🧠:<think>Okay, so I'm trying to solve this problem about the function ( f(x) = sum_{n=1}^{infty} frac{cos(nx)}{n^p} ). The philosopher is using this function to model the convergence of scientific and metaphysical ideas. The parameter ( p ) bridges the two realms, with ( p=2 ) representing the scientific and ( p=3 ) the metaphysical. First, I need to figure out the conditions on ( x ) under which the series converges absolutely for both ( p=2 ) and ( p=3 ). Then, I have to discuss how these conditions reflect the philosophical exploration of infinity and convergence. Alright, starting with part 1. Absolute convergence means that the series of absolute values converges. So, for ( f(x) ), the absolute series would be ( sum_{n=1}^{infty} left| frac{cos(nx)}{n^p} right| ). Since ( |cos(nx)| leq 1 ), each term is bounded by ( frac{1}{n^p} ). So, the absolute series is dominated by ( sum_{n=1}^{infty} frac{1}{n^p} ), which is a p-series. I remember that a p-series converges if and only if ( p > 1 ). Given that ( p=2 ) and ( p=3 ) are both greater than 1, the series ( sum_{n=1}^{infty} frac{1}{n^p} ) converges for both cases. Therefore, by the comparison test, the absolute series ( sum_{n=1}^{infty} left| frac{cos(nx)}{n^p} right| ) also converges for both ( p=2 ) and ( p=3 ), regardless of the value of ( x ). Wait, but hold on. The convergence of the series ( f(x) ) itself is different from its absolute convergence. For conditional convergence, we might have different conditions, but since the question specifically asks about absolute convergence, I think my initial thought is correct. So, for absolute convergence, the series ( f(x) ) converges absolutely for all real numbers ( x ) when ( p=2 ) and ( p=3 ). But let me double-check. The function ( cos(nx) ) oscillates between -1 and 1, so taking absolute value makes it between 0 and 1. So, the absolute series is always less than or equal to a convergent p-series. Hence, regardless of ( x ), the series converges absolutely. Therefore, the conditions on ( x ) are that it can be any real number. There are no restrictions on ( x ) for absolute convergence when ( p=2 ) or ( p=3 ). Now, reflecting on the philosophical exploration of infinity and convergence. Absolute convergence here suggests that no matter the context ( x ), the series converges. This might symbolize that the convergence of scientific and metaphysical ideas is robust and not dependent on specific contexts, which could be seen as a harmonious integration of both perspectives. The parameter ( p ) acts as a bridge, ensuring that the convergence is stable regardless of the philosophical context. Moving on to part 2. I need to find the value of ( f(x) ) when ( x = frac{pi}{2} ) and ( p ) takes on prime values less than 10. The prime numbers less than 10 are 2, 3, 5, and 7. So, I need to compute ( fleft( frac{pi}{2} right) ) for ( p=2, 3, 5, 7 ).First, let's recall that ( f(x) = sum_{n=1}^{infty} frac{cos(nx)}{n^p} ). So, substituting ( x = frac{pi}{2} ), we get:( fleft( frac{pi}{2} right) = sum_{n=1}^{infty} frac{cosleft( n cdot frac{pi}{2} right)}{n^p} ).Now, let's analyze the cosine term. ( cosleft( frac{npi}{2} right) ) cycles through 0, -1, 0, 1, etc., depending on the value of ( n ). Specifically:- When ( n equiv 0 mod 4 ), ( cosleft( frac{npi}{2} right) = 1 ).- When ( n equiv 1 mod 4 ), ( cosleft( frac{npi}{2} right) = 0 ).- When ( n equiv 2 mod 4 ), ( cosleft( frac{npi}{2} right) = -1 ).- When ( n equiv 3 mod 4 ), ( cosleft( frac{npi}{2} right) = 0 ).So, every even ( n ) will contribute either 1 or -1, depending on whether ( n ) is a multiple of 4 or 2 mod 4. Odd ( n ) will contribute 0. Therefore, we can rewrite the series as:( fleft( frac{pi}{2} right) = sum_{k=1}^{infty} frac{cosleft( frac{(2k)pi}{2} right)}{(2k)^p} + sum_{k=1}^{infty} frac{cosleft( frac{(2k-1)pi}{2} right)}{(2k-1)^p} ).But since the odd terms are zero, the second sum is zero. So, we have:( fleft( frac{pi}{2} right) = sum_{k=1}^{infty} frac{cos(kpi)}{(2k)^p} ).But ( cos(kpi) = (-1)^k ). So, substituting that in:( fleft( frac{pi}{2} right) = sum_{k=1}^{infty} frac{(-1)^k}{(2k)^p} = frac{1}{2^p} sum_{k=1}^{infty} frac{(-1)^k}{k^p} ).I recognize that ( sum_{k=1}^{infty} frac{(-1)^k}{k^p} ) is the Dirichlet eta function ( eta(p) ), which is related to the Riemann zeta function ( zeta(p) ) by the formula:( eta(p) = (1 - 2^{1 - p}) zeta(p) ).Therefore, substituting back:( fleft( frac{pi}{2} right) = frac{1}{2^p} (1 - 2^{1 - p}) zeta(p) ).Simplify this expression:( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - p}}{2^p} zeta(p) = left( frac{1}{2^p} - frac{2^{1 - p}}{2^p} right) zeta(p) = left( frac{1}{2^p} - frac{1}{2^{2p - 1}} right) zeta(p) ).Wait, let me double-check that algebra. Starting from:( frac{1 - 2^{1 - p}}{2^p} = frac{1}{2^p} - frac{2^{1 - p}}{2^p} = frac{1}{2^p} - frac{2^{1 - p}}{2^p} = frac{1}{2^p} - frac{1}{2^{p - 1}} ).Wait, that doesn't seem right. Let me compute ( frac{2^{1 - p}}{2^p} ):( frac{2^{1 - p}}{2^p} = 2^{1 - p - p} = 2^{1 - 2p} ). So, actually:( fleft( frac{pi}{2} right) = frac{1}{2^p} - 2^{1 - 2p} ) multiplied by ( zeta(p) ).Wait, no, that's not correct. Let me go back.We have:( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - p}}{2^p} zeta(p) ).Let me factor out ( 2^{-p} ):( fleft( frac{pi}{2} right) = 2^{-p} (1 - 2^{1 - p}) zeta(p) ).Alternatively, ( 1 - 2^{1 - p} = 1 - frac{2}{2^p} = 1 - 2^{1 - p} ). So, perhaps it's better to leave it as is.So, ( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - p}}{2^p} zeta(p) ).Alternatively, factor out ( 2^{-p} ):( fleft( frac{pi}{2} right) = 2^{-p} (1 - 2^{1 - p}) zeta(p) = (2^{-p} - 2^{-p + 1 - p}) zeta(p) ).Wait, that seems messy. Maybe it's better to just write it as ( frac{1 - 2^{1 - p}}{2^p} zeta(p) ).Alternatively, note that ( 1 - 2^{1 - p} = 1 - frac{2}{2^p} = 1 - 2^{1 - p} ). So, perhaps leave it as ( frac{1 - 2^{1 - p}}{2^p} zeta(p) ).Alternatively, factor out ( 2^{-p} ):( fleft( frac{pi}{2} right) = 2^{-p} (1 - 2^{1 - p}) zeta(p) = (2^{-p} - 2^{-p + 1 - p}) zeta(p) = (2^{-p} - 2^{1 - 2p}) zeta(p) ).But I think the expression ( frac{1 - 2^{1 - p}}{2^p} zeta(p) ) is the simplest.So, now, for each prime ( p ) less than 10, which are 2, 3, 5, 7, we can compute ( fleft( frac{pi}{2} right) ).Let's compute each case:1. For ( p=2 ):( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - 2}}{2^2} zeta(2) = frac{1 - 2^{-1}}{4} zeta(2) = frac{1 - 1/2}{4} zeta(2) = frac{1/2}{4} zeta(2) = frac{1}{8} zeta(2) ).We know that ( zeta(2) = frac{pi^2}{6} ), so:( fleft( frac{pi}{2} right) = frac{1}{8} cdot frac{pi^2}{6} = frac{pi^2}{48} ).2. For ( p=3 ):( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - 3}}{2^3} zeta(3) = frac{1 - 2^{-2}}{8} zeta(3) = frac{1 - 1/4}{8} zeta(3) = frac{3/4}{8} zeta(3) = frac{3}{32} zeta(3) ).( zeta(3) ) is known as Apery's constant, approximately 1.2020569...3. For ( p=5 ):( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - 5}}{2^5} zeta(5) = frac{1 - 2^{-4}}{32} zeta(5) = frac{1 - 1/16}{32} zeta(5) = frac{15/16}{32} zeta(5) = frac{15}{512} zeta(5) ).4. For ( p=7 ):( fleft( frac{pi}{2} right) = frac{1 - 2^{1 - 7}}{2^7} zeta(7) = frac{1 - 2^{-6}}{128} zeta(7) = frac{1 - 1/64}{128} zeta(7) = frac{63/64}{128} zeta(7) = frac{63}{8192} zeta(7) ).So, summarizing:- ( p=2 ): ( frac{pi^2}{48} )- ( p=3 ): ( frac{3}{32} zeta(3) )- ( p=5 ): ( frac{15}{512} zeta(5) )- ( p=7 ): ( frac{63}{8192} zeta(7) )Now, interpreting the significance of these results in the context of the philosopher's exploration. Prime numbers are considered fundamental yet elusive in mathematics, often seen as the building blocks of the number system. In the context of the function ( f(x) ), using prime values for ( p ) might symbolize the exploration of fundamental aspects of reality, where each prime represents a unique perspective or layer of metaphysical thought. The results show that as ( p ) increases (taking on larger prime values), the coefficient in front of ( zeta(p) ) decreases, but ( zeta(p) ) itself also decreases as ( p ) increases. However, the overall behavior depends on the interplay between the coefficient and the zeta function. For ( p=2 ), the result is a neat expression involving ( pi^2 ), which is a well-known constant in mathematics and physics, symbolizing the harmony between different fields. For ( p=3 ), we get a multiple of Apery's constant, which is less well-known but still significant in number theory. As ( p ) increases to 5 and 7, the expressions become more complex, involving higher zeta values which are less understood and more abstract. This might reflect the philosopher's journey from more concrete, scientifically grounded ideas (like ( p=2 ) with its connection to ( pi )) to more abstract, metaphysical explorations (like ( p=7 ) with its higher zeta function). The decreasing coefficients could symbolize the diminishing influence or clarity as one delves deeper into metaphysical realms, while the increasing complexity of the zeta functions might represent the richer, more intricate nature of metaphysical thought. In summary, the function ( f(x) ) at ( x = frac{pi}{2} ) with prime ( p ) values less than 10 yields results that transition from concrete mathematical constants to more abstract and less understood values, mirroring the philosopher's exploration from scientific to metaphysical concepts.</think>

🧠:<think>Alright, so I've got these two problems to solve, both related to emergency response during a multi-vehicle accident. Let me take them one at a time.Starting with the first problem: It involves a firefighter, a paramedic, and a trauma nurse working together to stabilize patients. Each has their own rate of stabilizing patients. The firefighter can stabilize one every 15 minutes, the paramedic every 10 minutes, and the trauma nurse every 8 minutes. They work independently but simultaneously, and I need to find the average time taken to stabilize one patient by the team working together.Hmm, okay. So, this seems like a classic combined work rate problem. I remember that when people work together, their rates add up. So, if I can find each person's rate of stabilizing patients per minute, then add them together, I can find the combined rate. Then, the average time per patient would be the reciprocal of that combined rate.Let me write that down. The firefighter's rate is 1 patient per 15 minutes, so that's 1/15 per minute. The paramedic's rate is 1/10 per minute, and the trauma nurse's rate is 1/8 per minute. So, adding these together:Firefighter: 1/15 per minuteParamedic: 1/10 per minuteTrauma Nurse: 1/8 per minuteCombined rate = 1/15 + 1/10 + 1/8I need to compute this sum. To add these fractions, I need a common denominator. Let's see, 15, 10, and 8. The least common multiple of these numbers is... 120, I think.So, converting each fraction:1/15 = 8/1201/10 = 12/1201/8 = 15/120Adding them together: 8 + 12 + 15 = 35, so 35/120 per minute.Simplify that: 35/120 can be reduced. Both numerator and denominator are divisible by 5. So, 7/24 per minute.So, the combined rate is 7/24 patients per minute. Therefore, the time per patient is the reciprocal of that, which is 24/7 minutes per patient.Calculating that, 24 divided by 7 is