💡Okay, so I have this trigonometry problem here. It says that α is an angle in the first quadrant, and cosα is equal to 2√5 over 5. Then I need to find the value of cos2α minus cosα over sinα. The options are A: -13/5, B: -7/5, C: 13/5, D: 1/10.Alright, let's break this down step by step. First, I know that in the first quadrant, both sine and cosine are positive, so that's good to remember. The problem gives me cosα, which is 2√5/5. I need to find cos2α and then subtract cosα divided by sinα.Hmm, to find cos2α, I can use one of the double-angle formulas. The double-angle formula for cosine is cos2α = 2cos²α - 1. That seems straightforward since I already know cosα. Let me write that down:cos2α = 2*(2√5/5)² - 1.Wait, let me compute (2√5/5) squared. That would be (2√5)^2 over 5^2, which is 4*5 over 25, right? So that's 20/25, which simplifies to 4/5. So, cos2α = 2*(4/5) - 1. That's 8/5 - 1. Hmm, 8/5 is 1 and 3/5, so subtracting 1 gives me 3/5. So cos2α is 3/5.Okay, now I need to find cosα over sinα. I have cosα, which is 2√5/5, but I don't have sinα yet. I can find sinα using the Pythagorean identity, which is sin²α + cos²α = 1. So, sin²α = 1 - cos²α. Let me compute that.cos²α is (2√5/5)^2, which we already found to be 4/5. So sin²α = 1 - 4/5 = 1/5. Therefore, sinα is the square root of 1/5. Since α is in the first quadrant, sinα is positive, so sinα = √(1/5) = √5/5.Now, cosα over sinα is (2√5/5) divided by (√5/5). Dividing these fractions, the denominators are the same, so they cancel out, and we're left with 2√5 divided by √5. The √5 in the numerator and denominator cancel each other, leaving just 2. So cosα over sinα is 2.Alright, so now I have cos2α as 3/5 and cosα over sinα as 2. The problem asks for cos2α minus cosα over sinα, so that's 3/5 - 2.Wait, 3/5 minus 2. Hmm, 2 is the same as 10/5, so 3/5 - 10/5 is -7/5. So the result is -7/5.Let me double-check my steps to make sure I didn't make a mistake. Starting with cosα = 2√5/5, I found sinα correctly using the identity. Then, using the double-angle formula, I calculated cos2α as 3/5. Then, cosα over sinα simplified to 2. Subtracting 2 from 3/5 gives me -7/5, which is option B.I think that's correct. I don't see any errors in my calculations. So yeah, the answer should be B: -7/5.**Final Answer**boxed{B}

💡Okay, so Spot has a doghouse with a regular hexagonal base, each side is two yards long. He's tied to a vertex with a three-yard rope. There's also an overhang that creates a shaded region, which is a sector with a 90-degree angle and a radius of one yard. This sector is located at a vertex two vertices away from where he's tethered. I need to find the total area Spot can reach, including this overhang.First, let me visualize the setup. A regular hexagon has all sides equal and all internal angles equal to 120 degrees. Each side is two yards, so the distance between adjacent vertices is two yards.Spot is tied to one vertex with a three-yard rope. Since the rope is longer than the side of the hexagon, he can go around the adjacent vertices. The rope is three yards, and each side is two yards, so when he goes around the corner, he has some extra rope left.Now, without considering the overhang, Spot can graze in a circular area around the tethering point. But because the doghouse is a hexagon, parts of this circle will be blocked by the doghouse itself. So, the area he can cover is a combination of sectors around the hexagon.Let me break it down step by step.1. **Main Sector from Tethering Point:** Spot is tied to a vertex with a 3-yard rope. The side of the hexagon is 2 yards. So, when he moves around the adjacent vertices, the rope will wrap around the hexagon. The angle at each vertex of a regular hexagon is 120 degrees. So, when Spot moves from the tethering point, he can swing around 240 degrees (which is 360 minus 120) without the rope being obstructed by the doghouse. This forms a 240-degree sector with a radius of 3 yards. The area of this sector is (240/360) * π * (3)^2 = (2/3) * π * 9 = 6π square yards.2. **Additional Sectors Around Adjacent Vertices:** When Spot reaches an adjacent vertex, he can go around it with the remaining rope. Since the rope is 3 yards and each side is 2 yards, after going around one vertex, he has 1 yard of rope left (3 - 2 = 1). But wait, actually, when he goes around a vertex, the rope wraps around the corner, so the remaining rope is 3 - 2 = 1 yard, but he can swing around the next vertex as well. However, since the hexagon has internal angles of 120 degrees, when Spot goes around one vertex, he can swing around 60 degrees (since 180 - 120 = 60) with the remaining rope. So, for each adjacent vertex, he can cover a 60-degree sector with a radius of 1 yard. Since there are two adjacent vertices, he can cover two such sectors. The area of one 60-degree sector is (60/360) * π * (1)^2 = (1/6) * π * 1 = π/6 square yards. Therefore, two such sectors would be 2 * (π/6) = π/3 square yards.3. **Overhang Sector:** There's an overhang creating a shaded region, which is a 90-degree sector with a radius of 1 yard. This sector is located at a vertex two vertices away from the tethering point. So, Spot can also access this sector. The area of this sector is (90/360) * π * (1)^2 = (1/4) * π * 1 = π/4 square yards.4. **Total Area:** Now, adding up all the areas: - Main sector: 6π - Additional sectors: π/3 - Overhang sector: π/4 Total area = 6π + π/3 + π/4 To add these fractions, I need a common denominator. The denominators are 1, 3, and 4. The least common denominator is 12. Converting each term: - 6π = 72π/12 - π/3 = 4π/12 - π/4 = 3π/12 Adding them up: 72π/12 + 4π/12 + 3π/12 = 79π/12 Wait, that doesn't match the initial calculation. Let me check my steps again. Oh, I think I made a mistake in calculating the additional sectors. When Spot goes around the adjacent vertices, he has 3 yards of rope, and each side is 2 yards. So, after going around one vertex, he has 1 yard left, but he can swing around the next vertex as well. However, the angle he can cover around the second vertex might be different. Let me reconsider. When Spot is at the first adjacent vertex, he has 1 yard of rope left. The angle he can cover around that vertex is 60 degrees, as the internal angle is 120 degrees, so the external angle is 60 degrees. So, he can cover a 60-degree sector with a radius of 1 yard. But since the overhang is at the vertex two away, maybe he can also reach that vertex with the remaining rope. Wait, the overhang is a separate sector, so maybe I don't need to consider it as part of the main grazing area. It's an additional area he can access. So, perhaps the initial calculation was correct, and the total area is 6π + π/3 + π/4 = (72π + 4π + 3π)/12 = 79π/12. But in the initial thought process, the user got 89π/12. Hmm, maybe I missed something. Let me recalculate: - Main sector: 240 degrees with radius 3: (240/360)*π*9 = 6π - Two sectors of 60 degrees each with radius 1: 2*(60/360)*π*1 = 2*(π/6) = π/3 - Overhang sector: 90 degrees with radius 1: (90/360)*π*1 = π/4 Total: 6π + π/3 + π/4 Converting to twelfths: 6π = 72π/12 π/3 = 4π/12 π/4 = 3π/12 Total: 72π/12 + 4π/12 + 3π/12 = 79π/12 So, I think the correct total area is 79π/12. But the initial thought process had 22/3π + 1/4π = (88/12 + 3/12) = 91/12π, which is different. Wait, maybe I made a mistake in the initial sectors. Let me check the main sector again. If Spot is tied to a vertex with a 3-yard rope, and the hexagon has sides of 2 yards, when he moves around the adjacent vertices, the angle he can cover is 240 degrees, as the internal angle is 120 degrees, so the external angle is 60 degrees, but the total angle he can cover without obstruction is 360 - 120 = 240 degrees. So, the main sector is 240 degrees with radius 3: (240/360)*π*9 = 6π. Then, when he goes around each adjacent vertex, he can cover a 60-degree sector with the remaining rope. Since the rope is 3 yards, and each side is 2 yards, after going around one vertex, he has 1 yard left. So, he can cover a 60-degree sector with radius 1 yard at each adjacent vertex. So, two such sectors: 2*(60/360)*π*1 = π/3. Then, the overhang is a separate sector: 90 degrees with radius 1 yard: π/4. So, total area: 6π + π/3 + π/4 = 6π + (4π + 3π)/12 = 6π + 7π/12. Wait, 6π is 72π/12, plus 7π/12 is 79π/12. So, I think the correct answer is 79π/12. But in the initial thought process, the user got 89π/12, which is different. Maybe they considered the overhang as part of the main area. Let me check the overhang again. The overhang is a sector with