💡Okay, so I have this problem here where I need to find the probability that the equation (4x^2 + m^2 y^2 = 1) represents an ellipse with foci on the y-axis. The number (m) is randomly selected from the interval ([1, 5]). The options given are A: (dfrac{1}{5}), B: (dfrac{1}{4}), C: (dfrac{3}{5}), and D: (dfrac{3}{4}).First, I need to recall what the standard form of an ellipse looks like. I remember that the standard equation of an ellipse is (dfrac{x^2}{a^2} + dfrac{y^2}{b^2} = 1), where (a) and (b) are the lengths of the semi-major and semi-minor axes. If the foci are on the y-axis, that means the major axis is along the y-axis, so (b > a).Looking at the given equation (4x^2 + m^2 y^2 = 1), I can rewrite it in standard form by dividing both sides by 1. That gives me (dfrac{x^2}{1/4} + dfrac{y^2}{1/m^2} = 1). So, comparing this with the standard form, (a^2 = 1/4) and (b^2 = 1/m^2).Since the foci are on the y-axis, the major axis is along the y-axis, which means (b > a). Therefore, (b^2 > a^2), so (1/m^2 > 1/4). To solve for (m), I can take reciprocals on both sides, remembering that this reverses the inequality if both sides are positive. So, (m^2 < 4). Taking square roots, we get (m < 2).But wait, (m) is selected from the interval ([1, 5]). So, (m) has to be greater than or equal to 1 and less than 2. Therefore, the range of (m) that satisfies the condition is ([1, 2)).Now, to find the probability, I need to calculate the length of this interval where the condition is satisfied and divide it by the total length of the interval from which (m) is selected.The length of the interval ([1, 2)) is (2 - 1 = 1). The total length of the interval ([1, 5]) is (5 - 1 = 4). So, the probability is (dfrac{1}{4}).Let me double-check my steps to make sure I didn't make any mistakes. I converted the given equation into the standard form of an ellipse, identified the conditions for the foci to be on the y-axis, solved the inequality, found the interval for (m), and then calculated the probability by dividing the lengths of the intervals. Everything seems to check out.Therefore, the probability is (dfrac{1}{4}), which corresponds to option B.**Final Answer**boxed{B}

💡Alright, so I have this quadratic equation: ( ax^2 + bx + c = 0 ). The problem says that one root is triple the other, and I need to find the relationship between the coefficients ( a ), ( b ), and ( c ). Hmm, okay, let's see.First, I remember that for a quadratic equation, the roots can be found using the quadratic formula, but maybe I can use Vieta's formulas here since they relate the coefficients directly to the sum and product of the roots. Vieta's formulas say that if the roots are ( alpha ) and ( beta ), then:1. ( alpha + beta = -frac{b}{a} )2. ( alpha beta = frac{c}{a} )So, if one root is triple the other, let's say ( beta = 3alpha ). That should help me set up some equations.Substituting ( beta = 3alpha ) into the first equation:( alpha + 3alpha = -frac{b}{a} )Simplifying that:( 4alpha = -frac{b}{a} )So, ( alpha = -frac{b}{4a} )Okay, now that I have ( alpha ), I can find ( beta ) since ( beta = 3alpha ):( beta = 3 times left( -frac{b}{4a} right) = -frac{3b}{4a} )Now, let's use the second Vieta's formula, which is the product of the roots:( alpha beta = frac{c}{a} )Substituting the values of ( alpha ) and ( beta ):( left( -frac{b}{4a} right) times left( -frac{3b}{4a} right) = frac{c}{a} )Multiplying the left side:First, multiply the numerators: ( (-b) times (-3b) = 3b^2 )Then, multiply the denominators: ( 4a times 4a = 16a^2 )So, the left side becomes ( frac{3b^2}{16a^2} )Setting that equal to ( frac{c}{a} ):( frac{3b^2}{16a^2} = frac{c}{a} )To solve for the relationship between ( a ), ( b ), and ( c ), I can cross-multiply to eliminate the denominators:Multiply both sides by ( 16a^2 ):( 3b^2 = 16a^2 times frac{c}{a} )Simplify the right side:( 16a^2 times frac{c}{a} = 16a times c = 16ac )So, now I have:( 3b^2 = 16ac )That seems to be the relationship. Let me check if I did everything correctly.Starting from the beginning: roots are ( alpha ) and ( 3alpha ). Sum is ( 4alpha = -b/a ), so ( alpha = -b/(4a) ). Then product is ( alpha times 3alpha = 3alpha^2 = c/a ). Substituting ( alpha ):( 3 times left( frac{b^2}{16a^2} right) = frac{c}{a} )Which simplifies to ( frac{3b^2}{16a^2} = frac{c}{a} ), and cross-multiplying gives ( 3b^2 = 16ac ). Yep, that looks right.So, the relationship is ( 3b^2 = 16ac ), which corresponds to option C.