approximately 3.42857 minutes, which is about 3 minutes and 25.7 seconds. But since the question asks for the average time, I can express it as a fraction.So, 24/7 minutes is the exact value. Let me check if that makes sense. Each individual has a rate, and together they should be faster than the fastest individual, which is the trauma nurse at 8 minutes per patient. Since 24/7 is approximately 3.428 minutes, which is indeed faster than 8 minutes, that seems reasonable.Wait, hold on. Actually, 24/7 is about 3.428 minutes, which is faster than each individual's time. But that seems counterintuitive because if each person is working on different patients, the time to stabilize one patient should be less than the slowest individual, but more than the fastest. Wait, no, actually, when they work together, the combined rate is additive, so the time per patient is less than the time it takes the fastest individual.Wait, no, that's not right. Because if they're each stabilizing different patients, the time to stabilize one patient is actually determined by the slowest person, but since they can work on different patients simultaneously, the overall rate is additive. So, the time per patient is determined by the combined rate.Wait, maybe I'm confusing myself. Let me think again.If the firefighter can do 1 patient every 15 minutes, that's a rate of 1/15 per minute. Similarly, the paramedic is 1/10, and the trauma nurse is 1/8. So, together, they can stabilize 1/15 + 1/10 + 1/8 patients per minute. So, the total rate is 7/24 per minute, meaning that every minute, they can stabilize 7/24 of a patient. Therefore, the time to stabilize one patient is 1 divided by (7/24), which is 24/7 minutes, or approximately 3.428 minutes.But wait, that seems too fast because each person is working on their own patient. So, if they all start at the same time, the time taken to stabilize one patient would actually be the minimum of their individual times? No, that doesn't make sense because they are working on different patients.Wait, perhaps I need to model this differently. Maybe it's not about the time to stabilize one patient, but the time until the first patient is stabilized. But the question says, "the average time taken to stabilize one patient by the team working together." Hmm.Alternatively, maybe it's about the expected time until one patient is stabilized, considering all three working simultaneously. But that might be a different approach.Wait, perhaps I should think in terms of Poisson processes or something. If each person is stabilizing patients at their own rate, the combined process would have a rate equal to the sum of their individual rates. Therefore, the time until the first patient is stabilized would follow an exponential distribution with rate equal to the sum of their individual rates.But the question is asking for the average time taken to stabilize one patient, so that would be the expected value of the exponential distribution, which is 1 divided by the total rate.So, in that case, the total rate is 7/24 per minute, so the expected time is 24/7 minutes, which is approximately 3.428 minutes.But wait, that seems conflicting with my initial thought. Let me verify.If each person is working on their own patient, then the time to stabilize one patient would be the minimum of their individual times. But since they are working simultaneously, the time until the first patient is stabilized is indeed the minimum of their individual times. However, the question is about the average time taken to stabilize one patient by the team working together. So, perhaps it's asking for the expected time until one patient is stabilized, which would be the expectation of the minimum of their individual times.Wait, but that's a different calculation. The minimum of exponential variables is also exponential with rate equal to the sum of the individual rates. So, if each person's time is exponentially distributed with their respective rates, then the minimum time would have a rate equal to the sum of the individual rates.But in this case, the problem doesn't specify that the stabilization times are exponential. It just says they can stabilize one patient every 15, 10, and 8 minutes. So, perhaps we can assume that each person can stabilize a patient in a fixed time, and they can work on different patients simultaneously.In that case, the time to stabilize one patient would be the minimum of their individual times, but since they are working on different patients, the time to stabilize one patient is actually the minimum time among the three. But that doesn't make sense because they are working on different patients, so the time to stabilize one patient would be the time it takes for the first person to finish stabilizing a patient.Wait, now I'm confused. Let me clarify.If they are all working on different patients simultaneously, then the time to stabilize one patient is the minimum time it takes for any one of them to finish stabilizing a patient. So, if the firefighter takes 15 minutes, the paramedic 10 minutes, and the trauma nurse 8 minutes, then the first patient would be stabilized in 8 minutes by the trauma nurse. But that's the time for the first patient. However, the question is about the average time taken to stabilize one patient by the team working together.Alternatively, maybe it's about the combined rate. So, if each person can stabilize a patient in their respective times, working together, how many patients can they stabilize per minute, and thus the average time per patient.Wait, that's the approach I took earlier. So, the