🧠:<think>Okay, so I've got this problem about a rural village in France with strict conservative Christian values. The population is structured in a way that their social connections form a tree. That means it's a connected acyclic graph, right? So, no cycles, which makes sense because they don't want extramarital affairs, which could create cycles in the social graph.The first part of the problem is asking for the minimum number of individuals ( m ) required such that each married couple forms a distinct edge of the tree, and all other individuals maintain the tree structure without creating cycles or extramarital connections. The village has ( k ) married couples, and the total population is ( n ).Hmm, let me think. So, in a tree structure, the number of edges is always one less than the number of nodes. That's a fundamental property of trees. So, if each married couple is an edge, then the number of edges ( k ) should be equal to ( m - 1 ), where ( m ) is the number of individuals. Wait, but the total population is ( n ). So, does that mean ( m ) is the number of individuals in the tree, and ( n ) is the total population, which might include other people?Wait, the problem says "the total population of the village is ( n )", so maybe ( n ) is the total number of individuals, which includes both the married couples and the other individuals. So, if each married couple is an edge, and the tree is formed by these edges, then the number of edges is ( k ), so the number of nodes in the tree is ( k + 1 ). But the total population is ( n ), so the number of individuals not in the tree would be ( n - (k + 1) ). But the problem says all individuals must be part of the tree. Wait, no, maybe not. Let me read again."Each individual in the village can have a direct social connection with any other individual, but due to the community’s values condemning extramarital affairs, the connections must form a tree structure (a connected acyclic graph). Given that the village has ( k ) married couples, find the minimum number of individuals ( m ) required such that each married couple forms a distinct edge of the tree, and all other individuals maintain the tree structure while ensuring no cycles or extramarital connections exist."Wait, so the entire village's social connections form a tree. Each married couple is an edge, so the number of edges is ( k ). Therefore, the number of nodes in the tree is ( k + 1 ). But the total population is ( n ), so ( n = k + 1 ). But that can't be right because the problem is asking for the minimum number of individuals ( m ) required. Maybe I'm misunderstanding.Wait, perhaps the tree is formed by the married couples as edges, but each couple consists of two individuals, so each edge is a couple, meaning each edge has two nodes. So, if there are ( k ) edges, each edge is a couple, so the number of nodes is at least ( k + 1 ) because in a tree, the number of nodes is one more than the number of edges. But each edge is a couple, so each edge connects two nodes. So, if we have ( k ) edges, each edge is a couple, so the number of nodes is ( k + 1 ). But each couple is two people, so the total number of individuals would be ( 2k ). But wait, that can't be because in a tree, the number of nodes is ( k + 1 ), but each node is a person, so the total number of people is ( k + 1 ). But each couple is two people, so how does that reconcile?Wait, maybe each edge is a couple, so each edge represents a connection between two people, but each person can be part of multiple edges? No, because in a tree, each edge is unique, and each node can have multiple edges, but each edge is a couple, so each edge must connect two distinct nodes. So, if we have ( k ) edges, each edge is a couple, so the number of nodes is at least ( k + 1 ), but each edge is two people, so the total number of people is ( 2k ). But that would mean the number of nodes is ( 2k ), but in a tree, the number of nodes is ( k + 1 ). That doesn't add up.Wait, perhaps I'm overcomplicating. Let me think differently. Each married couple is an edge in the tree. So, each edge is a couple, so each edge connects two people. So, the number of edges is ( k ), so the number of nodes is ( k + 1 ). But each edge is a couple, so each edge is two people, but in the tree, each edge is a connection between two people, so each edge is a couple. So, the number of people is ( k + 1 ), but each couple is two people, so the total number of people would be ( 2k ). But that's a contradiction because ( k + 1 ) can't equal ( 2k ) unless ( k = 1 ). So, maybe I'm misunderstanding.Wait, perhaps the tree is formed by the social connections, and each married couple is an edge, but each person can be part of multiple social connections, but only one marriage. So, each person is a node, and each married couple is an edge between two nodes. So, the number of edges is ( k ), so the number of nodes is ( k + 1 ). But each edge is a couple, so each edge is two people, so the number of people is ( 2k ). But that would mean ( k + 1 = 2k ), which implies ( k = 1 ). That can't be right because ( k ) can be any number.Wait, maybe the tree is formed by the individuals, and the edges are the marriages. So, each marriage is an edge between two individuals, and the entire social graph is a tree. So, the number of edges is ( k ), so the number of nodes is ( k + 1 ). But each edge is a couple, so each edge is two people, so the number of people is ( 2k ). But again, that would mean ( k + 1 = 2k ), which only holds for ( k = 1 ). So, that's a problem.Wait, perhaps the tree is formed by the individuals, and each edge is a social connection, but the marriages are a subset of these edges. So, the number of edges in the tree is ( n - 1 ), where ( n ) is the number of individuals. But the number of married couples is ( k ), so ( k ) of these edges are marriages, and the rest are non-marital connections. But the problem says "each married couple forms a distinct edge of the tree", so all marriages are edges, but the tree can have other edges as well. Wait, no, because the problem says "each individual in the village can have a direct social connection with any other individual, but due to the community’s values condemning extramarital affairs, the connections must form a tree structure". So, the entire social graph is a tree, and each married couple is an edge in that tree. So, the number of edges is ( k ), so the number of nodes is ( k + 1 ). But each edge is a couple, so each edge is two people, so the number of people is ( 2k ). But again, ( k + 1 = 2k ) implies ( k = 1 ). So, that's not possible.Wait, maybe the tree is formed by the individuals, and each edge is a social connection, but the marriages are a subset of these edges. So, the number of edges in the tree is ( n - 1 ), where ( n ) is the number of individuals. But the number of married couples is ( k ), so ( k ) of these edges are marriages, and the rest are non-marital connections. But the problem says "each married couple forms a distinct edge of the tree", so all marriages are edges, but the tree can have other edges as well. Wait, but the problem is asking for the minimum number of individuals ( m ) required such that each married couple forms a distinct edge of the tree. So, perhaps ( m ) is the number of individuals, and the tree has ( m - 1 ) edges, of which ( k ) are married couples. So, the minimum ( m ) is such that ( m - 1 geq k ), so ( m geq k + 1 ). But since each married couple is an edge, and each edge is two people, but in the tree, each edge is a connection between two people, so the number of people is ( m ), and the number of edges is ( m - 1 ). So, if ( k ) of those edges are marriages, then ( k leq m - 1 ). So, the minimum ( m ) is ( k + 1 ). But wait, each marriage is an edge, so each marriage is two people, so the number of people in marriages is ( 2k ). But the total number of people is ( m ), so ( m geq 2k ). But also, ( m - 1 geq k ) because the tree has ( m - 1 ) edges. So, which one is more restrictive? If ( 2k geq k + 1 ) for ( k geq 1 ), which it is, then the minimum ( m ) is ( 2k ). But wait, let me check for ( k = 1 ). If ( k = 1 ), then ( m = 2 ), which makes sense because you have one couple, two people, and the tree has one edge, which is the couple. For ( k = 2 ), ( m = 4 ). But wait, can you have a tree with 4 nodes and 3 edges, but only 2 of those edges are marriages? Wait, no, because the problem says each married couple forms a distinct edge of the tree, so all edges must be marriages. Wait, no, the problem says "each married couple forms a distinct edge of the tree", but it doesn't say that all edges are marriages. So, perhaps the tree can have other edges that are not marriages, but the marriages are a subset of the edges. So, the number of edges in the tree is ( m - 1 ), and ( k ) of those are marriages. So, the minimum ( m ) is such that ( m - 1 geq k ), so ( m geq k + 1 ). But each marriage is two people, so the number of people in marriages is ( 2k ). So, the total number of people ( m ) must be at least ( 2k ). But also, ( m - 1 geq k ), which is ( m geq k + 1 ). So, which is larger? For ( k geq 1 ), ( 2k geq k + 1 ) when ( k geq 1 ). So, the minimum ( m ) is ( 2k ). But wait, let me test with ( k = 1 ). ( m = 2 ), which works. For ( k = 2 ), ( m = 4 ). Can we have a tree with 4 nodes and 3 edges, with 2 of those edges being marriages? Yes, for example, a tree where two edges are marriages and one edge is a non-marital connection. So, that works. So, the minimum number of individuals ( m ) required is ( 2k ).Wait, but let me think again. If each married couple is an edge, and each edge is two people, then the number of people is ( 2k ). But in a tree, the number of edges is ( m - 1 ), so ( m - 1 geq k ), which is ( m geq k + 1 ). But ( 2k geq k + 1 ) for ( k geq 1 ), so the minimum ( m ) is ( 2k ). So, the answer is ( m = 2k ).Wait, but let me think about ( k = 3 ). Then ( m = 6 ). So, a tree with 6 nodes has 5 edges. If 3 of those edges are marriages, that's possible. Each marriage