more? Send 'em