combined rate is 1/15 + 1/10 + 1/8 = 7/24 per minute. Therefore, the average time per patient is 24/7 minutes.But let me think about it differently. Suppose they work for a certain amount of time, say t minutes. In that time, the firefighter can stabilize t/15 patients, the paramedic t/10, and the trauma nurse t/8. So, the total number of patients stabilized is t/15 + t/10 + t/8.Therefore, the rate is (1/15 + 1/10 + 1/8) patients per minute, which is 7/24 per minute. So, the time to stabilize one patient is 1 divided by (7/24) = 24/7 minutes.Yes, that makes sense. So, the average time taken to stabilize one patient by the team working together is 24/7 minutes, which is approximately 3.428 minutes.Okay, so I think that's the answer for the first problem.Moving on to the second problem: After stabilization, the team needs to evacuate patients to a hospital. The evacuation route has a variable traffic flow that affects the travel time. The probability density function of the travel time T (in minutes) is given by f(T) = k(30 - T) for 0 ≤ T ≤ 30, where k is a constant. I need to calculate the expected travel time for the evacuation route and determine the value of k.Alright, so first, I need to find the value of k. Since f(T) is a probability density function (pdf), it must satisfy the condition that the integral of f(T) over its domain equals 1. The domain here is from 0 to 30 minutes.So, the integral from 0 to 30 of k(30 - T) dT = 1.Let me compute that integral.First, factor out the constant k:k * ∫₀³⁰ (30 - T) dT = 1Compute the integral:∫ (30 - T) dT = 30T - (1/2)T² + CEvaluate from 0 to 30:At T=30: 30*30 - (1/2)*(30)^2 = 900 - (1/2)*900 = 900 - 450 = 450At T=0: 0 - 0 = 0So, the integral from 0 to 30 is 450 - 0 = 450Therefore, k * 450 = 1 => k = 1/450So, k is 1/450.Now, to find the expected travel time, E[T], which is the integral of T*f(T) dT from 0 to 30.So, E[T] = ∫₀³⁰ T * f(T) dT = ∫₀³⁰ T * (1/450)(30 - T) dTFactor out 1/450:E[T] = (1/450) ∫₀³⁰ T(30 - T) dTExpand the integrand:T(30 - T) = 30T - T²So, E[T] = (1/450) ∫₀³⁰ (30T - T²) dTCompute the integral:∫ (30T - T²) dT = 15T² - (1/3)T³ + CEvaluate from 0 to 30:At T=30: 15*(30)^2 - (1/3)*(30)^3 = 15*900 - (1/3)*27000 = 13500 - 9000 = 4500At T=0: 0 - 0 = 0So, the integral is 4500 - 0 = 4500Therefore, E[T] = (1/450) * 4500 = 10So, the expected travel time is 10 minutes.Let me double-check the calculations.First, for k:∫₀³⁰ k(30 - T) dT = k * [30T - (1/2)T²] from 0 to 30At 30: 30*30 - 0.5*900 = 900 - 450 = 450So, k*450 = 1 => k=1/450. Correct.For E[T]:∫₀³⁰ T*(1/450)(30 - T) dT = (1/450) ∫₀³⁰ (30T - T²) dTIntegral of 30T is 15T², integral of T² is (1/3)T³At 30: 15*900 = 13500, (1/3)*27000 = 9000So, 13500 - 9000 = 4500Multiply by 1/450: 4500 / 450 = 10. Correct.So, the expected travel time is 10 minutes, and k is 1/450.I think that's solid.Final Answer1. The average time taken to stabilize one patient is boxed{dfrac{24}{7}} minutes.2. The expected travel time is boxed{10} minutes, and the value of ( k ) is boxed{dfrac{1}{450}}.</think>

🧠:<think>Okay, so I have this problem about a cricket batsman's performance over 10 matches. The commentator says his runs follow a quadratic sequence. The first three scores are 45, 50, and 57. I need to find the general formula for the nth match and then calculate the total runs over the next 15 matches. Hmm, quadratic sequence, so the runs can be modeled by a quadratic function, right?Alright, let me recall what a quadratic sequence is. In a quadratic sequence, the difference between consecutive terms is not constant, but the second difference is constant. So, if I denote the runs scored in the nth match as ( a_n ), then ( a_n ) is a quadratic function of n. That means it can be written as:( a_n = An^2 + Bn + C )where A, B, and C are constants that we need to find.Given that, we have the first three terms:- When n = 1, ( a_1 = 45 )- When n = 2, ( a_2 = 50 )- When n = 3, ( a_3 = 57 )So, plugging these into the quadratic formula, we can set up a system of equations.First equation: For n=1,( A(1)^2 + B(1) + C = 45 )Simplifies to:( A + B + C = 45 ) ...(1)Second equation: For n=2,( A(2)^2 + B(2) + C = 50 )Simplifies to:( 4A + 2B + C = 50 ) ...(2)Third equation: For n=3,( A(3)^2 + B(3) + C = 57 )Simplifies to:( 9A + 3B + C = 57 ) ...(3)Now, I have three equations:1. ( A + B + C = 45 )2. ( 4A + 2B + C = 50 )3. ( 9A + 3B + C = 57 )I need to solve this system for A, B, and C.Let me subtract equation (1) from equation (2):Equation (2) - Equation (1):( (4A + 2B + C) - (A + B + C) = 50 - 45 )Simplify:( 3A + B = 5 ) ...(4)Similarly, subtract equation (2) from equation (3):Equation (3) - Equation (2):( (9A + 3B + C) - (4A + 2B + C) = 57 - 50 )Simplify:( 5A + B = 7 ) ...(5)Now, I have two equations:4. ( 3A + B = 5 )5. ( 5A + B = 7 )Subtract equation (4) from equation (5):( (5A + B) - (3A + B) = 7 - 5 )Simplify:( 2A = 2 )So, ( A = 1 )Now, plug A = 1 into equation (4):( 3(1) + B = 5 )So, ( 3 + B = 5 )Thus, ( B = 2 )Now, plug A = 1 and B = 2 into equation (1):( 1 + 2 + C = 45 )So, ( 3 + C = 45 )Hence, ( C = 42 )So, the quadratic formula is:( a_n = n^2 + 2n + 42 )Wait, let me check if this works for the given terms.For n=1: 1 + 2 + 42 = 45 ✔️For n=2: 4 + 4 + 42 = 50 ✔️For n=3: 9 + 6 + 42 = 57 ✔️Perfect, so the general formula is correct.So, part 1 is done. The general formula is ( a_n = n^2 + 2n + 42 ).Now, moving on to part 2: Calculate the total number of runs over the next 15 matches. Wait, the batsman has already played 10 matches, so the next 15 would be matches 11 to 25. So, we need to find the sum from n=11 to n=25 of ( a_n ).Alternatively, since