💡Okay, so I have this problem about two concentric circles with radii 10 meters and 20 meters. An aardvark starts at point A on the outer circle and goes through several movements: halfway around the outer circle, moves inward to the smaller circle, runs the full circumference there, moves back outward, runs along a chord that subtends a 60-degree angle, and finally returns to point A. I need to calculate the total distance the aardvark travels.First, let me visualize the setup. There are two circles with the same center. The outer circle has a radius of 20 meters, and the inner one is 10 meters. The aardvark starts at point A on the outer circle.1. **Halfway around the outer circle**: So, the circumference of a circle is 2πr. For the outer circle, that's 2π*20 = 40π meters. Half of that would be 20π meters. So, the aardvark runs 20π meters along the outer circle.2. **Moves radially inward to the smaller circle**: Radially means along a straight line towards the center. Since the outer radius is 20 and the inner is 10, the distance here is 20 - 10 = 10 meters. So, that's 10 meters.3. **Runs a full circumference of the smaller circle**: The circumference of the smaller circle is 2π*10 = 20π meters. So, that's another 20π meters.4. **Moves radially outward back to the outer circle**: Again, moving from the inner circle to the outer, so that's another 10 meters.5. **Runs along a chord that subtends a 60-degree angle at the center**: Hmm, okay, so a chord is a straight line connecting two points on the circle. The length of a chord can be found using the formula: chord length = 2r sin(θ/2), where θ is the angle in radians. But here, the angle is given in degrees, 60 degrees. So, first, I need to convert that to radians? Wait, actually, the formula works with degrees as well if I remember correctly. Let me confirm: chord length = 2r sin(θ/2). So, θ is 60 degrees, r is 20 meters.So, chord length = 2*20*sin(60/2) = 40*sin(30 degrees). Sin(30 degrees) is 0.5, so 40*0.5 = 20 meters. So, that's 20 meters.6. **Finally returns to point A**: Wait, after running along the chord, does it need to return to point A? Or is the chord itself the path back to A? Let me think. The aardvark starts at A, goes halfway around the outer circle, moves inward, runs the smaller circle, moves outward, runs the chord, and then returns to A. So, after the chord, it needs to get back to A. But wait, if the chord subtends a 60-degree angle, does that mean the chord connects back to A? Or is it another point?Wait, maybe I misread. Let me check the problem again: "runs along a chord that subtends a 60-degree angle at the circle's center, and finally returns to point A." So, after running along the chord, it returns to A. So, does that mean the chord is from the current position back to A? Or is the chord a separate segment?Wait, perhaps I need to clarify the path. Let me break it down step by step.- Start at A on the outer circle.- Runs halfway around the outer circle: so from A, goes 180 degrees around the outer circle to the point directly opposite A, let's call it point B.- Then moves radially inward to the smaller circle: from B on the outer circle to the corresponding point on the inner circle, let's say point C.- Runs a full circumference of the smaller circle: so from C, goes all the way around the inner circle, ending up back at C.- Then moves radially outward back to the outer circle: from C to the corresponding point on the outer circle, which is point B again.- Runs along a chord that subtends a 60-degree angle at the center: so from point B, runs along a chord that subtends 60 degrees. Since the center is O, the chord would connect B to another point D on the outer circle such that angle BOD is 60 degrees.- Finally returns to point A.Wait, so after running the chord from B to D, it needs to return to A. So, is the return from D to A along another path? Or is the chord from B to A? But the chord subtends 60 degrees, so if A is 180 degrees from B, then the chord from B to A would subtend 180 degrees, which is not 60 degrees. So, perhaps the chord is from B to another point D, 60 degrees away from B, and then from D back to A.But that might complicate things. Alternatively, maybe the chord is from B to A, but that subtends 180 degrees, which contradicts the 60-degree angle. So, perhaps the chord is from B to another point, and then from there back to A.Wait, maybe I need to think about the entire path.Let me try to map the aardvark's journey:1. Start at A on the outer circle.2. Runs halfway around the outer circle: so 180 degrees, ending at point B, which is diametrically opposite A.3. Moves inward along the radius to the inner circle, ending at point C, which is directly below B on the inner circle.4. Runs the full circumference of the inner circle: so starting at C, goes all the way around the inner circle, ending back at C.5. Moves outward along the radius back to the outer circle, ending at point B again.6. Runs along a chord that subtends a 60-degree angle at the center: so from B, moves along a chord to a new point D on the outer circle, such that angle BOD is 60 degrees.7. Finally, returns to point A.Wait, so after step 6, the aardvark is at point D, and then needs to return to A. So, the last segment is from D back to A. But how? Is that another chord or along the circumference?Wait, the problem says "runs along a