answer:Note that [x^{2^n} - x^{-2^n} = frac{x^{2^{n + 1}} - 1}{x^{2^n}}.]Thus, begin{align*} sum_{n = 0}^infty frac{1}{x^{2^n} - x^{-2^n}} &= sum_{n = 0}^infty frac{x^{2^n}}{x^{2^{n + 1}} - 1} &= sum_{n = 0}^infty frac{x^{2^n}}{(x^{2^n} - 1)(x^{2^n} + 1)} &= sum_{n = 0}^infty frac{1}{x^{2^n} + 1} &= frac{1}{x - 1}, end{align*}where we have used the geometric series formula [frac{1}{1 - r} = 1 + r + r^2 + dotsb = sum_{k = 0}^infty r^k]for |r| < 1. Since x > 1, frac{1}{x - 1} = boxed{frac{1}{x - 1}}.The answer is: frac{1}{x-1}

answer:If each room holds a family of 3, then there are a total of 10 * 3 = 30 people in the hotel. If each person receives 2 towels, then the hotel hands out a total of 30 * 2 = 60 towels. 60 The answer is: 60

answer:The officer is eligible for a 20% discount, so the discounted price would be 85 - (20/100) * 85 = 68 The officer is also eligible for an additional 25% off the discounted price, so the final price would be 68 - (25/100) * 68 = 51 The officer has to pay 51 for the duty shoes 51 The answer is: 51

answer:First, we can simplify the inequality by subtracting 5.2 from both sides: 1.2n - 4.4 - 5.2 < 0 1.2n - 9.6 < 0 Next, we can divide both sides by 1.2: n - 8 < 0 n < 8 So, the positive integers that satisfy the inequality are 1, 2, 3, 4, 5, 6, and 7. The sum of these integers is 1 + 2 + 3 + 4 + 5 + 6 + 7 = boxed{28} The answer is: 28