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 are the solutions to your math problems. 1. An arithmetic progression (AP) is given as 600 + 650 + 700 + 750 + a) the 30^th term of the AP; Step 1: Identify the first term (a) and the common difference (d). The first term is a = 600. The common difference is d = 650 - 600 = 50. Step 2: Use the formula for the n^th term of an AP, T_n = a + (n-1)d. For the 30^th term, n=30. T_30 = 600 + (30-1)50 Step 3: Calculate the 30^th term. T_30 = 600 + (29)50 T_30 = 600 + 1450 T_30 = 2050 The 30^th term of the AP is 2050. b) the sum of the first 30 terms of the AP. Step 1: Use the formula for the sum of the first n terms of an AP, S_n = (n)/(2)(a + T_n). We know n=30, a=600, and from part (a), T_30 = 2050. S_30 = (30)/(2)(600 + 2050) Step 2: Calculate the sum of the first 30 terms. S_30 = 15(2650) S_30 = 39750 The sum of the first 30 terms of the AP is 39750. 2. The quadratic equation 5x^2 + kx + 20 = 0 has only one root. Determine the possible values of k. Step 1: Identify the coefficients a, b, and c from the quadratic equation ax^2 + bx + c = 0. For 5x^2 + kx + 20 = 0: a = 5 b = k c = 20 Step 2: For a quadratic equation to have only one root (or repeated roots), its discriminant must be equal to zero. The discriminant is given by = b^2 - 4ac. Set the discriminant to zero: k^2 - 4(5)(20) = 0 Step 3: Solve the equation for k. k^2 - 400 = 0 k^2 = 400 k = ±sqrt(400) k = ± 20 The possible values of k are 20 and -20. That's 2 down. 3 left today — send the next one.