is two people, so 3 marriages account for 6 people, which is the total population. So, that works. So, yes, the minimum number of individuals is ( 2k ).Okay, so for part 1, the minimum number of individuals ( m ) is ( 2k ).Now, moving on to part 2. Each married couple is represented by an edge in the tree, and each couple can have 0, 1, or 2 children, who are also part of the village. The function ( c(e) ) represents the number of children for each edge ( e ). The total number of children in the village is ( C ). We need to demonstrate that the sum of all possible configurations of ( c(e) ) can be expressed using a generating function and determine the generating function ( G(x) ) as a closed-form expression.So, each edge (married couple) can have 0, 1, or 2 children. So, for each edge, the generating function for the number of children is ( 1 + x + x^2 ). Since each edge is independent, the generating function for the entire tree would be the product of the generating functions for each edge. So, if there are ( k ) edges, the generating function would be ( (1 + x + x^2)^k ).But wait, the problem says "the sum of all possible configurations of ( c(e) ) can be expressed using a generating function". So, the generating function would be the sum over all possible configurations, weighted by ( x^C ), where ( C ) is the total number of children. So, each configuration corresponds to choosing ( c(e) ) for each edge ( e ), where ( c(e) in {0, 1, 2} ). The total number of children is ( C = sum_{e} c(e) ). So, the generating function is ( G(x) = sum_{C} left( text{number of configurations with total children } C right) x^C ).Since each edge contributes independently, the generating function is indeed ( (1 + x + x^2)^k ). So, that's the generating function. But the problem asks to express ( G(x) ) as a closed-form expression. So, we can write it as ( G(x) = (1 + x + x^2)^k ). Alternatively, we can express it in terms of a generating function for each edge and then raise it to the power of ( k ).Wait, but perhaps we can write it in a more closed-form expression. Let me think. The generating function ( 1 + x + x^2 ) can be written as ( frac{1 - x^3}{1 - x} ). So, ( G(x) = left( frac{1 - x^3}{1 - x} right)^k ). That's another way to write it, which might be considered a closed-form expression.Alternatively, we can expand it using the binomial theorem, but that would give a sum, not a closed-form. So, perhaps the most concise closed-form is ( (1 + x + x^2)^k ) or ( left( frac{1 - x^3}{1 - x} right)^k ).So, I think either of those would be acceptable, but perhaps the problem expects ( (1 + x + x^2)^k ) as the generating function.Wait, but let me confirm. The generating function for each edge is ( 1 + x + x^2 ), so for ( k ) edges, it's ( (1 + x + x^2)^k ). So, that's the generating function. So, that's the answer.So, summarizing:1. The minimum number of individuals ( m ) is ( 2k ).2. The generating function ( G(x) ) is ( (1 + x + x^2)^k ), which can also be written as ( left( frac{1 - x^3}{1 - x} right)^k ).But I think the first form is simpler and more direct, so I'll go with ( (1 + x + x^2)^k ).Final Answer1. The minimum number of individuals required is boxed{2k}.2. The generating function is boxed{(1 + x + x^2)^k}.</think>

🧠:<think>Alright, so I have this problem about a Dutch naming system where each name is assigned a unique number forming an arithmetic progression. There are two parts to the problem. Let me tackle them one by one.Starting with part 1: I need to express the sum ( S ) of all assigned numbers in terms of ( k ), ( x_1 ), and ( d ). Hmm, okay. I remember that in an arithmetic progression, each term increases by a common difference ( d ). The first term is ( x_1 ), the second is ( x_1 + d ), the third is ( x_1 + 2d ), and so on, up to the ( k )-th term, which would be ( x_1 + (k-1)d ).To find the sum ( S ) of an arithmetic progression, the formula is ( S = frac{k}{2} times (2x_1 + (k - 1)d) ). Wait, is that right? Let me recall. Yes, the sum of the first ( k ) terms of an AP is given by ( S = frac{k}{2} times (first term + last term) ). The last term is ( x_1 + (k - 1)d ), so substituting that in, it becomes ( S = frac{k}{2} times (x_1 + [x_1 + (k - 1)d]) ). Simplifying inside the parentheses: ( x_1 + x_1 + (k - 1)d = 2x_1 + (k - 1)d ). So, yes, ( S = frac{k}{2} times (2x_1 + (k - 1)d) ). That seems correct.Moving on to part 2: I need to calculate ( P = sum_{1 leq i < j leq k} x_i x_j ) given ( k = 10 ), ( x_1 = 5 ), and ( d = 3 ). Hmm, okay. So ( P ) is the sum of the products of each pair of numbers in the arithmetic progression. I remember that there's a formula related to the square of the sum of terms and the sum of the squares of the terms. Specifically, ( (sum x_i)^2 = sum x_i^2 + 2sum_{i < j} x_i x_j ). So, if I can find ( (sum x_i)^2 ) and ( sum x_i^2 ), I can solve for ( P ).First, let's compute ( sum x_i ). From part 1, we have ( S = frac{k}{2}(2x_1 + (k - 1)d) ). Plugging in the values: ( k = 10 ), ( x_1 = 5 ), ( d = 3 ). So, ( S = frac{10}{2}(2*5 + (10 - 1)*3) ). Calculating inside: ( 2*5 = 10 ), ( 9*3 = 27 ), so ( 10 + 27 = 37 ). Then, ( S = 5 * 37 = 185 ). So, the sum of all ( x_i ) is 185.Next, I need ( (sum x_i)^2 ), which is ( 185^2 ). Let me compute that: 185 squared. 