chord that subtends a 60-degree angle at the circle's center, and finally returns to point A." So, perhaps the chord itself is from D back to A, but that would mean the chord subtends a 60-degree angle. But the angle from D to A would depend on where D is.Alternatively, maybe the chord is from B to D, and then from D back to A along another path. But the problem doesn't specify, so perhaps I need to assume that after running the chord, it's already back at A. But that might not make sense because the chord subtends 60 degrees, not 180.Wait, maybe I'm overcomplicating. Let me try to think of each segment separately and calculate their lengths.1. Halfway around the outer circle: circumference is 2π*20 = 40π, so half is 20π.2. Radial inward: 20 - 10 = 10 meters.3. Full circumference of the smaller circle: 2π*10 = 20π.4. Radial outward: 10 meters.5. Chord subtending 60 degrees: chord length is 2*20*sin(60/2) = 40*sin(30) = 40*0.5 = 20 meters.6. Return to A: After the chord, it needs to get back to A. But where is it after the chord? If it started at B, ran a chord subtending 60 degrees, it would end up at a point D, which is 60 degrees from B along the circumference. Since B is 180 degrees from A, D would be 180 - 60 = 120 degrees from A. So, to get back to A, it would need to run another chord or along the circumference.But the problem says "finally returns to point A," so perhaps the last segment is along the circumference from D back to A, which would be 120 degrees. The length of that arc would be (120/360)*2π*20 = (1/3)*40π = 40π/3 ≈ 13.33π meters. But that's not one of the options, so maybe I'm misunderstanding.Alternatively, maybe the chord is from B to A, but that subtends 180 degrees, which contradicts the 60-degree angle. So, perhaps the chord is from B to another point, and then from there back to A via another chord or arc.Wait, maybe the chord is from B to a point D such that angle BOD is 60 degrees, and then from D back to A along another chord or arc. But the problem doesn't specify, so perhaps the total distance is just the sum of all the segments mentioned, without needing to return along another path.Wait, let me read the problem again: "runs along a chord that subtends a 60-degree angle at the circle's center, and finally returns to point A." So, perhaps the chord itself is the path from the current position back to A, but that would mean the chord subtends 60 degrees, which would mean A is 60 degrees from the current position. But the current position after moving outward from the inner circle is point B, which is 180 degrees from A. So, that doesn't add up.Alternatively, maybe the chord is from B to a point D, which is 60 degrees from B, and then from D back to A along another chord or arc. But the problem doesn't specify, so perhaps I'm overcomplicating.Wait, maybe the chord is from B to A, but that would be a diameter, subtending 180 degrees, which contradicts the 60-degree angle. So, perhaps the chord is from B to another point, and then the aardvark takes another path back to A.Wait, maybe the chord is the last segment before returning to A, so the aardvark runs the chord and then is already at A. But that would mean the chord is from the current position back to A, which would subtend 60 degrees, meaning the current position is 60 degrees from A. But after moving outward from the inner circle, the aardvark is at point B, which is 180 degrees from A. So, unless the chord is from B to a point 60 degrees from B towards A, which would place it at 120 degrees from A, and then from there back to A.But the problem doesn't specify, so perhaps I need to assume that the chord is the final segment that brings the aardvark back to A, subtending 60 degrees. So, the chord would be from the current position (point B) to a point D such that angle BOD is 60 degrees, and then from D to A. But that would require another segment.Wait, maybe I'm overcomplicating. Let me try to calculate each segment as per the problem statement, without assuming additional paths.So, the segments are:1. Half circumference of outer circle: 20π.2. Radial inward: 10 meters.3. Full circumference of inner circle: 20π.4. Radial outward: 10 meters.5. Chord subtending 60 degrees: 20 meters.6. Return to A: ?Wait, after the chord, the aardvark is at a new point, say D, and needs to return to A. So, how does it return? Is it along another chord or along the circumference?The problem says "finally returns to point A," but it doesn't specify the path. So, perhaps the return is along the straight line from D to A, which would be another chord. But the problem didn't mention that, so maybe I'm supposed to assume that the chord itself is the path back to A, but that would mean the chord subtends 60 degrees, which would mean A is 60 degrees from D, but D is 60 degrees from B, which is 180 degrees from A. So, the angle from A to D would be 180 - 60 = 120 degrees. So, the chord from D to A would subtend 120 degrees, which would have a length of 2*20*sin(60) = 40*(√3/2) = 20√3 meters. But that's not one of the options, so maybe that's not the case.Alternatively, maybe the aardvark doesn't need to return along another chord, but instead, the chord itself is the final segment that brings it back to A. But that would mean the chord subtends 60 degrees, so the angle from the current position to A is 60 degrees. But the current position