the quadratic formula is known, we can compute the sum from n=1 to n=25 and subtract the sum from n=1 to n=10.But let me think about which approach is better.First, let me recall that the sum of a quadratic sequence can be found by summing each term, which is a quadratic function. Alternatively, we can use the formula for the sum of a quadratic series.The general formula for the sum of the first N terms of a quadratic sequence ( a_n = An^2 + Bn + C ) is:( S_N = A cdot frac{N(N+1)(2N+1)}{6} + B cdot frac{N(N+1)}{2} + C cdot N )So, if I compute ( S_{25} ) and ( S_{10} ), then subtract ( S_{10} ) from ( S_{25} ) to get the total runs from match 11 to 25.Given that, let me compute ( S_{25} ) and ( S_{10} ).First, let's note that A = 1, B = 2, C = 42.Compute ( S_{25} ):( S_{25} = 1 cdot frac{25 cdot 26 cdot 51}{6} + 2 cdot frac{25 cdot 26}{2} + 42 cdot 25 )Let me compute each term step by step.First term: ( frac{25 cdot 26 cdot 51}{6} )Compute numerator: 25 * 26 = 650; 650 * 51 = let's compute 650*50=32,500 and 650*1=650, so total 32,500 + 650 = 33,150.Divide by 6: 33,150 / 6 = 5,525.Second term: ( 2 cdot frac{25 cdot 26}{2} )Simplify: The 2 in the numerator and denominator cancels out, so it's 25 * 26 = 650.Third term: 42 * 25 = 1,050.So, adding all three terms:5,525 + 650 + 1,050 = let's compute step by step.5,525 + 650 = 6,1756,175 + 1,050 = 7,225So, ( S_{25} = 7,225 )Now, compute ( S_{10} ):( S_{10} = 1 cdot frac{10 cdot 11 cdot 21}{6} + 2 cdot frac{10 cdot 11}{2} + 42 cdot 10 )Compute each term.First term: ( frac{10 cdot 11 cdot 21}{6} )Compute numerator: 10 * 11 = 110; 110 * 21 = 2,310.Divide by 6: 2,310 / 6 = 385.Second term: ( 2 cdot frac{10 cdot 11}{2} )Again, 2 cancels out, so 10 * 11 = 110.Third term: 42 * 10 = 420.Adding all three terms:385 + 110 + 420 = let's compute.385 + 110 = 495495 + 420 = 915So, ( S_{10} = 915 )Therefore, the total runs from match 11 to 25 is ( S_{25} - S_{10} = 7,225 - 915 = 6,310 )Wait, let me verify the calculations because 7,225 - 915 is 6,310? Wait, 7,225 - 900 is 6,325, so subtract 15 more: 6,310. Yes, that's correct.Alternatively, just to be thorough, maybe I should compute the sum from n=11 to n=25 directly.But since the formula is correct, and the steps seem right, I think 6,310 is the total runs.But just to be absolutely sure, let me compute ( S_{25} ) and ( S_{10} ) again.Compute ( S_{25} ):First term: ( frac{25 times 26 times 51}{6} )25*26=650, 650*51=33,150. 33,150 /6=5,525.Second term: ( 2 times frac{25 times 26}{2} )25*26=650, so 2*(650/2)=650.Third term: 42*25=1,050.Total: 5,525 + 650 + 1,050 = 7,225. Correct.Compute ( S_{10} ):First term: ( frac{10 times 11 times 21}{6} )10*11=110, 110*21=2,310. 2,310 /6=385.Second term: ( 2 times frac{10 times 11}{2} )10*11=110, so 2*(110/2)=110.Third term: 42*10=420.Total: 385 + 110 + 420 = 915. Correct.So, 7,225 - 915 = 6,310.Therefore, the total runs expected over the next 15 matches is 6,310.Wait, but just to make sure, maybe I can compute the sum from n=11 to n=25 by another method.Alternatively, since the formula is quadratic, the sum from n=11 to n=25 is the same as the sum from n=1 to n=25 minus the sum from n=1 to n=10, which is exactly what I did. So, that should be correct.Alternatively, I can compute the average of the 15 terms and multiply by 15, but that might be more complicated.Alternatively, I can compute the first term of the next 15 matches, which is n=11, and the last term is n=25, and use the formula for the sum of an arithmetic series, but wait, this is a quadratic sequence, not arithmetic. So, the difference isn't constant, so that approach won't work.Alternatively, maybe compute the sum term by term, but that would be tedious for 15 terms.Alternatively, I can use the formula for the sum of a quadratic sequence.Wait, the formula I used earlier is correct, so I think 6,310 is the right answer.But just to double-check, let me compute the sum from n=11 to n=25 using the quadratic formula.Compute each term from n=11 to n=25:But that would take time, but maybe I can compute a few terms to see if the trend is correct.Compute a_11: 11^2 + 2*11 + 42 = 121 + 22 + 42 = 185a_12: 144 + 24 + 42 = 210a_13: 169 + 26 + 42 = 237a_14: 196 + 28 + 42 = 266a_15: 225 + 30 + 42 = 297a_16: 256 + 32 + 42 = 330a_17: 289 + 34 + 42 = 365a_18: 324 + 36 + 42 = 402a_19: 361 + 38 + 42 = 441a_20: 400 + 40 + 42 = 482a_21: 441 + 42 + 42 = 525a_22: 484 + 44 + 42 = 570a_23: 529 + 46 + 42 = 617a_24: 576 + 48 + 42 = 666a_25: 625 + 50 + 42 = 717Now, let me sum these up:185, 210, 237, 266, 297, 330, 365, 402, 441, 482, 525, 570, 617, 666, 717.Let me add them step by step:Start with 185.185 + 210 = 395395 + 237 = 632632 + 266 = 898898 + 297 = 1,1951,195 + 330 = 1,5251,525 + 365 = 1,8901,890 + 402 = 2,2922,292 + 441 = 2,7332,733 + 482 = 3,2153,215 + 525 = 3,7403,740 + 570 = 4,3104,310 + 617 = 4,9274,927 + 666 = 5,5935,593 + 717 = 6,310Wow, that matches exactly with the previous result. So, the total runs from n=11 to n=25 is indeed 6,310.Therefore, the answer is 6,310 runs.But just to make sure, let me recount the addition step by step:1. 1852. 185 + 210 = 3953. 395 + 237 = 6324. 632 + 266 = 8985. 898 + 297 = 1,1956. 1,195 + 330 = 1,5257. 1,525 + 365 = 1,8908. 1,890 + 402 = 2,2929. 2,292 + 441 = 2,73310. 2,733 + 482 = 3,21511. 3,215 + 525 = 3,74012. 3,740 + 570 = 4,31013. 4,310 + 617 = 4,92714. 4,927 + 666 = 5,59315. 5,593 + 717 = 6,310Yes, that's correct. So, both methods confirm that the total is 6,310.Therefore, the batsman is expected to score 6,310 runs over the next 15 matches.Final Answer1. The general formula is boxed{n^2 + 2n + 42}.2. The total number of runs expected over the next 15 matches is boxed{6310}.</think>