180 squared is 32400, and 5 squared is 25. Then, cross term is 2*180*5 = 1800. So, 32400 + 1800 + 25 = 34225. So, ( (sum x_i)^2 = 34225 ).Now, I need ( sum x_i^2 ). Hmm, how do I compute the sum of the squares of an arithmetic progression? I think there's a formula for that. Let me recall. The sum of squares of an AP can be expressed as ( sum_{i=1}^{k} (x_1 + (i - 1)d)^2 ). Expanding that, it becomes ( sum x_1^2 + 2x_1 d (i - 1) + d^2 (i - 1)^2 ). So, summing over ( i ) from 1 to ( k ), it's ( kx_1^2 + 2x_1 d sum_{i=1}^{k} (i - 1) + d^2 sum_{i=1}^{k} (i - 1)^2 ).Simplify each term:1. ( kx_1^2 ): straightforward.2. ( 2x_1 d sum_{i=1}^{k} (i - 1) ): The sum ( sum_{i=1}^{k} (i - 1) ) is the same as ( sum_{j=0}^{k - 1} j ), which is ( frac{(k - 1)k}{2} ).3. ( d^2 sum_{i=1}^{k} (i - 1)^2 ): Similarly, ( sum_{j=0}^{k - 1} j^2 = frac{(k - 1)k(2k - 1)}{6} ).So, putting it all together:( sum x_i^2 = kx_1^2 + 2x_1 d cdot frac{(k - 1)k}{2} + d^2 cdot frac{(k - 1)k(2k - 1)}{6} ).Simplify each term:First term: ( kx_1^2 ).Second term: ( 2x_1 d cdot frac{(k - 1)k}{2} = x_1 d (k - 1)k ).Third term: ( d^2 cdot frac{(k - 1)k(2k - 1)}{6} ).So, combining these:( sum x_i^2 = kx_1^2 + x_1 d k(k - 1) + frac{d^2 k(k - 1)(2k - 1)}{6} ).Now, plugging in the values ( k = 10 ), ( x_1 = 5 ), ( d = 3 ):First term: ( 10 * 5^2 = 10 * 25 = 250 ).Second term: ( 5 * 3 * 10 * 9 = 15 * 90 = 1350 ).Third term: ( 3^2 * 10 * 9 * 19 / 6 ). Wait, let me compute step by step.First, ( d^2 = 9 ).Then, ( k(k - 1)(2k - 1) = 10 * 9 * 19 ). Let's compute that: 10*9=90, 90*19. 90*20=1800, minus 90 is 1710.So, ( 9 * 1710 / 6 ). Compute 1710 / 6 first: 1710 divided by 6 is 285. Then, 9 * 285 = 2565.So, third term is 2565.Now, summing all three terms:250 + 1350 = 1600.1600 + 2565 = 4165.So, ( sum x_i^2 = 4165 ).Now, going back to the earlier equation:( (sum x_i)^2 = sum x_i^2 + 2P ).We have ( 34225 = 4165 + 2P ).So, subtract 4165 from both sides: ( 34225 - 4165 = 2P ).Compute 34225 - 4165:34225 - 4000 = 30225.30225 - 165 = 30060.So, 30060 = 2P.Therefore, P = 30060 / 2 = 15030.Wait, let me double-check that subtraction:34225 - 4165.Let me compute 34225 - 4000 = 30225.Then, subtract 165: 30225 - 165.30225 - 100 = 30125.30125 - 65 = 30060. Yes, correct.So, 30060 = 2P => P = 15030.Therefore, the value of P is 15030.But wait, just to make sure I didn't make any mistakes in the calculations, let me verify each step.First, computing ( S = 185 ), that seems correct: 10 terms, starting at 5, difference 3. The terms are 5, 8, 11, 14, 17, 20, 23, 26, 29, 32. Let me add them up:5 + 8 = 1313 + 11 = 2424 + 14 = 3838 + 17 = 5555 + 20 = 7575 + 23 = 9898 + 26 = 124124 + 29 = 153153 + 32 = 185. Yes, correct.Then, ( S^2 = 185^2 = 34225 ). Correct.Sum of squares: 4165. Let me compute the sum of squares of the terms:5^2 = 258^2 = 6411^2 = 12114^2 = 19617^2 = 28920^2 = 40023^2 = 52926^2 = 67629^2 = 84132^2 = 1024Adding these up:25 + 64 = 8989 + 121 = 210210 + 196 = 406406 + 289 = 695695 + 400 = 10951095 + 529 = 16241624 + 676 = 23002300 + 841 = 31413141 + 1024 = 4165. Yes, correct.So, ( sum x_i^2 = 4165 ).Then, ( 34225 = 4165 + 2P ).So, ( 2P = 34225 - 4165 = 30060 ).Thus, ( P = 15030 ). That seems correct.Alternatively, another way to compute ( P ) is using the formula:( P = frac{S^2 - sum x_i^2}{2} ).Which is exactly what I did. So, that's consistent.Therefore, I think the answer is 15030.Final AnswerThe sum ( S ) is expressed as ( boxed{frac{k}{2}(2x_1 + (k - 1)d)} ), and the value of ( P ) is ( boxed{15030} ).</think>

🧠:<think>Okay, so I have this problem about a retired news presenter who wants to analyze their viewership over 30 years. The problem is divided into two parts. The first part is solving a differential equation to find the explicit form of the viewership function V(t), and the second part is calculating the total viewership by integrating V(t) from t=0 to t=30. Let me try to work through this step by step.Starting with part 1. The differential equation given is:[frac{dV}{dt} + alpha V = beta e^{gamma t}]This looks like a linear first-order differential equation. I remember that the standard form for such equations is:[frac{dy}{dt} + P(t)y = Q(t)]In this case, P(t) is α, which is a constant, and Q(t) is β e^{γ t}, which is an exponential function. To solve this, I think I need to use an integrating factor. The integrating factor μ(t) is given by:[mu(t) = e^{int P(t) dt} = e^{int alpha dt} = e^{alpha t}]Wait, hold on, is that right? Because if P(t) is α, then integrating α with respect to t would be α t, so yes, the integrating factor is e^{α t}.Once I have the integrating factor, I can multiply both sides of the differential equation by μ(t):[e^{alpha t} frac{dV}{dt} + alpha e^{alpha t} V = beta e^{gamma t} e^{alpha t}]Simplifying the right-hand side, that becomes