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 for Question 3 with the corrected terms. The first three terms of the geometric progression are T_1 = 3b-2, T_2 = 3b+6, and T_3 = 3b+3. For a geometric progression, the ratio of consecutive terms is constant. This means (T_2)/(T_1) = (T_3)/(T_2). The value of b Step 1: Set up the equation using the common ratio property. (3b+6)/(3b-2) = (3b+3)/(3b+6) Step 2: Cross-multiply to eliminate the denominators. (3b+6)(3b+6) = (3b-2)(3b+3) (3b+6)^2 = (3b-2)(3b+3) Step 3: Expand both sides of the equation. 9b^2 + 36b + 36 = 9b^2 + 9b - 6b - 6 9b^2 + 36b + 36 = 9b^2 + 3b - 6 Step 4: Rearrange the terms to form a linear equation. Notice that the 9b^2 terms cancel out. 9b^2 - 9b^2 + 36b - 3b + 36 + 6 = 0 33b + 42 = 0 Step 5: Solve for b. 33b = -42 b = (-42)/(33) b = -(14)/(11) The value of b is -(14)/(11). The common ratio Step 1: Substitute the value of b = -(14)/(11) into the expressions for the terms. First term T_1 = 3b-2 = 3(-(14)/(11)) - 2 = -(42)/(11) - (22)/(11) = -(64)/(11). Second term T_2 = 3b+6 = 3(-(14)/(11)) + 6 = -(42)/(11) + (66)/(11) = (24)/(11). Step 2: Calculate the common ratio r = (T_2)/(T_1). r = (24)/(11)-(64)/(11) r = (24)/(11) × (-(11)/(64)) r = -(24)/(64) r = -(3)/(8) The common ratio is -(3)/(8). What's next?