a 90-degree angle and radius 1 yard, located at a vertex two vertices away from the tethering point. So, Spot is tied at vertex A. The overhang is at vertex C, which is two vertices away. So, can Spot reach vertex C with his 3-yard rope? The distance from A to C in a regular hexagon is 2 * side length * sin(60 degrees) = 2*2*(√3/2) = 2√3 ≈ 3.464 yards. But Spot's rope is only 3 yards, which is less than 3.464 yards. So, he cannot reach vertex C directly. However, the overhang is a sector at vertex C, which Spot can access because the overhang is adjacent to the doghouse. So, maybe the overhang is accessible through the doghouse structure, even though Spot can't reach vertex C directly. Alternatively, perhaps the overhang is connected in such a way that Spot can reach it by going around the doghouse. Wait, if Spot is tied at vertex A, and the overhang is at vertex C, which is two vertices away, maybe he can reach it by going around vertex B. Let me calculate the path. From A, he goes to B, which is 2 yards, then from B to C is another 2 yards, but his rope is 3 yards. So, from A to B is 2 yards, leaving 1 yard of rope. Then, from B, he can go 1 yard towards C, but C is 2 yards away from B. So, he can only reach 1 yard towards C, which is 1 yard from B, which is 3 yards from A. But the overhang is at C, which is 2 yards from B, so he can't reach it directly. However, the overhang is a sector with radius 1 yard, so maybe he can reach the edge of the overhang. Alternatively, perhaps the overhang is connected in such a way that Spot can access it without having to go all the way to C. This is getting a bit confusing. Maybe I should consider that the overhang is an additional area that Spot can access, regardless of whether he can reach vertex C directly. So, perhaps the overhang is a separate sector that Spot can access, and we just add its area to the total. Therefore, the total area would be the main sector (6π) plus the two smaller sectors (π/3) plus the overhang sector (π/4), totaling 6π + π/3 + π/4 = 79π/12. But the initial thought process had 89π/12, which is larger. Maybe I missed something. Wait, perhaps when Spot goes around the first vertex, he can cover more than just a 60-degree sector. Let me think again. When Spot is at vertex B, which is adjacent to A, he has 1 yard of rope left. The internal angle at B is 120 degrees, so the external angle is 60 degrees. So, he can swing around B for 60 degrees with the remaining 1 yard of rope. But also, from B, he can go towards C, but as we saw, he can't reach C. However, the overhang is at C, so maybe he can access part of the overhang by swinging around B. Alternatively, perhaps the overhang is connected in such a way that Spot can reach it by going around B and then extending into the overhang. But I'm not sure. Maybe the overhang is a separate area that Spot can access, so we just add its area regardless. Alternatively, perhaps the overhang is part of the main area, so we don't need to add it separately. Wait, the problem says the overhang creates a shaded region that Spot can also explore. So, it's an additional area he can access, in addition to the areas he can reach from the tethering point. Therefore, we should add the area of the overhang sector to the total area. So, the total area is 6π + π/3 + π/4 = 79π/12. But the initial thought process had 89π/12, which is different. Maybe I made a mistake in calculating the additional sectors. Let me recalculate the additional sectors. When Spot is at vertex B, he has 1 yard of rope left. The internal angle at B is 120 degrees, so the external angle is 60 degrees. Therefore, he can swing around B for 60 degrees with the remaining 1 yard of rope. So, the area is (60/360)*π*(1)^2 = π/6. Since there are two adjacent vertices (B and F, assuming the hexagon is labeled A, B, C, D, E, F), he can cover two such sectors, totaling 2*(π/6) = π/3. So, that part is correct. Then, the overhang is a separate sector: 90 degrees with radius 1 yard, so π/4. Therefore, total area is 6π + π/3 + π/4 = 79π/12. So, I think the correct answer is 79π/12. But the initial thought process had 89π/12, which is different. Maybe the user considered the overhang as part of the main area, but I think it's separate. Alternatively, maybe the