β e^{(gamma + alpha) t}.Now, the left-hand side should be the derivative of (V * μ(t)) with respect to t. Let me check:[frac{d}{dt} [V e^{alpha t}] = e^{alpha t} frac{dV}{dt} + alpha e^{alpha t} V]Yes, that's exactly the left-hand side. So, we can write:[frac{d}{dt} [V e^{alpha t}] = beta e^{(gamma + alpha) t}]Now, to solve for V(t), I need to integrate both sides with respect to t:[int frac{d}{dt} [V e^{alpha t}] dt = int beta e^{(gamma + alpha) t} dt]The left side simplifies to V e^{alpha t} + C, where C is the constant of integration. The right side is the integral of β e^{(gamma + α) t} dt. Let me compute that:Let me denote k = γ + α for simplicity. Then, the integral becomes:[int beta e^{k t} dt = frac{beta}{k} e^{k t} + C]So, putting it all together:[V e^{alpha t} = frac{beta}{gamma + alpha} e^{(gamma + alpha) t} + C]Now, solve for V(t):[V(t) = frac{beta}{gamma + alpha} e^{gamma t} + C e^{-alpha t}]Okay, so that's the general solution. Now, we need to apply the initial condition V(0) = V_0 to find the constant C.Plugging t = 0 into the equation:[V(0) = frac{beta}{gamma + alpha} e^{0} + C e^{0} = frac{beta}{gamma + alpha} + C = V_0]So, solving for C:[C = V_0 - frac{beta}{gamma + alpha}]Therefore, the explicit solution for V(t) is:[V(t) = frac{beta}{gamma + alpha} e^{gamma t} + left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t}]Let me double-check this solution. If I plug t = 0, I should get V(0) = V_0. Plugging in:[V(0) = frac{beta}{gamma + alpha} + left( V_0 - frac{beta}{gamma + alpha} right) = V_0]Yes, that works. Also, if I take the derivative of V(t):[frac{dV}{dt} = frac{beta gamma}{gamma + alpha} e^{gamma t} - alpha left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t}]Then, plug into the original differential equation:[frac{dV}{dt} + alpha V = frac{beta gamma}{gamma + alpha} e^{gamma t} - alpha left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t} + alpha left( frac{beta}{gamma + alpha} e^{gamma t} + left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t} right)]Simplify term by term:First term: (frac{beta gamma}{gamma + alpha} e^{gamma t})Second term: (- alpha left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t})Third term: ( alpha frac{beta}{gamma + alpha} e^{gamma t} )Fourth term: ( alpha left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t} )Now, combining the first and third terms:[left( frac{beta gamma}{gamma + alpha} + frac{alpha beta}{gamma + alpha} right) e^{gamma t} = frac{beta (gamma + alpha)}{gamma + alpha} e^{gamma t} = beta e^{gamma t}]Combining the second and fourth terms:[- alpha left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t} + alpha left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t} = 0]So, altogether, we get:[beta e^{gamma t} + 0 = beta e^{gamma t}]Which matches the right-hand side of the original differential equation. So, the solution seems correct.Alright, so part 1 is done. Now, moving on to part 2: calculating the total viewership over 30 years by integrating V(t) from t=0 to t=30.So, the total viewership T is:[T = int_{0}^{30} V(t) dt = int_{0}^{30} left[ frac{beta}{gamma + alpha} e^{gamma t} + left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t} right] dt]I can split this integral into two parts:[T = frac{beta}{gamma + alpha} int_{0}^{30} e^{gamma t} dt + left( V_0 - frac{beta}{gamma + alpha} right) int_{0}^{30} e^{-alpha t} dt]Let me compute each integral separately.First integral:[int_{0}^{30} e^{gamma t} dt = left[ frac{1}{gamma} e^{gamma t} right]_0^{30} = frac{1}{gamma} (e^{30 gamma} - 1)]Second integral:[int_{0}^{30} e^{-alpha t} dt = left[ -frac{1}{alpha} e^{-alpha t} right]_0^{30} = -frac{1}{alpha} (e^{-30 alpha} - 1) = frac{1}{alpha} (1 - e^{-30 alpha})]So, plugging these back into T:[T = frac{beta}{gamma + alpha} cdot frac{1}{gamma} (e^{30 gamma} - 1) + left( V_0 - frac{beta}{gamma + alpha} right) cdot frac{1}{alpha} (1 - e^{-30 alpha})]Simplify each term:First term:[frac{beta}{gamma (gamma + alpha)} (e^{30 gamma} - 1)]Second term:[frac{V_0 - frac{beta}{gamma + alpha}}{alpha} (1 - e^{-30 alpha}) = frac{V_0}{alpha} (1 - e^{-30 alpha}) - frac{beta}{alpha (gamma + alpha)} (1 - e^{-30 alpha})]So, combining all together:[T = frac{beta}{gamma (gamma + alpha)} (e^{30 gamma} - 1) + frac{V_0}{alpha} (1 - e^{-30 alpha}) - frac{beta}{alpha (gamma + alpha)} (1 - e^{-30 alpha})]I can factor out the terms involving β:[T = frac{beta}{gamma + alpha} left( frac{e^{30 gamma} - 1}{gamma} - frac{1 - e^{-30 alpha}}{alpha} right) + frac{V_0}{alpha} (1 - e^{-30 alpha})]Alternatively, I can leave it as is, but perhaps it's better to write it in a more compact form.Alternatively, let me see if I can combine the terms:Note that the first term is:[frac{beta}{gamma (gamma + alpha)} (e^{30 gamma} - 1)]And the third term is:[- frac{beta}{alpha (gamma + alpha)} (1 - e^{-30 alpha})]So, combining these two:[frac{beta}{gamma + alpha} left( frac{e^{30 gamma} - 1}{gamma} - frac{1 - e^{-30 alpha}}{alpha} right)]So, the total T is:[T = frac{beta}{gamma + alpha} left( frac{e^{30 