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: Define the terms and formulas for a geometric series. Let the first term of the geometric series be a and the common ratio be r. The sum of an infinite geometric series is given by the formula: S_ = (a)/(1-r) for |r| < 1 The sum of the first two terms is a + ar. Step 2: Formulate equations from the given information. Given that the sum of the infinite geometric series is (16)/(3): (a)/(1-r) = (16)/(3) (1) Given that the sum of the first two terms is 5: a + ar = 5 a(1+r) = 5 (2) Step 3: Solve the system of equations for r. From equation (2), express a in terms of r: a = (5)/(1+r) Substitute this expression for a into equation (1): (5)/(1+r)1-r = (16)/(3) (5)/((1+r)(1-r)) = (16)/(3) Using the difference of squares formula, (1+r)(1-r) = 1-r^2: (5)/(1-r^2) = (16)/(3) Cross-multiply: 3 × 5 = 16 × (1-r^2) 15 = 16 - 16r^2 Rearrange the equation to solve for r^2: 16r^2 = 16 - 15 16r^2 = 1 r^2 = (1)/(16) Take the square root of both sides: r = ± sqrt((1)/(16)) r = ± (1)/(4) Both values, r = (1)/(4) and r = -(1)/(4), satisfy the condition |r| < 1 for an infinite geometric series. The common ratio of the series can be (1)/(4) or -(1)/(4). The final answer is ± (1)/(4). Send me the next one 📸