🧠:<think>Okay, so I have this problem about Alex, a single parent who runs a blog. They want to understand how their blog's visitor growth works and figure out a posting strategy. There are two parts to this problem.Starting with the first part: The number of unique visitors V(t) is modeled by V(t) = V0 * e^(kt). They currently have 50 posts and 1,000 unique visitors per day. The visitors double after 10 additional posts. I need to find the growth rate k.Hmm, let's parse this. So V(t) is the number of visitors after t new posts. V0 is the initial number of visitors, which is 1,000. So when t = 0, V(0) = 1,000. After 10 additional posts, t = 10, the visitors double, so V(10) = 2,000.So plugging into the equation: 2,000 = 1,000 * e^(k*10). I can divide both sides by 1,000 to get 2 = e^(10k). Then take the natural logarithm of both sides: ln(2) = 10k. So k = ln(2)/10.Calculating that, ln(2) is approximately 0.6931, so 0.6931 divided by 10 is about 0.06931. So k is approximately 0.0693 per post.Wait, let me make sure. The model is V(t) = V0 * e^(kt), where t is the number of new posts. So each post contributes a multiplicative factor of e^k. After 10 posts, it's e^(10k) = 2. So yes, k = ln(2)/10. That seems right.Moving on to the second part. Alex has 40 hours per week to allocate between writing blog posts and job searching. Writing one post takes 2 hours, and each job application takes 1 hour. They want to increase their blog's unique visitors to 5,000 within 4 weeks. I need to find the minimum number of job applications they can submit each week while still achieving the visitor goal.First, let's figure out how many posts Alex needs to write to get from 1,000 visitors to 5,000 visitors. Using the growth model V(t) = 1,000 * e^(kt). We found k = ln(2)/10 ≈ 0.0693.So we need V(t) = 5,000. So 5,000 = 1,000 * e^(0.0693t). Dividing both sides by 1,000: 5 = e^(0.0693t). Take natural log: ln(5) = 0.0693t. So t = ln(5)/0.0693.Calculating ln(5): approximately 1.6094. So t ≈ 1.6094 / 0.0693 ≈ 23.22 posts. Since Alex can't write a fraction of a post, they need to write at least 24 posts.But wait, Alex currently has 50 posts. So the number of new posts needed is 24. But wait, hold on. The model is V(t) = V0 * e^(kt), where t is the number of new posts. So starting from 50 posts, they need to write t more posts to reach 5,000 visitors.Wait, no. Wait, actually, in the first part, the model is V(t) = V0 * e^(kt), where t is the number of new posts. So V0 is 1,000 visitors when t = 0, which is after 50 posts. So each additional post increases t by 1. So to get to 5,000 visitors, they need t such that 1,000 * e^(kt) = 5,000.So yes, t = ln(5)/k ≈ ln(5)/(ln(2)/10) ≈ (1.6094)/(0.0693) ≈ 23.22. So they need approximately 23.22 additional posts. So since they can't write a fraction, they need 24 posts.But wait, let me confirm. If they write 23 posts, then V(23) = 1,000 * e^(0.0693*23). Let's compute 0.0693*23 ≈ 1.5939. e^1.5939 ≈ 4.91. So 1,000 * 4.91 ≈ 4,910 visitors. That's close to 5,000 but not quite there. So 23 posts would get them to about 4,910, which is just under 5,000. So they need 24 posts.So t = 24. So they need to write 24 additional posts over 4 weeks. So per week, that's 24 / 4 = 6 posts per week.Each post takes 2 hours, so 6 posts per week would take 6 * 2 = 12 hours per week.Since Alex has 40 hours per week, the remaining time can be spent on job searching. So 40 - 12 = 28 hours per week can be spent on job applications. Each job application takes 1 hour, so they can submit 28 job applications per week.But wait, the question is asking for the minimum number of job applications they can submit each week while still achieving their blog's visitor goal. So if they want to minimize job applications, they need to maximize the time spent on writing posts. But wait, they have a fixed time of 40 hours per week. So if they spend more time on writing posts, they can write more posts, but in this case, they need exactly 24 posts over 4 weeks, which is 6 per week, taking 12 hours. So the remaining 28 hours must be spent on job searching. So the minimum number of job applications is 28 per week.Wait, but hold on. Is there a way to write more posts in less time? No, because each post takes 2 hours. So to write 6 posts, it's 12 hours. They can't write more posts without exceeding the 40-hour limit. So 28 job applications per week is the minimum if they want to write exactly 6 posts per week.But wait, maybe they can write more posts in some weeks and fewer in others? But the problem says they have 40 hours per week to allocate each week. So they can't borrow time from other weeks. Each week is independent. So each week, they can choose how many posts to write and how many job applications to submit, but the total time per week must be 40 hours.So to achieve 24 posts over 4 weeks, they need to write an average of 6 posts per week. So each week, they can write 6 posts, taking 12 hours, and submit 28 job applications. Alternatively, they could write more in some weeks and less in others, but since the total number of posts needed is 24, they have to write at least 6 per week on average. But if they write more in some weeks, they might have to write fewer in others, but the total would still be 24. However, since the growth is exponential, writing more posts earlier would lead to more growth earlier, but the problem doesn't specify any time constraints on when the visitors need to reach 5,000, just within 4 weeks. So writing 6 per week is