gamma} - 1}{gamma} - frac{1 - e^{-30 alpha}}{alpha} right) + frac{V_0}{alpha} (1 - e^{-30 alpha})]I think this is as simplified as it can get unless there's a specific relationship between α and γ, but since the problem doesn't specify, I think this is the final expression.Let me just double-check the integration steps to make sure I didn't make a mistake.First integral: ∫ e^{γ t} dt from 0 to 30 is (e^{30 γ} - 1)/γ. Correct.Second integral: ∫ e^{-α t} dt from 0 to 30 is (1 - e^{-30 α})/α. Correct.Then, substituting back into T:First term: β/(γ + α) * (e^{30 γ} - 1)/γSecond term: (V0 - β/(γ + α)) * (1 - e^{-30 α})/αWhich expands to V0*(1 - e^{-30 α})/α - β/(γ + α)*(1 - e^{-30 α})/αSo, combining the two beta terms:β/(γ + α) [ (e^{30 γ} - 1)/γ - (1 - e^{-30 α})/α ]Yes, that's correct.So, the total viewership T is:[T = frac{beta}{gamma + alpha} left( frac{e^{30 gamma} - 1}{gamma} - frac{1 - e^{-30 alpha}}{alpha} right) + frac{V_0}{alpha} (1 - e^{-30 alpha})]I think this is the correct expression for the total viewership over 30 years.Just to recap, the steps were:1. Recognize the differential equation as linear and use integrating factor.2. Solve the differential equation using integrating factor method.3. Apply initial condition to find the constant.4. Express V(t) explicitly.5. Set up the integral for total viewership from 0 to 30.6. Compute the integral by splitting into two parts.7. Evaluate each integral separately.8. Combine the results to get the total viewership.I think I followed all the steps correctly, and the algebra seems to check out. I don't see any mistakes in the process.Final Answer1. The explicit form of ( V(t) ) is ( boxed{V(t) = frac{beta}{gamma + alpha} e^{gamma t} + left( V_0 - frac{beta}{gamma + alpha} right) e^{-alpha t}} ).2. The total viewership over 30 years is ( boxed{frac{beta}{gamma + alpha} left( frac{e^{30 gamma} - 1}{gamma} - frac{1 - e^{-30 alpha}}{alpha} right) + frac{V_0}{alpha} (1 - e^{-30 alpha})} ).</think>

🧠:<think>Okay, so I've got this problem about a fashion photographer using an app to enhance their portfolio images. The app uses some matrix transformations and a nonlinear scaling function. I need to figure out two things: first, express the transformed color vector after a matrix multiplication, and second, determine the range of a constant k so that the scaling function keeps the color components within 0 to 255.Starting with the first part. The transformation is given by a matrix T, which is a 3x3 matrix. The initial color vector is c0 = (R0, G0, B0). So, to find the transformed vector c1, I need to multiply the matrix T by the vector c0.Let me write down the matrix T:T = [ [0.8, 0.1, 0.1], [0.2, 1.0, -0.2], [-0.1, 0.1, 1.0] ]So, c1 = T * c0. That means each component of c1 is a linear combination of R0, G0, and B0.Let me compute each component step by step.First component of c1 (R1):R1 = 0.8*R0 + 0.1*G0 + 0.1*B0Second component (G1):G1 = 0.2*R0 + 1.0*G0 + (-0.2)*B0Third component (B1):B1 = (-0.1)*R0 + 0.1*G0 + 1.0*B0So, putting it all together, the transformed vector c1 is:c1 = (0.8R0 + 0.1G0 + 0.1B0, 0.2R0 + G0 - 0.2B0, -0.1R0 + 0.1G0 + B0)I think that's straightforward. Just matrix multiplication.Now, moving on to the second part. The photographer uses a nonlinear scaling function f(x) = 255 / (1 + e^{-k(x - 128)}). They want to ensure that after applying this function to each component of c1, the result stays within 0 and 255 for any initial c0 with components between 0 and 255.Hmm. So, f(x) is a sigmoid function, right? It's an S-shaped curve that asymptotically approaches 0 as x approaches negative infinity and approaches 255 as x approaches positive infinity. The parameter k controls the steepness of the curve.But in our case, x is the transformed color component, which is a result of the linear transformation T applied to c0. So, we need to ensure that for any c0 in [0,255]^3, after applying T, the resulting c1 has components such that f(c1_i) is still in [0,255].Wait, but f(x) is defined as 255 / (1 + e^{-k(x - 128)}). Let's analyze this function.First, when x = 128, f(x) = 255 / (1 + e^{0}) = 255 / 2 ≈ 127.5.As x increases beyond 128, the exponent becomes positive, so e^{-k(x - 128)} decreases, making the denominator approach 1, so f(x) approaches 255.As x decreases below 128, the exponent becomes negative, so e^{-k(x - 128)} increases, making the denominator larger, so f(x) approaches 0.Therefore, f(x) maps x from (-infty, +infty) to (0, 255). But in our case, x is the transformed color component, which is c1_i. However, c1_i is a linear combination of R0, G0, B0, each in [0,255].Wait, but c1_i could potentially be outside [0,255], right? Because the matrix T can have negative entries, so for example, B1 = -0.1R0 + 0.1G0 + B0. If R0 is large and G0 is small, B1 could be negative. Similarly, if R0 is small and G0 is large, B1 could be larger than 255.But the photographer wants to apply f to each component of c1, and have f(c1_i) in [0,255]. Since f(x) is always between 0 and 255, regardless of