is point B, which is 180 degrees from A, so that's not possible.Wait, maybe I'm misunderstanding the path. Let me try to think differently.After moving outward from the inner circle to point B, the aardvark runs along a chord that subtends 60 degrees. So, from B, it moves along a chord to a new point D such that angle BOD is 60 degrees. Then, from D, it needs to return to A. So, the path from D to A would be another segment.But the problem says "finally returns to point A," so perhaps the total path is:A -> halfway around outer circle -> B -> inward to C -> full inner circle -> back to B -> chord to D -> back to A.So, the chord from B to D is 20 meters, and then from D back to A. But how?If angle BOD is 60 degrees, then point D is 60 degrees from B. Since B is 180 degrees from A, point D is 180 - 60 = 120 degrees from A. So, the angle AOD is 120 degrees. Therefore, the chord from D to A would subtend 120 degrees, and its length would be 2*20*sin(60) = 40*(√3/2) = 20√3 meters.But 20√3 is approximately 34.64 meters, which isn't one of the options. The options are all in terms of π and integers, so maybe I'm overcomplicating.Alternatively, perhaps the chord from B to D is 60 degrees, and then the aardvark takes the shorter arc from D back to A, which would be 120 degrees. The length of that arc would be (120/360)*2π*20 = (1/3)*40π = 40π/3 ≈ 13.33π meters. But again, that's not an option.Wait, maybe the chord is the only segment after moving outward, and the aardvark doesn't need to return along another path because it's already back at A. But that doesn't make sense because the chord subtends 60 degrees, not 180.Wait, perhaps the chord is from B to A, but that's a diameter, subtending 180 degrees, which contradicts the 60-degree angle. So, maybe the chord is from B to a point D, 60 degrees away, and then from D back to A along the circumference, which would be 120 degrees. But again, that's not one of the options.Wait, maybe the problem is simpler. Let me just calculate each segment as described, without worrying about the return path, because the problem says "finally returns to point A," but doesn't specify the path, so perhaps the chord is the last segment that brings it back to A, and the total distance is just the sum of all the segments mentioned.So, let's list them:1. Half circumference of outer circle: 20π.2. Radial inward: 10 meters.3. Full circumference of inner circle: 20π.4. Radial outward: 10 meters.5. Chord subtending 60 degrees: 20 meters.6. Return to A: ?Wait, but if the chord is the last segment, then the aardvark is already at A, so maybe the return is included in the chord. But that would mean the chord is from the current position back to A, which would subtend 60 degrees, but the current position is point B, which is 180 degrees from A, so that's not possible.Alternatively, maybe the chord is from B to a point D, 60 degrees away, and then the aardvark takes the straight line from D back to A, which would be another chord. But that would add another segment, which isn't mentioned.Wait, maybe the problem is that after running the chord, the aardvark is at a point that allows it to return to A via another path, but the problem doesn't specify, so perhaps I'm supposed to assume that the chord is the final segment that brings it back to A, and the total distance is just the sum of all the segments up to that point.But that would mean the aardvark is at A after the chord, which would require the chord to be from the current position back to A, subtending 60 degrees, but that's not possible because the current position is 180 degrees from A.Wait, maybe I'm overcomplicating. Let me try to think of the total distance as the sum of all the segments mentioned, without worrying about the return path, because the problem says "finally returns to point A," but doesn't specify the path, so perhaps the total distance is just the sum of the segments up to the chord.But that would leave the aardvark at a new point, not back at A, which contradicts the problem statement.Wait, maybe the chord is the path from the current position back to A, and the angle subtended is 60 degrees, so the chord length is 20 meters, as calculated earlier. So, the total distance would be:20π (half outer circle) + 10 (inward) + 20π (inner circle) + 10 (outward) + 20 (chord) + ?Wait, but after the chord, it's already back at A, so the total distance is 20π + 10 + 20π + 10 + 20 = 40π + 40 meters.Wait, that's one of the options, option B: 40π + 40.But wait, does that make sense? Because after the chord, the aardvark is back at A, so the total distance is just the sum of all those segments.But let me double-check:1. Half outer circle: 20π.2. Inward: 10.3. Inner circle: 20π.4. Outward: 10.5. Chord: 20.Total: 20π + 10 + 20π + 10 + 20 = 40π + 40.Yes, that seems to be the case. So, the total distance is 40π + 40 meters, which is option B.I think I was overcomplicating earlier by trying to figure out the return path, but the problem states that after the chord, it returns to A, so the chord must be the final segment that brings it back, even though geometrically it might not make sense. But since the problem says it returns to A after the chord, we can assume that the total distance is the sum of all the segments mentioned, including the chord, which brings it back to A.So, the answer should be 40π + 40 meters, which is option B.

