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
Welcome back yo — been a while! Let's pick up where you left off. Step 1: Understand the property of an Arithmetic Progression (A.P.). If three terms x, y, z are in A.P., then the middle term is the average of the other two, or equivalently, the common difference between consecutive terms is constant. This means y - x = z - y, which simplifies to 2y = x + z. Step 2: Apply the A.P. property to the given terms. Given that (1)/(a), (1)/(b), (1)/(c) are in A.P., we can write: 2 ((1)/(b)) = (1)/(a) + (1)/(c) Simplify the right side by finding a common denominator: (2)/(b) = (c+a)/(ac) Step 3: Express the common difference. The common difference, d, can be found by subtracting any term from its succeeding term. Let's use the first two terms: d = (1)/(b) - (1)/(a) Step 4: Substitute the relationship from Step 2 into the common difference expression and simplify. From Step 2, we have (2)/(b) = (c+a)/(ac). This means (1)/(b) = (c+a)/(2ac). Substitute this expression for (1)/(b) into the equation for d: d = (c+a)/(2ac) - (1)/(a) To subtract these fractions, find a common denominator, which is 2ac: d = (c+a)/(2ac) - (2c)/(2ac) Combine the numerators: d = (c+a-2c)/(2ac) d = (a-c)/(2ac) Thus, the common difference is (a-c)/(2ac). Send me the next one 📸