x, because as x approaches negative infinity, f(x) approaches 0, and as x approaches positive infinity, f(x) approaches 255.Wait, but the question is to determine the range of k such that for any c0 in [0,255]^3, after applying T, and then f to each component, the result is within [0,255]. But since f(x) is always between 0 and 255, regardless of k, as long as k is positive, right?Wait, hold on. Let me think again. If k is positive, then as x increases, f(x) increases. If k is negative, then f(x) would decrease as x increases, which might invert the function. But in our case, we want f(x) to map x to [0,255], regardless of x. So, as long as k is positive, f(x) will always be between 0 and 255, regardless of x.But wait, the question is more specific: it says "for any transformed color vector c1, after applying f to each component, the result remains within the bounds of 0 and 255 for any initial color vector c0 with components between 0 and 255."But since f(x) is designed such that it maps any real number x to (0,255), regardless of k, as long as k is positive, right? Because if k is positive, the function is increasing, and as x approaches infinity, f(x) approaches 255, and as x approaches negative infinity, it approaches 0.But if k is negative, then f(x) would be decreasing. So, as x approaches infinity, f(x) approaches 0, and as x approaches negative infinity, f(x) approaches 255. That would invert the colors, which might not be desirable, but the problem doesn't specify anything about the direction, just that the result remains within 0 and 255.Wait, but even if k is negative, f(x) is still between 0 and 255. So, regardless of k, f(x) is bounded between 0 and 255.But that can't be right because the problem is asking for the range of k such that the result remains within 0 and 255. So, maybe I'm missing something.Wait, perhaps the issue is that the function f(x) is applied after the transformation T. So, we need to ensure that f(c1_i) is within [0,255], but c1_i could be outside [0,255], but f(x) is designed to map any real number to [0,255]. So, regardless of k, as long as k is positive, f(x) will map c1_i to [0,255]. But if k is zero, f(x) becomes 255/2, which is 127.5, which is within [0,255], but it's a constant function, which might not be desired.Wait, but the problem is about the range of k such that for any c0, f(c1_i) is within [0,255]. Since f(x) is always between 0 and 255 regardless of x, as long as k is positive, right? Because if k is positive, f(x) is a sigmoid function that maps x to (0,255). If k is zero, f(x) is 127.5, which is within [0,255]. If k is negative, f(x) is a decreasing function, mapping x to (0,255) as well.Wait, but if k is negative, then f(x) is decreasing, so for x increasing, f(x) decreases. But the problem doesn't specify anything about the direction, just that the result is within 0 and 255. So, perhaps k can be any real number except zero? But if k is zero, f(x) is 127.5, which is still within [0,255].Wait, but the problem says "nonlinear scaling function". If k is zero, it's a linear function, so maybe k cannot be zero. But the problem says "nonlinear", so k cannot be zero. So, k must be non-zero.But wait, the function is nonlinear for any k ≠ 0, because if k=0, it's linear (constant). So, the photographer wants a nonlinear scaling, so k ≠ 0.But the main point is that f(x) is always between 0 and 255, regardless of k, as long as k is a real number. So, does that mean that any k is acceptable? But the problem is asking for the range of k such that for any c0, f(c1_i) is within [0,255].Wait, but maybe the issue is that c1_i could be outside [0,255], but f(x) is designed to map any x to [0,255]. So, regardless of k, f(c1_i) is within [0,255]. Therefore, k can be any real number except zero, but the problem might be more about ensuring that f(x) doesn't cause any issues, but since it's a sigmoid, it's always bounded.But perhaps the question is more about ensuring that the function f is well-defined and doesn't cause overflow or something, but in terms of mathematical bounds, f(x) is always between 0 and 255.Wait, but maybe the problem is that if k is too large, the function becomes too steep, and might cause the output to be either 0 or 255 for some inputs, but since the inputs c1_i are real numbers, f(x) will never actually reach 0 or 255, just approach them asymptotically.But the problem says "the result remains within the bounds of 0 and 255", so as long as f(x) doesn't exceed these bounds, which it doesn't, regardless of k.Wait, but maybe the problem is that the transformation T could potentially cause c1_i to be outside [0,255], but f(x) is designed to map any real number to [0,255]. So, regardless of k, f(c1_i) is within [0,255]. Therefore, the range of k is all real numbers except zero, but since k is a constant that controls the intensity, it's usually positive.Wait, but the problem says "determine the range of values for k such that for any transformed color vector c1, after applying f to each component, the