sufficient.But actually, wait, the growth is continuous with each post. So if they write more posts earlier, the growth would compound more. But since the model is V(t) = 1,000 * e^(kt), where t is the total number of posts, not the time. So the total number of posts is what affects the visitors, not the timing of when the posts are written. So whether they write 6 posts each week or front-load the posts, the total number after 4 weeks is 24, so the visitors would be the same. So the timing doesn't matter for the total visitors, only the total number of posts.Therefore, they need to write 24 posts over 4 weeks, which is 6 per week. So each week, they spend 12 hours writing, leaving 28 hours for job applications. So the minimum number of job applications per week is 28.Wait, but the question says "the minimum number of job applications they can submit each week while still achieving their blog's visitor goal." So if they can write more posts in some weeks, they might be able to write fewer in others, but the total would still be 24. But since the growth is based on total posts, not the distribution, they could potentially write all 24 posts in one week, but that would take 24*2=48 hours, which exceeds their 40-hour limit. So they can't do that. So the maximum number of posts they can write in a week is 20, since 20*2=40 hours. But they only need 24 over 4 weeks, so 6 per week is the minimum required per week.Wait, no, actually, they can vary the number of posts per week as long as the total is 24. For example, they could write 7 posts in week 1, 7 in week 2, 5 in week 3, and 5 in week 4. But the total would still be 24. However, the time spent each week would vary. For example, writing 7 posts takes 14 hours, leaving 26 hours for job applications. Writing 5 posts takes 10 hours, leaving 30 hours for job applications. So if they front-load the posts, they can have fewer job applications in the first weeks and more in the later weeks, but the total job applications over 4 weeks would be the same as if they wrote 6 per week.But the question is about the minimum number of job applications each week. So if they write more posts in some weeks, they can have fewer job applications in those weeks, but more in others. But the question is asking for the minimum number per week, not the total. So to minimize the number of job applications in a week, they would need to maximize the number of posts in that week, thus minimizing job applications. But since they have a total of 24 posts over 4 weeks, the maximum number of posts they can write in a single week is limited by their time.Each post takes 2 hours, so in a week, they can write up to 20 posts (40 hours / 2 hours per post). But they only need 24 posts total. So if they write 20 posts in week 1, that would take 40 hours, leaving 0 hours for job applications. Then in the remaining 3 weeks, they only need 4 more posts. So in week 2, they write 4 posts, taking 8 hours, leaving 32 hours for job applications. Weeks 3 and 4, they could write 0 posts, leaving 40 hours each week for job applications.But wait, the problem is that the growth is based on the total number of posts. So if they write 20 posts in week 1, then V(t) = 1,000 * e^(0.0693*20). Let's compute that: 0.0693*20 ≈ 1.386, e^1.386 ≈ 4. So V(t) ≈ 4,000 visitors after week 1. Then in week 2, they write 4 more posts: t = 24, so V(t) = 1,000 * e^(0.0693*24). 0.0693*24 ≈ 1.663, e^1.663 ≈ 5.28. So V(t) ≈ 5,280 visitors, which exceeds 5,000. So they actually reach the goal in week 2.But the problem says they want to reach 5,000 visitors within 4 weeks. So if they front-load the posts, they can reach the goal earlier, but the question is about the minimum number of job applications each week. So in week 1, they wrote 20 posts, 0 job applications. Week 2, they wrote 4 posts, 32 job applications. Weeks 3 and 4, they wrote 0 posts, 40 job applications each week. So the minimum number of job applications per week is 0 in week 1, but the question is about the minimum number they can submit each week while still achieving the goal. So if they can have 0 in some weeks, that's the minimum. But the question is asking for the minimum number per week, not the total. So if they can have 0 in some weeks, that's the minimum. But the problem is that they need to maintain the blog's growth, so they have to write the posts in time. Wait, no, the growth is based on the total number of posts, not the timing. So as long as they have 24 posts by week 4, they can reach 5,000 visitors. So they could write all 24 posts in week 4, but that would take 48 hours, which is more than their 40-hour limit. So they can't do that. So the maximum number of posts they can write in a single week is 20, as 20*2=40 hours.But to reach 24 posts over 4 weeks, they can distribute the posts in such a way that in some weeks they write more, and in others less. For example, writing 20 posts in week 1, 4 posts in week 2, and 0 in weeks 3 and 4. But as we saw, that would reach the goal in week 2, but they still have to write the posts within the 4 weeks. So the minimum number of job applications per week would be 0 in week 1, 32 in week 2, and 40 in weeks 3 and 4. But the question is asking for the minimum number of job applications they can submit each week while still achieving their blog's visitor goal. So if they can have 0 in some weeks, that's the minimum. But I think the question is asking for the minimum number per week, not the minimum across weeks. So perhaps they need to have a consistent number each week.Wait, the problem says "the minimum number of job applications