answer:Proof: (1) When n=1, the left side = dfrac{1}{3}, the right side = dfrac{1}{3}. The equation holds true. (2) Assume when n=k, the equation holds true, that is dfrac{1}{1 times 3} + dfrac{1}{3 times 5} + ldots + dfrac{1}{(2k-1)(2k+1)} = dfrac{k}{2k+1}. Then, when n=k+1, the left side = dfrac{1}{1 times 3} + dfrac{1}{3 times 5} + ldots + dfrac{1}{(2k-1)(2k+1)} + dfrac{1}{(2k+1)(2k+3)} = dfrac{k}{2k+1} + dfrac{1}{(2k+1)(2k+3)} = dfrac{k+1}{2k+3}. This means that the equation also holds true when n=k+1. Based on (1) and (2), we know that the equation holds true for any n in mathbb{N}^*. boxed{text{End}}

answer:Given the problem statement, we can start by establishing the relationship between the unit prices of type A and type B labor tools. The unit price of type B labor tools is 4 yuan more expensive than that of type A. If we denote the unit price of type A labor tools as x yuan, then the unit price of type B labor tools can be expressed as (x + 4) yuan. The total cost for type A labor tools is given as 1000 yuan, and for type B labor tools, it's 2400 yuan. Furthermore, the quantity of type B tools purchased is twice the quantity of type A tools. This information allows us to set up a ratio comparing the total costs of type A and type B tools in relation to their quantities and unit prices. For type A tools, the total cost divided by the unit price gives us the quantity of type A tools purchased, which is frac{1000}{x}. Similarly, for type B tools, the total cost divided by the unit price gives us the quantity of type B tools purchased, which is frac{2400}{x + 4}. Since the quantity of type B tools is twice that of type A tools, we can set up the equation: [ frac{2400}{x + 4} = 2 times frac{1000}{x} ] This equation represents the relationship between the unit prices of type A and type B labor tools and their quantities, based on the given total costs and the fact that the unit price of type B is 4 yuan more than that of type A. Therefore, the fractional equation satisfied by x is: [ boxed{frac{2400}{x + 4} = 2 times frac{1000}{x}} ]

answer:To calculate the measure of the obtuse angle, which is given as 60% larger than a right angle: [ frac{60}{100} cdot 90^circ = 54^circ ] [ 90^circ + 54^circ = 144^circ ] This leaves a total of: [ 180^circ - 144^circ = 36^circ ] for the other two equal angles in the isosceles triangle. Since these angles are equal: [ frac{36^circ}{2} = boxed{18^circ} ]

answer:First, let's calculate the average cost per year for both the repaired shoes and the new shoes. For the repaired shoes: The cost to repair the shoes is 10.50, and they will last for 1 year. So the average cost per year is 10.50 / 1 year = 10.50 per year. For the new shoes: The cost to purchase the new shoes is 30.00, and they will last for 2 years. So the average cost per year is 30.00 / 2 years = 15.00 per year. Now, let's calculate the percentage increase from the cost of repairing the used shoes to the cost of the new shoes per year. The difference in cost per year is 15.00 (new shoes) - 10.50 (repaired shoes) = 4.50. To find the percentage increase, we divide the difference by the original cost (the cost of repairing the used shoes) and then multiply by 100 to get the percentage: Percentage increase = (4.50 / 10.50) * 100 = 0.42857 * 100 ≈ 42.86% So, the average cost per year of the new shoes is approximately boxed{42.86%} greater than the cost of repairing the used shoes.