overhang is connected in such a way that Spot can reach it by going around two vertices, but given the rope length, he can't reach vertex C directly. Therefore, I think the correct total area is 79π/12. Wait, but let me check the initial calculation again. The user had: - Main sector: 240 degrees with radius 3: 6π - Two sectors of 60 degrees each with radius 2: 2*(60/360)*π*4 = 4π/3 - Overhang sector: 90 degrees with radius 1: π/4 Total: 6π + 4π/3 + π/4 = (72π + 16π + 3π)/12 = 91π/12 Wait, that's different from both my calculations. So, the user considered the additional sectors as having radius 2 yards, but that doesn't make sense because after going around the first vertex, Spot only has 1 yard of rope left. So, I think the user made a mistake in considering the radius for the additional sectors. They should be radius 1 yard, not 2 yards. Therefore, the correct additional sectors are radius 1 yard, so the area is π/3, not 4π/3. Therefore, the total area should be 6π + π/3 + π/4 = 79π/12. So, I think the initial thought process had an error in calculating the additional sectors, considering the radius as 2 yards instead of 1 yard. Therefore, the correct total area is 79π/12. But wait, let me double-check the distance from A to C. In a regular hexagon, the distance between two vertices separated by one vertex (i.e., two vertices apart) is 2 * side length * sin(60 degrees) = 2*2*(√3/2) = 2√3 ≈ 3.464 yards. Since Spot's rope is 3 yards, which is less than 3.464 yards, he cannot reach vertex C directly. Therefore, the overhang at C is not reachable by Spot unless he can go around the doghouse. However, the overhang is adjacent to the doghouse, so maybe it's connected in such a way that Spot can access it by going around the doghouse. Alternatively, perhaps the overhang is a separate area that Spot can access through the doghouse structure, even though he can't reach vertex C directly. If that's the case, then the overhang sector is an additional area that Spot can access, so we should add its area to the total. Therefore, the total area is 6π + π/3 + π/4 = 79π/12. So, I think the correct answer is 79π/12. However, the initial thought process had 89π/12, which is different. I think the user made a mistake in considering the radius for the additional sectors as 2 yards instead of 1 yard. Therefore, the correct total area is 79π/12.

💡First, I'll denote the height of the first person as ( H ).Since the heights increase in order, the second person is ( H + 2 ) inches tall, the third person is ( H + 4 ) inches tall, and the fourth person is ( H + 10 ) inches tall.Given that the average height of the four people is 79 inches, I can set up the equation:[frac{H + (H + 2) + (H + 4) + (H + 10)}{4} = 79]Multiplying both sides by 4 to eliminate the denominator:[4H + 16 = 316]Subtracting 16 from both sides:[4H = 300]Dividing both sides by 4:[H = 75]Finally, the height of the fourth person is:[H + 10 = 75 + 10 = 85 text{ inches}]

💡Okay, so I have this geometry problem here. Let me read it again to make sure I understand.We have circle O with the equation x² + y² = 1. That's a standard circle centered at the origin with radius 1. Then there's a point A at (-2, 0). So, point A is outside the circle since the circle has a radius of 1 and A is at (-2, 0), which is two units to the left of the origin.Now, there's another point B with coordinates (b, 0), and b is not equal to -2. So, point B is somewhere on the x-axis, not overlapping with point A. There's also a constant λ (lambda) such that for any point M on circle O, the distance from M to B is λ times the distance from M to A. So, |MB| = λ|MA| for all M on the circle.The question is asking for λ - b. So, I need to find the values of λ and b that satisfy this condition and then subtract b from λ.Hmm, okay. Let me think about how to approach this. Since this condition must hold for any point M on the circle, it's a general condition. That suggests that the equation derived from |MB| = λ|MA| should hold true for all points (x, y) on the circle x² + y² = 1.Let me write down the equation |MB| = λ|MA| in terms of coordinates.Point M is (x, y) on the circle, so:|MB| = sqrt[(x - b)² + y²]|MA| = sqrt[(x + 2)² + y²]So, according to the condition:sqrt[(x - b)² + y²] = λ sqrt[(x + 