result remains within the bounds of 0 and 255 for any initial color vector c0 with components between 0 and 255."So, perhaps the answer is that k can be any positive real number, because if k is positive, f(x) is increasing, mapping x to (0,255). If k is negative, f(x) is decreasing, but still maps x to (0,255). So, k can be any real number except zero.But the problem says "nonlinear scaling function", so k cannot be zero, because that would make it a linear function (constant). So, k can be any real number except zero.But wait, the function f(x) is defined as 255 / (1 + e^{-k(x - 128)}). If k is positive, it's a sigmoid function increasing with x. If k is negative, it's a sigmoid function decreasing with x. Both cases map x to (0,255). So, regardless of k ≠ 0, f(x) is within (0,255). Therefore, the range of k is all real numbers except zero.But the problem says "the result remains within the bounds of 0 and 255". Since f(x) approaches 0 and 255 asymptotically, it never actually reaches them, but gets arbitrarily close. So, technically, f(x) is always strictly between 0 and 255, regardless of k ≠ 0.Therefore, the range of k is all real numbers except zero. So, k ∈ ℝ {0}.But let me double-check. Suppose k is positive: f(x) is increasing, mapping x to (0,255). If k is negative: f(x) is decreasing, mapping x to (0,255). So, in both cases, the output is within (0,255). Therefore, k can be any real number except zero.But the problem says "the result remains within the bounds of 0 and 255". Since f(x) never actually reaches 0 or 255, but stays strictly within them, it's always within the bounds. So, the answer is that k can be any real number except zero.But wait, the problem says "determine the range of values for k such that for any transformed color vector c1, after applying f to each component, the result remains within the bounds of 0 and 255 for any initial color vector c0 with components between 0 and 255."So, if k is zero, f(x) is 127.5, which is within [0,255], but it's a constant function, which is linear, not nonlinear. Since the function is called a nonlinear scaling function, k cannot be zero. So, k must be any real number except zero.But the problem might be expecting a specific range, like positive real numbers, but I think both positive and negative k are acceptable because f(x) will still map to (0,255). So, the range is all real numbers except zero.But let me think again. If k is positive, f(x) is increasing; if k is negative, f(x) is decreasing. Both cases, the function is bounded between 0 and 255. So, yes, k can be any real number except zero.Therefore, the range of k is k ∈ ℝ {0}.But let me check if there's any constraint on k based on the transformation T. Because c1_i could be outside [0,255], but f(x) is designed to handle any real number. So, regardless of how T transforms c0, f(x) will always map it to (0,255). Therefore, the only constraint is that k ≠ 0.So, the answer is that k can be any real number except zero.Wait, but the problem says "determine the range of values for k such that for any transformed color vector c1, after applying f to each component, the result remains within the bounds of 0 and 255 for any initial color vector c0 with components between 0 and 255."So, yes, as long as k ≠ 0, f(x) will map any real number to (0,255). Therefore, the range of k is all real numbers except zero.But maybe the problem expects k to be positive, because usually in sigmoid functions, k is positive to have an increasing function. But the problem doesn't specify anything about the direction, just that the result is within 0 and 255. So, both positive and negative k are acceptable.Therefore, the range of k is all real numbers except zero.But let me think again. If k is positive, f(x) is increasing; if k is negative, f(x) is decreasing. Both cases, the function is bounded between 0 and 255. So, yes, k can be any real number except zero.So, to sum up:1. The transformed color vector c1 is:c1 = (0.8R0 + 0.1G0 + 0.1B0, 0.2R0 + G0 - 0.2B0, -0.1R0 + 0.1G0 + B0)2. The range of k is all real numbers except zero.But wait, let me check if k can be zero. If k=0, f(x) = 255 / (1 + e^{0}) = 255 / 2 = 127.5, which is a constant function. So, it's linear, not nonlinear. Since the problem mentions a nonlinear scaling function, k cannot be zero. So, k must be any real number except zero.Therefore, the range of k is k ∈ ℝ {0}.But the problem might expect a specific interval, like k > 0, but I think both positive and negative are acceptable because the function still maps to (0,255). So, the answer is k can be any real number except zero.But let me check if there's any other constraint. For example, if k is too large, does f(x) cause any issues? But since f(x) is a sigmoid, it's smooth and bounded regardless of k. So, no, k can be any real number except zero.Therefore, the final answers are:1. c1 = (0.8R0 + 0.1G0 + 0.1B0, 0.2R0 + G0 - 0.2B0, -0.1R0 + 0.1G0 + B0)2. k can be any real number except zero.But let me write them in the required format.</think>