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 are right to double-check! Let's clarify the terms from the image for the geometric progression. The terms given in the image are: First term: 3b-2 Second term: 5b+6 Third term: 3b+3 As you can see, the second term does contain 5b (specifically, 5b+6). My previous solution used these exact terms for the calculations. Let's re-confirm the steps for finding the value of b and the common ratio based on these terms: The value of b Step 1: Set up the equation using the common ratio property (T_2)/(T_1) = (T_3)/(T_2). (5b+6)/(3b-2) = (3b+3)/(5b+6) Step 2: Cross-multiply. (5b+6)(5b+6) = (3b-2)(3b+3) (5b+6)^2 = (3b-2)(3b+3) Step 3: Expand both sides. 25b^2 + 60b + 36 = 9b^2 + 9b - 6b - 6 25b^2 + 60b + 36 = 9b^2 + 3b - 6 Step 4: Rearrange to form a quadratic equation. 16b^2 + 57b + 42 = 0 Step 5: Solve for b using the quadratic formula. b = -57 ± sqrt(57^2 - 4(16)(42))2(16) b = -57 ± sqrt(3249 - 2688)32 b = -57 ± sqrt(561)32 The value of b is -57 ± sqrt(561)32. The common ratio Using these values of b, the common ratio r is: r = 15 ± sqrt(561)56 My calculations were based on the terms as written in the image. 4 down, 1 left today — send the next one.