💡To determine the probability that the product of two distinct numbers selected from the set ({1, 2, 3, 4, 5, 6, 7}) is even, we can use complementary counting.1. **Total Possible Combinations**: The total number of ways to choose 2 distinct numbers from the set is given by the binomial coefficient: [ binom{7}{2} = 21 ]2. **Odd Product Calculation**: For the product to be odd, both numbers must be odd. The set contains four odd numbers: ({1, 3, 5, 7}). The number of ways to choose 2 odd numbers is: [ binom{4}{2} = 6 ]3. **Probability of Odd Product**: The probability that both numbers are odd is: [ frac{6}{21} = frac{2}{7} ]4. **Probability of Even Product**: Using complementary counting, the probability that the product is even is: [ 1 - frac{2}{7} = frac{5}{7} ]Thus, the probability that the product of the two selected numbers is even is (boxed{frac{5}{7}}).

💡Okay, so I need to show that 29 divides (3a + 2b) if and only if 29 divides (11a + 17b), where (a) and (b) are integers. Hmm, this seems like a problem involving modular arithmetic. Let me break it down step by step.First, I know that if 29 divides a number, that number is congruent to 0 modulo 29. So, I can rephrase the problem as: (3a + 2b equiv 0 pmod{29}) if and only if (11a + 17b equiv 0 pmod{29}).To tackle this, I think I need to find a relationship between the two expressions (3a + 2b) and (11a + 17b). Maybe I can express one in terms of the other modulo 29. Let me try to find a multiplier that can convert (3a + 2b) into (11a + 17b) modulo 29.Let’s assume there exists some integer (k) such that:[k(3a + 2b) equiv 11a + 17b pmod{29}]Expanding this, we get:[3k a + 2k b equiv 11a + 17b pmod{29}]This should hold for all integers (a) and (b), so the coefficients of (a) and (b) must be congruent modulo 29. Therefore:[3k equiv 11 pmod{29}]and[2k equiv 17 pmod{29}]Now, I need to solve for (k) in these two congruences.Starting with the first equation:[3k equiv 11 pmod{29}]To solve for (k), I need the multiplicative inverse of 3 modulo 29. Let me find an integer (x) such that (3x equiv 1 pmod{29}).Trying out values:- (3 times 10 = 30 equiv 1 pmod{29})So, the inverse of 3 modulo 29 is 10.Multiplying both sides of (3k equiv 11 pmod{29}) by 10:[k equiv 11 times 10 pmod{29}][k equiv 110 pmod{29}]Calculating (110 div 29):- (29 times 3 = 87)- (110 - 87 = 23)So, (110 equiv 23 pmod{29}), which means (k equiv 23 pmod{29}).Now, let's check the second congruence:[2k equiv 17 pmod{29}]Substituting (k = 23):[2 times 23 = 46][46 div 29 = 1) with a remainder of 17, so (46 equiv 17 pmod{29})]Perfect, it satisfies the second equation as well. So, (k = 23) works.This means:[23(3a + 2b) equiv 11a + 17b pmod{29}]Therefore, if (3a + 2b equiv 0 pmod{29}), then multiplying both sides by 23 gives:[11a + 17b equiv 0 pmod{29}]Conversely, if (11a + 17b equiv 0 pmod{29}), since 23 has an inverse modulo 29 (which we found earlier as 10), we can multiply both sides by 10 to get:[3a + 2b equiv 0 pmod{29}]Thus, we've shown both directions, proving that 29 divides (3a + 2b) if and only if it divides (11a + 17b).I think that covers it. I started by assuming a multiplier (k) to relate the two expressions, found that (k = 23) works, and then used the inverse to show the converse. This seems solid, but let me double-check my steps to be sure.First, finding the inverse of 3 modulo 29: 3 × 10 = 30 ≡ 1 mod29, correct.Then, solving for (k) in 3k ≡11 mod29: k ≡11×10=110≡23 mod29, correct.Checking 2k ≡17 mod29 with k=23: 2×23=46≡17 mod29, correct.Therefore, the logic holds, and the proof is complete.

