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.
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Here are the solutions to your problems. Problem 1: Solve the equation 1x^2 - 8 = 0 Step 1: Isolate the x^2 term. x^2 - 8 = 0 x^2 = 8 Step 2: Take the square root of both sides. Remember to include both positive and negative roots. x = ±sqrt(8) x = ±sqrt(4 × 2) x = ± 2sqrt(2) The solutions are x = 2sqrt(2) or x = -2sqrt(2). Problem 2: Arithmetic Progression The formula for the n^th term of an arithmetic progression is T_n = 2n-3. i) Find the first term Step 1: To find the first term, substitute n=1 into the formula for T_n. T_1 = 2(1) - 3 T_1 = 2 - 3 T_1 = -1 The first term is -1. ii) Find the common difference if the sum of 5 terms is 45. Step 1: We use the first term a = T_1 = -1 found in part (i). We are given n=5 and S_5 = 45. The formula for the sum of the first n terms of an arithmetic progression is: S_n = (n)/(2)(2a + (n-1)d) Step 2: Substitute the known values into the sum formula. 45 = (5)/(2)(2(-1) + (5-1)d) 45 = (5)/(2)(-2 + 4d) Step 3: Solve for d. Multiply both sides by 2: 45 × 2 = 5(-2 + 4d) 90 = 5(-2 + 4d) Divide both sides by 5: (90)/(5) = -2 + 4d 18 = -2 + 4d Add 2 to both sides: 18 + 2 = 4d 20 = 4d Divide by 4: d = (20)/(4) d = 5 The common difference is 5. Send me the next one 📸