2)² + y²]To eliminate the square roots, I can square both sides:(x - b)² + y² = λ²[(x + 2)² + y²]Now, let's expand both sides.Left side: (x - b)² + y² = x² - 2bx + b² + y²Right side: λ²[(x + 2)² + y²] = λ²(x² + 4x + 4 + y²)So, putting it all together:x² - 2bx + b² + y² = λ²x² + 4λ²x + 4λ² + λ²y²Now, since this equation must hold for all points (x, y) on the circle x² + y² = 1, we can substitute x² + y² = 1 into the equation. That might simplify things.Let me do that. Replace x² + y² with 1:1 - 2bx + b² = λ²(1) + 4λ²x + 4λ²Wait, hold on. Let me make sure I do that correctly.Original equation after expansion:x² - 2bx + b² + y² = λ²x² + 4λ²x + 4λ² + λ²y²Bring all terms to the left side:x² - 2bx + b² + y² - λ²x² - 4λ²x - 4λ² - λ²y² = 0Factor like terms:(1 - λ²)x² + (-2b - 4λ²)x + (1 - λ²)y² + (b² - 4λ²) = 0Hmm, this is a quadratic equation in x and y. Since this must hold for all (x, y) on the circle x² + y² = 1, the coefficients of x², y², x, and the constant term must satisfy certain conditions.In other words, for the equation to hold for all points on the circle, the coefficients of x² and y² must be equal, and the coefficients of x must be zero, and the constant term must adjust accordingly.Wait, actually, since x² + y² = 1, we can express y² as 1 - x². Maybe substituting that in would help.Let me try that. Replace y² with 1 - x² in the equation:(1 - λ²)x² + (-2b - 4λ²)x + (1 - λ²)(1 - x²) + (b² - 4λ²) = 0Let me expand this:(1 - λ²)x² + (-2b - 4λ²)x + (1 - λ²) - (1 - λ²)x² + (b² - 4λ²) = 0Now, let's combine like terms:The x² terms: (1 - λ²)x² - (1 - λ²)x² = 0The x term: (-2b - 4λ²)xThe constants: (1 - λ²) + (b² - 4λ²) = 1 - λ² + b² - 4λ² = 1 + b² - 5λ²So, the equation simplifies to:(-2b - 4λ²)x + (1 + b² - 5λ²) = 0Now, this equation must hold for all x on the circle. But x can vary between -1 and 1 because the circle has radius 1. For the equation to hold for all x, the coefficient of x must be zero, and the constant term must also be zero.So, we have two equations:1. Coefficient of x: -2b - 4λ² = 02. Constant term: 1 + b² - 5λ² = 0So, let's write them down:Equation 1: -2b - 4λ² = 0Equation 2: 1 + b² - 5λ² = 0Let me solve Equation 1 first.Equation 1: -2b - 4λ² = 0Let me rearrange it:-2b = 4λ²Divide both sides by -2:b = -2λ²Okay, so b is expressed in terms of λ. Now, let's substitute this into Equation 2.Equation 2: 1 + b² - 5λ² = 0Substitute b = -2λ²:1 + (-2λ²)² - 5λ² = 0Compute (-2λ²)²:(-2λ²)² = 4λ⁴So, the equation becomes:1 + 4λ⁴ - 5λ² = 0Let me write it as:4λ⁴ - 5λ² + 1 = 0Hmm, this is a quartic equation, but it's quadratic in terms of λ². Let me set u = λ². Then the equation becomes:4u² - 5u + 1 = 0Now, let's solve for u.Using quadratic formula:u = [5 ± sqrt(25 - 16)] / 8Because the quadratic is 4u² -5u +1, so a=4, b=-5, c=1.Discriminant D = (-5)² - 4*4*1 = 25 - 16 = 9So,u = [5 ± 3]/8So, two solutions:u = (5 + 3)/8 = 8/8 = 1u = (5 - 3)/8 = 2/8 = 1/4So, u = 1 or u = 1/4But u = λ², so λ² = 1 or λ² = 1/4Therefore, λ can be ±1 or ±1/2.But let's think about the context. Since λ is a scaling factor for distances, it should be positive. So, λ = 1 or λ = 1/2.Now, let's consider each case.Case 1: λ² = 1, so λ = 1Then from Equation 1: b = -2λ² = -2*1 = -2But wait, the problem states that b ≠ -2. So, this solution is invalid because it would make b = -2, which is the same as point A. So, we discard this solution.Case 2: λ² = 1/4, so λ = 1/2Then from Equation 1: b = -2*(1/4) = -2*(1/4) = -1/2So, b = -1/2 and λ = 1/2Perfect, this satisfies the condition that b ≠ -2.Therefore, we have found that λ = 1/2 and b = -1/2.So, the problem asks for λ - b.Compute λ - b:λ - b = (1/2) - (-1/2) = 1/2 + 1/2 = 1So, λ - b = 1Let me double-check my steps to make sure I didn't make any mistakes.1. I started by writing the distance equations and squared both sides to eliminate the square roots.2. Expanded both sides and then substituted x² + y² = 1 to simplify.3. Replaced y² with 1 - x² and simplified the equation, which led to a linear equation in x.4. Since this equation must hold for all x on the circle, the coefficients of x and the constant term must both be zero, giving two equations.5. Solved the first equation for b in terms of λ, substituted into the second equation, and solved the resulting quadratic in λ².6. Discarded the solution where b = -2 since it's not allowed, and took the other solution.7. Calculated λ - b and got 1.Everything seems to check out. I think the answer is correct.**Final Answer**boxed{1}