they can submit each week while still achieving their blog's visitor goal." So it's about the minimum number per week, not the minimum total. So if they can have some weeks with 0 job applications, that's the minimum. But they have to ensure that over the 4 weeks, they write 24 posts. So the way to minimize the number of job applications per week is to maximize the number of posts in some weeks, thus minimizing job applications in those weeks, while having more job applications in other weeks.But the question is about the minimum number they can submit each week. So the minimum per week is 0, but they have to ensure that over the 4 weeks, they write 24 posts. So in some weeks, they can have 0 job applications, but in others, they have to have more. However, the question is asking for the minimum number they can submit each week, which would be 0. But that might not be the case because they have to write the posts over the 4 weeks, so they can't write all 24 posts in one week because that would take 48 hours, which is more than 40. So the maximum they can write in a week is 20 posts, taking 40 hours, leaving 0 for job applications. Then in the next week, they can write 4 posts, taking 8 hours, leaving 32 hours for job applications. Then in the remaining two weeks, they can write 0 posts, leaving 40 hours for job applications each week.But the question is about the minimum number of job applications they can submit each week. So in week 1, they submit 0, which is the minimum. But the problem is that they have to achieve the visitor goal within 4 weeks. So if they write 20 posts in week 1, they reach 4,000 visitors. Then in week 2, they write 4 more posts, reaching 5,280 visitors, which is over 5,000. So they've achieved the goal by week 2. But the question is about the minimum number of job applications each week while still achieving the goal within 4 weeks. So they can have 0 job applications in week 1, but they still have to write the posts in time. So the minimum number of job applications per week is 0, but that might not be the answer intended because the question might be expecting a consistent number each week.Alternatively, if they spread the posts evenly, writing 6 posts per week, taking 12 hours, leaving 28 hours for job applications. So 28 job applications per week. That would be the minimum if they have to maintain a consistent number each week. But the question doesn't specify that they have to maintain a consistent number, just the minimum number each week. So technically, they can have 0 in some weeks, but the question is about the minimum number they can submit each week. So if they can have 0 in some weeks, that's the minimum. But I think the question is expecting the minimum number they have to submit each week, meaning the minimum number they can't go below, regardless of distribution.Wait, let me read the question again: "determine the minimum number of job applications they can submit each week while still achieving their blog's visitor goal." So it's about the minimum number per week, not the total. So if they can have 0 in some weeks, that's the minimum. But they have to write the posts within 4 weeks, so they can't write all 24 posts in one week because that would take 48 hours, which is more than 40. So the maximum they can write in a week is 20 posts, taking 40 hours, leaving 0 for job applications. Then in week 2, they can write 4 posts, taking 8 hours, leaving 32 hours for job applications. Weeks 3 and 4, they can write 0 posts, leaving 40 hours for job applications. So in this case, the minimum number of job applications per week is 0, but they have to write the posts in the first two weeks. However, the question is about the minimum number they can submit each week, so if they can have 0 in some weeks, that's the minimum. But I think the question is expecting the minimum number they have to submit each week, meaning the minimum number they can't go below, regardless of distribution. So perhaps they have to write at least 6 posts per week, taking 12 hours, leaving 28 hours for job applications. So 28 job applications per week is the minimum they have to submit each week to ensure they reach the goal within 4 weeks.Wait, but if they write 6 posts per week, they reach the goal by week 4. If they write more in some weeks and less in others, they can have some weeks with fewer job applications, but the question is about the minimum number they can submit each week. So if they can have 0 in some weeks, that's the minimum, but they have to ensure that over the 4 weeks, they write 24 posts. So the minimum number of job applications per week is 0, but that might not be the answer intended. Alternatively, if they have to maintain a consistent number each week, it's 28.I think the answer is 28 job applications per week because if they have to write 6 posts each week, they can't go below that, so they have to spend 12 hours on posts, leaving 28 for job applications. So the minimum number of job applications they can submit each week is 28.Wait, but let me think again. If they write 6 posts per week, they can't write more than that without exceeding the 40-hour limit. So 6 posts per week is the maximum they can write without reducing job applications below 28. So if they want to minimize job applications, they have to write as many posts as possible, which is 6 per week, leaving 28 hours for job applications. So 28 is the minimum number of job applications they can submit each week while still achieving the goal.Yes, that makes sense. So the answer is 28 job applications per week.</think>