answer：First, I need to determine how many puppies, adult dogs, and kittens remain after the sales.For the puppies:45 puppies were initially present, and 39 were sold.45 - 39 = 6 puppies remaining.For the adult dogs:30 adult dogs were initially present, and 15 were sold.30 - 15 = 15 adult dogs remaining.For the kittens:25 kittens were initially present, and 10 were sold.25 - 10 = 15 kittens remaining.Next, I need to calculate the number of cages required for each type of animal based on the given capacities.For the puppies:Each cage can hold 3 puppies.6 puppies ÷ 3 puppies per cage = 2 cages.For the adult dogs:Each cage can hold 2 adult dogs.15 adult dogs ÷ 2 adult dogs per cage = 7.5 cages.Since we can't have half a cage, we'll need to round up to 8 cages.For the kittens:Each cage can hold 2 kittens.15 kittens ÷ 2 kittens per cage = 7.5 cages.Again, rounding up, we'll need 8 cages.Finally, I'll add up the total number of cages needed:2 cages (puppies) + 8 cages (adult dogs) + 8 cages (kittens) = 18 cages.Therefore, the pet store used a total of 18 cages.

answer：Okay, so I have this problem about a regular tetrahedron. I need to figure out the ratio of the volume of a smaller tetrahedron to the original one after scaling down the coordinates of its vertices by half. Then, express this ratio as a fraction p/q where p and q are coprime, and find p + q. Hmm, okay, let me break this down step by step.First, I should recall what a regular tetrahedron is. It's a three-dimensional shape with four triangular faces, each of which is an equilateral triangle. All edges are equal in length, and all faces are congruent. So, it's a very symmetrical shape.Now, the problem mentions scaling down the coordinates of its vertices by half. I think this means that each coordinate of every vertex is multiplied by 1/2. So, if a vertex was at (x, y, z), it becomes (x/2, y/2, z/2). But wait, the problem says "determine the vertices of the tetrahedron formed by connecting these points." Hmm, does this mean that the new tetrahedron is formed by connecting these scaled-down points? I think so.But before I get into that, maybe I should consider how scaling affects the volume of a three-dimensional object. I remember that when you scale a shape by a factor, the volume scales by the cube of that factor. So, if you scale each dimension by 1/2, the volume becomes (1/2)^3 = 1/8 of the original. So, is the answer just 1/8? That seems too straightforward, but maybe it is.Wait, but let me make sure. Maybe I need to actually compute the volumes to confirm. Let's see. To compute the volume of a tetrahedron, I can use the formula involving the determinant of a matrix constructed from its vertices. But that might be complicated. Alternatively, since it's a regular tetrahedron, there's a simpler formula for its volume in terms of its edge length.The volume V of a regular tetrahedron with edge length a is given by V = (a^3)/(6√2). So, if I can find the edge lengths of both the original and the scaled tetrahedron, I can compute their volumes and find the ratio.Alright, let's assign coordinates to the original tetrahedron. Since it's regular, it's often convenient to place it in a coordinate system such that its vertices are easy to define. One common way is to place the vertices at (1, 1, 1), (-1, -1, 1), (-1, 1, -1), and (1, -1, -1). But wait, that might complicate things because the edge lengths would need to be calculated.Alternatively, maybe I can use a more straightforward coordinate system. Let me think. If I place one vertex at (0, 0, 0), another at (1, 0, 0), another at (1/2, √3/2, 0), and the last one at (1/2, √3/6, √6/3). Hmm, that might be a regular tetrahedron. Let me verify the edge lengths.The distance between (0, 0, 0) and (1, 0, 0) is 1. The distance between (0, 0, 0) and (1/2, √3/2, 0) is sqrt[(1/2)^2 + (√3/2)^2] = sqrt[1/4 + 3/4] = sqrt[1] = 1. Similarly, the distance between (0, 0, 0) and (1/2, √3/6, √6/3) is sqrt[(1/2)^2 + (√3/6)^2 + (√6/3)^2]. Let me compute that:(1/2)^2 = 1/4(√3/6)^2 = 3/36 = 1/12(√6/3)^2 = 6/9 = 2/3Adding these up: 1/4 + 1/12 + 2/3 = 3/12 + 1/12 + 8/12 = 12/12 = 1. So, that distance is also 1. Great, so all edges from the origin are length 1. Now, let's check the distance between (1, 0, 0) and (1/2, √3/2, 0). That's sqrt[(1 - 1/2)^2 + (0 - √3/2)^2] = sqrt[(1/2)^2 + (√3/2)^2] = sqrt[1/4 + 3/4] = sqrt[1] = 1. Similarly, the distance between (1, 0, 0) and (1/2, √3/6, √6/3) is sqrt[(1 - 1/2)^2 + (0 - √3/6)^2 + (0 - √6/3)^2] = sqrt[(1/2)^2 + (√3/6)^2 + (√6/3)^2] = same as before, which is 1. So, all edges are indeed length 1. So, this is a regular tetrahedron with edge length 1.Now, if I scale each coordinate by 1/2, the new coordinates become:(0, 0, 0) scaled by 1/2 is still (0, 0, 0).