💡Okay, so I have this problem about a cone where the lateral surface is unfolded into a semicircle, and I need to find the apex angle of the cone. The options are 30°, 45°, 60°, or 90°. Hmm, let me try to visualize this.First, I remember that when you unfold the lateral surface of a cone, you get a sector of a circle. The radius of this sector is equal to the slant height (or generatrix) of the cone. In this case, it's unfolded into a semicircle, so the sector must be a semicircle. That means the arc length of the semicircle is equal to the circumference of the base of the cone.Let me denote the slant height (generatrix) of the cone as ( l ). Since it's a semicircle, the radius of the semicircle is ( l ), so the circumference of the full circle would be ( 2pi l ). But since it's a semicircle, the length of the arc is half of that, which is ( pi l ).Now, this arc length ( pi l ) should be equal to the circumference of the base of the cone. The circumference of the base is ( 2pi r ), where ( r ) is the radius of the base of the cone. So, setting them equal:[pi l = 2pi r]Dividing both sides by ( pi ):[l = 2r]So, the slant height ( l ) is twice the radius of the base ( r ). That gives me a relationship between ( l ) and ( r ).Now, I need to find the apex angle of the cone. The apex angle is the angle between two generatrices in the axial section of the cone. In other words, it's the angle at the tip of the cone when you cut it along its axis.To find this angle, I can consider the triangle formed by the two generatrices and the diameter of the base. This is an isosceles triangle with two sides equal to ( l ) and the base equal to ( 2r ).I can use the Law of Cosines to find the apex angle ( theta ). The Law of Cosines states that:[c^2 = a^2 + b^2 - 2ab cos C]In this case, the sides ( a ) and ( b ) are both ( l ), and the side ( c ) is ( 2r ). So plugging in:[(2r)^2 = l^2 + l^2 - 2l^2 cos theta]Simplifying:[4r^2 = 2l^2 - 2l^2 cos theta]Divide both sides by 2:[2r^2 = l^2 - l^2 cos theta]From earlier, we know that ( l = 2r ). Let's substitute ( l ) with ( 2r ):[2r^2 = (2r)^2 - (2r)^2 cos theta]Calculating ( (2r)^2 ):[2r^2 = 4r^2 - 4r^2 cos theta]Subtract ( 4r^2 ) from both sides:[2r^2 - 4r^2 = -4r^2 cos theta]Simplify:[-2r^2 = -4r^2 cos theta]Divide both sides by ( -2r^2 ):[1 = 2 cos theta]So,[cos theta = frac{1}{2}]Taking the arccosine of both sides:[theta = arccosleft(frac{1}{2}right)]I remember that ( arccosleft(frac{1}{2}right) ) is 60°, since ( cos 60° = 0.5 ).Therefore, the apex angle of the cone is 60°, which corresponds to option C.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the arc length, I correctly equated it to the base circumference. Then, using the Law of Cosines, I substituted ( l = 2r ) correctly. The algebra seems right, and the angle calculation leads to 60°, which is one of the options. So, I think my answer is correct.**Final Answer**boxed{C}

💡To determine the number of integer values of ( n ) between 1 and 493 inclusive for which the decimal representation of ( frac{n}{425} ) terminates, we start by examining the prime factorization of the denominator. The prime factors of 425 are ( 5^2 ) and 17, since ( 425 = 5^2 cdot 17 ). For the decimal representation of ( frac{n}{425} ) to terminate, the simplified form of the fraction must have a denominator with only the prime factors 2 and/or 5. This means that ( n ) must be such that it cancels out the prime factor 17 in the denominator. Therefore, ( n ) must be a multiple of 17.Next, we count how many multiples of 17 exist between 1 and 493. The smallest multiple of 17 in this range is 17 itself, and the largest is ( 17 times 29 = 493 ). This forms an arithmetic sequence: 17, 34, 51, ..., 493. To find the number of terms in this sequence, we use the formula for the ( n )-th term of an arithmetic sequence:[a_n = a_1 + (n-1)d]where ( a_1 = 17 ), ( d = 17 ), and ( a_n = 493 ). Plugging in these values:[493 = 17 + 17(n-1)]Solving for ( n ):[493 = 17 + 17n - 17 493 = 17n n = frac{493}{17} n = 29]Thus, there are 29 integer values of ( n ) that satisfy the condition.