💡Alright, so I have this sequence problem here. The sequence is defined by ( a_1 = 1 ) and the recursive formula ( n a_{n+1} = (n+1) a_n ). I need to find the general term formula for ( a_n ) and choose the correct option among A, B, C, or D.First, let me try to understand what the recursive formula is telling me. It says that each term ( a_{n+1} ) is related to the previous term ( a_n ) by the equation ( n a_{n+1} = (n+1) a_n ). If I rearrange this equation to solve for ( a_{n+1} ), I get:[a_{n+1} = frac{n+1}{n} a_n]So, this tells me that each term is a multiple of the previous term, where the multiplier is ( frac{n+1}{n} ). That seems like a ratio that changes with each term, but maybe I can find a pattern or express it in a more manageable form.Given that ( a_1 = 1 ), I can try to compute the first few terms to see if a pattern emerges.Let's compute ( a_2 ):[a_2 = frac{2}{1} a_1 = 2 times 1 = 2]Now, ( a_3 ):[a_3 = frac{3}{2} a_2 = frac{3}{2} times 2 = 3]Next, ( a_4 ):[a_4 = frac{4}{3} a_3 = frac{4}{3} times 3 = 4]Hmm, I see a pattern here. Each term seems to be equal to its index. So, ( a_1 = 1 ), ( a_2 = 2 ), ( a_3 = 3 ), ( a_4 = 4 ), and so on. This suggests that ( a_n = n ) for all ( n ).But wait, let me make sure this isn't just a coincidence. Maybe I should test it for another term. Let's compute ( a_5 ):[a_5 = frac{5}{4} a_4 = frac{5}{4} times 4 = 5]Yep, it's 5. So, the pattern continues. It seems like each term is simply equal to its position in the sequence. That would mean ( a_n = n ).But just to be thorough, maybe I can approach this more formally. The recursive formula is:[a_{n+1} = frac{n+1}{n} a_n]This is a first-order linear recurrence relation. To solve it, I can use the method of iteration or telescoping products. Let's try telescoping.Starting from ( a_1 ), I can express ( a_n ) as a product of the ratios:[a_n = a_1 times prod_{k=1}^{n-1} frac{a_{k+1}}{a_k}]Substituting the recursive formula into the product:[a_n = 1 times prod_{k=1}^{n-1} frac{k+1}{k}]Let's write out the product:[prod_{k=1}^{n-1} frac{k+1}{k} = frac{2}{1} times frac{3}{2} times frac{4}{3} times ldots times frac{n}{n-1}]I notice that in this product, each numerator cancels with the denominator of the next fraction. So, all the intermediate terms cancel out, leaving only the first denominator and the last numerator:[frac{2}{1} times frac{3}{2} times frac{4}{3} times ldots times frac{n}{n-1} = frac{n}{1} = n]Therefore, ( a_n = n ). This confirms the pattern I observed earlier.Looking back at the options provided:A: ( a_n = frac{1}{n} ) B: ( a_n = 2^n - 1 ) C: ( a_n = n ) D: ( a_n = frac{n+1}{2n} )The correct answer is clearly option C: ( a_n = n ).Just to ensure there's no mistake, let me verify with the original recursive formula. If ( a_n = n ), then:[n a_{n+1} = n times (n+1) = n(n+1)][(n+1) a_n = (n+1) times n = n(n+1)]Both sides are equal, so the formula satisfies the recursive relation. Additionally, ( a_1 = 1 ) holds true as given. Therefore, I'm confident that option C is correct.**Final Answer**boxed{C}