(1, 0, 0) becomes (1/2, 0, 0).(1/2, √3/2, 0) becomes (1/4, √3/4, 0).(1/2, √3/6, √6/3) becomes (1/4, √3/12, √6/6).Wait, so the new tetrahedron has vertices at (0, 0, 0), (1/2, 0, 0), (1/4, √3/4, 0), and (1/4, √3/12, √6/6). Hmm, is this a regular tetrahedron? Let me check the edge lengths.First, the distance between (0, 0, 0) and (1/2, 0, 0) is 1/2.Distance between (0, 0, 0) and (1/4, √3/4, 0): sqrt[(1/4)^2 + (√3/4)^2] = sqrt[1/16 + 3/16] = sqrt[4/16] = sqrt[1/4] = 1/2.Distance between (0, 0, 0) and (1/4, √3/12, √6/6): sqrt[(1/4)^2 + (√3/12)^2 + (√6/6)^2]. Let's compute each term:(1/4)^2 = 1/16(√3/12)^2 = 3/144 = 1/48(√6/6)^2 = 6/36 = 1/6Adding these: 1/16 + 1/48 + 1/6. Let's convert to 48 denominators:1/16 = 3/481/48 = 1/481/6 = 8/48Total: 3 + 1 + 8 = 12/48 = 1/4. So sqrt(1/4) = 1/2. So, all edges from (0, 0, 0) are 1/2.Now, distance between (1/2, 0, 0) and (1/4, √3/4, 0): sqrt[(1/2 - 1/4)^2 + (0 - √3/4)^2] = sqrt[(1/4)^2 + (√3/4)^2] = sqrt[1/16 + 3/16] = sqrt[4/16] = 1/2.Distance between (1/2, 0, 0) and (1/4, √3/12, √6/6): sqrt[(1/2 - 1/4)^2 + (0 - √3/12)^2 + (0 - √6/6)^2] = sqrt[(1/4)^2 + (√3/12)^2 + (√6/6)^2]. We already computed this earlier as 1/2.Similarly, distance between (1/4, √3/4, 0) and (1/4, √3/12, √6/6): sqrt[(1/4 - 1/4)^2 + (√3/4 - √3/12)^2 + (0 - √6/6)^2] = sqrt[0 + ( (3√3/12 - √3/12) )^2 + (√6/6)^2] = sqrt[(2√3/12)^2 + (√6/6)^2] = sqrt[(√3/6)^2 + (√6/6)^2] = sqrt[3/36 + 6/36] = sqrt[9/36] = sqrt[1/4] = 1/2.So, all edges are indeed 1/2. Therefore, the scaled tetrahedron is also a regular tetrahedron with edge length 1/2.Now, since the original tetrahedron had edge length 1, its volume is V1 = (1^3)/(6√2) = 1/(6√2).The scaled tetrahedron has edge length 1/2, so its volume V2 = ((1/2)^3)/(6√2) = (1/8)/(6√2) = 1/(48√2).Therefore, the ratio V2/V1 = (1/(48√2)) / (1/(6√2)) = (1/48√2) * (6√2/1) = (6√2)/(48√2) = 6/48 = 1/8.So, the ratio is 1/8, which is already in simplest terms, so p = 1 and q = 8, hence p + q = 9.Wait, but hold on. The original tetrahedron I considered had edge length 1, but in the problem, it's a regular tetrahedron without specifying edge length. So, does this ratio hold regardless of the original edge length? Let me think.Yes, scaling affects volume proportionally regardless of the original size. So, if you scale by 1/2, the volume scales by (1/2)^3 = 1/8, regardless of the original edge length. So, even if the original tetrahedron had a different edge length, say 'a', scaling each coordinate by 1/2 would scale the edge lengths by 1/2, and hence the volume by (1/2)^3.Therefore, the ratio is indeed 1/8, so p + q = 1 + 8 = 9.But just to be thorough, let me consider another approach. Maybe using linear algebra and determinants.The volume of a tetrahedron can be found using the scalar triple product of vectors. If I have four points A, B, C, D, then the volume is |(AB · (AC × AD))| / 6.Let me assign coordinates again. Let me choose a regular tetrahedron with vertices at (1, 1, 1), (-1, -1, 1), (-1, 1, -1), and (1, -1, -1). Wait, is this a regular tetrahedron? Let me check the distances.Distance between (1,1,1) and (-1,-1,1): sqrt[(-2)^2 + (-2)^2 + 0^2] = sqrt[4 + 4] = sqrt[8] = 2√2.Distance between (1,1,1) and (-1,1,-1): sqrt[(-2)^2 + 0^2 + (-2)^2] = sqrt[4 + 4] = sqrt[8] = 2√2.Similarly, distance between (1,1,1) and (1,-1,-1): sqrt[0^2 + (-2)^2 + (-2)^2] = sqrt[4 + 4] = sqrt[8] = 2√2.So, all edges from (1,1,1) are 2√2. Similarly, distance between (-1,-1,1) and (-1,1,-1): sqrt[0^2 + (2)^2 + (-2)^2] = sqrt[4 + 4] = sqrt[8] = 2√2.Same for others. So, all edges are 