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
Step 1: Write the formulas for the sum of the first n terms of a geometric progression (GP). The sum of the first n terms of a GP is given by the formula: S_n = (a(1-r^n))/(1-r) where a is the first term and r is the common ratio. Step 2: Set up equations based on the given information. We are given that the sum of the first three terms (S_3) is 189: S_3 = (a(1-r^3))/(1-r) = 189 (*) We are also given that the sum of the first six terms (S_6) is 182: S_6 = (a(1-r^6))/(1-r) = 182 (**) Step 3: Rewrite the expression for S_6 using the difference of squares identity. We know that 1-r^6 = (1-r^3)(1+r^3). Substitute this into the equation for S_6: S_6 = (a(1-r^3)(1+r^3))/(1-r) = 182 This can be rearranged as: ((a(1-r^3))/(1-r))(1+r^3) = 182 Step 4: Substitute the value of S_3 into the rearranged equation for S_6. From equation (*), we know that (a(1-r^3))/(1-r) = 189. Substitute this into the equation from Step 3: 189(1+r^3) = 182 Step 5: Solve for r^3. Divide both sides by 189: 1+r^3 = (182)/(189) Subtract 1 from both sides: r^3 = (182)/(189) - 1 r^3 = (182 - 189)/(189) r^3 = (-7)/(189) Step 6: Simplify the fraction and solve for r. Both the numerator and the denominator are divisible by 7: r^3 = -(7 ÷ 7)/(189 ÷ 7) r^3 = -(1)/(27) Take the cube root of both sides to find r: r = [3]-(1)/(27) r = -(1)/(3) The common ratio is indeed r = -(1)/(3). The final answer is r = -(1)/(3).