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: Identify the given information for the geometric progression. Let the three terms of the geometric progression be (a)/(r), a, and ar, where a is the middle term and r is the common ratio. The sum of the first three terms is 26: (a)/(r) + a + ar = 26 (*) The product of the first three terms is 216: (a)/(r) · a · ar = 216 Step 2: Solve for a using the product equation. a^3 = 216 Take the cube root of both sides: a = [3]216 a = 6 Step 3: Substitute the value of a into the sum equation. Substitute a=6 into equation (*): (6)/(r) + 6 + 6r = 26 Step 4: Simplify and rearrange the equation to solve for r. Subtract 6 from both sides: (6)/(r) + 6r = 20 Multiply the entire equation by r to eliminate the fraction: 6 + 6r^2 = 20r Rearrange into a standard quadratic equation form Ar^2 + Br + C = 0: 6r^2 - 20r + 6 = 0 Divide the entire equation by 2 to simplify: 3r^2 - 10r + 3 = 0 Step 5: Solve the quadratic equation for r. Factor the quadratic equation: 3r^2 - 9r - r + 3 = 0 3r(r - 3) - 1(r - 3) = 0 (3r - 1)(r - 3) = 0 Set each factor to zero to find the possible values for r: 3r - 1 = 0 3r = 1 r = (1)/(3) r - 3 = 0 r = 3 The common ratio can be (1)/(3) or 3. The common ratio is (1)/(3) or 3. Step 1: Recall the difference of squares formula. The difference of squares formula states that x^2 - y^2 = (x - y)(x + y). Step 2: Substitute the given values into the formula. We are given: x^2 - y^2 = 12 x - y = 2 Substitute these values into the formula: (x - y)(x + y) = 12 (2)(x + y) = 12 Step 3: Solve for x + y. Divide both sides by 2: x + y = (12)/(2) x + y = 6 The value of x + y is 6. Send me the next one 📸