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 "13" in the expression S_5 = 2(1-(13^5)) is a typo. Based on the context of a geometric progression and the subsequent correct calculation steps, the common ratio r is (1)/(3). Therefore, the term (13^5) should actually be ((1)/(3))^5. Let's correct the line and show the intended calculation: The formula for the sum of the first n terms of a geometric progression is: S_n = (a(1-r^n))/(1-r) Given a=2, r=(1)/(3), and n=5: S_5 = (2(1-(1)/(3))^5)1-(1)/(3) S_5 = (2(1-1^5)/(3^5))(3)/(3)-(1)/(3) S_5 = (2(1-1)/(243))(2)/(3) S_5 = (2(243-1)/(243))(2)/(3) S_5 = (2(242)/(243))(2)/(3) To divide by a fraction, multiply by its reciprocal: S_5 = 2 × (242)/(243) × (3)/(2) S_5 = (2 × 242 × 3)/(243 × 2) S_5 = (242 × 3)/(243) S_5 = (726)/(243) S_5 = (242)/(81) The final answer is (242)/(81).