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
The formula for the n-th term of a Geometric Progression (GP) is T_n = ar^n-1, where a is the first term and r is the common ratio. Step 1: Set up equations for the first GP. Given: First term is a, common ratio is r, and the 6th term is 768. Using the formula T_n = ar^n-1: T_6 = ar^6-1 768 = ar^5 (*) Step 2: Set up equations for the second GP. Given: First term is a, common ratio is 6r, and the 3rd term is 3456. Using the formula T_n = ar^n-1: T_3 = a(6r)^3-1 3456 = a(6r)^2 3456 = a · 36r^2 (**) Step 3: Solve the system of equations for a and r. From equation (*), express a in terms of r: a = (768)/(r^5) Substitute this expression for a into equation (**): 3456 = ((768)/(r^5)) · 36r^2 3456 = (768 × 36)/(r^3) Multiply both sides by r^3: 3456r^3 = 768 × 36 3456r^3 = 27648 Divide by 3456: r^3 = (27648)/(3456) r^3 = 8 Step 4: Find the value of r. Take the cube root of both sides: r = [3]8 r = 2 Step 5: Substitute the value of r back into equation (*) to find a. 768 = a(2)^5 768 = a · 32 Divide by 32: a = (768)/(32) a = 24 The values are a = 24 and r = 2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.