2√2. So, edge length is 2√2.Now, scaling each coordinate by 1/2, the new coordinates become:(1/2, 1/2, 1/2), (-1/2, -1/2, 1/2), (-1/2, 1/2, -1/2), (1/2, -1/2, -1/2).Now, let's compute the edge lengths of the scaled tetrahedron.Distance between (1/2,1/2,1/2) and (-1/2,-1/2,1/2): sqrt[(-1)^2 + (-1)^2 + 0^2] scaled by 1/2? Wait, no, let's compute it directly.sqrt[(1/2 - (-1/2))^2 + (1/2 - (-1/2))^2 + (1/2 - 1/2)^2] = sqrt[(1)^2 + (1)^2 + 0] = sqrt[2].Similarly, distance between (1/2,1/2,1/2) and (-1/2,1/2,-1/2): sqrt[(1)^2 + 0 + (-1)^2] = sqrt[2].Same for others. So, all edges are sqrt[2]. So, edge length is sqrt[2].Original edge length was 2√2, scaled edge length is sqrt[2], which is 1/2 of the original. So, scaling factor is 1/2.Therefore, volume scales by (1/2)^3 = 1/8.Thus, the ratio is 1/8, so p + q = 9.Alternatively, let me compute the volumes using the scalar triple product.Original tetrahedron: Let me take points A(1,1,1), B(-1,-1,1), C(-1,1,-1), D(1,-1,-1).Vectors AB = B - A = (-2, -2, 0)Vectors AC = C - A = (-2, 0, -2)Vectors AD = D - A = (0, -2, -2)Compute the scalar triple product AB · (AC × AD).First, compute AC × AD.AC = (-2, 0, -2)AD = (0, -2, -2)Cross product:|i j k||-2 0 -2||0 -2 -2|= i*(0*(-2) - (-2)*(-2)) - j*(-2*(-2) - (-2)*0) + k*(-2*(-2) - 0*0)= i*(0 - 4) - j*(4 - 0) + k*(4 - 0)= (-4i - 4j + 4k)So, AC × AD = (-4, -4, 4)Now, AB · (AC × AD) = (-2, -2, 0) · (-4, -4, 4) = (-2)*(-4) + (-2)*(-4) + 0*4 = 8 + 8 + 0 = 16Therefore, volume V1 = |16| / 6 = 16/6 = 8/3.Wait, but earlier I thought the volume was 1/(6√2). Hmm, that's conflicting. Wait, maybe I messed up the coordinates.Wait, no, in the first approach, I considered a tetrahedron with edge length 1, but in this case, the edge length is 2√2, so the volume should be larger.Wait, let me compute the volume using the formula V = (a^3)/(6√2), where a is edge length.Here, a = 2√2, so V = ( (2√2)^3 ) / (6√2) = (16√2) / (6√2) = 16/6 = 8/3. Okay, that matches the scalar triple product result. So, V1 = 8/3.Now, the scaled tetrahedron has edge length sqrt[2], so its volume V2 = ( (sqrt[2])^3 ) / (6√2) = (2√2) / (6√2) = 2/6 = 1/3.Therefore, the ratio V2/V1 = (1/3)/(8/3) = 1/8. So, same result.Therefore, regardless of the approach, the ratio is 1/8, so p + q = 9.Wait, but let me think again. The problem says "scale down the coordinates of its vertices by half". So, does that mean scaling each coordinate by 1/2, which scales the edge lengths by 1/2, hence volume by 1/8? Yes, that's consistent.Alternatively, if I think about the centroid of the tetrahedron. Scaling about the centroid by 1/2 would also scale the volume by 1/8. But in this case, we're scaling all coordinates by 1/2, which is equivalent to scaling about the origin. But since the origin is one of the vertices in my first coordinate system, it's a bit different.Wait, in the first coordinate system, one vertex was at (0,0,0), and others were at (1,0,0), etc. So, scaling about the origin would scale the position vectors of the vertices, but the origin remains fixed. So, in that case, the new tetrahedron is similar to the original, scaled by 1/2, but with one vertex at the origin.But in the second coordinate system, all vertices are scaled, so the origin is not a vertex. Hmm, but regardless, the scaling factor is 1/2, so the volume scales by 1/8.I think I've confirmed this in multiple ways: using the volume formula, using the scalar triple product, and considering the scaling factor's effect on volume. So, I'm confident that the ratio is 1/8, so p + q = 9.Just to make sure, let me think of another way. Maybe using matrices. If I scale each coordinate by 1/2, the transformation matrix is a diagonal matrix with 1/2 on the diagonal. The determinant of this matrix is (1/2)^3 = 1/8, which is the scaling factor for volume. So, the volume of the transformed tetrahedron is 1/8 of the original. Therefore, the ratio is 1/8.Yes, that's another way to see it. The determinant of the scaling matrix gives the factor by which volumes are scaled. So, since we're scaling each axis by 1/2, the determinant is (1/2)^3 = 1/8.Therefore, regardless of the method, the ratio is 1/8, so p + q = 9.I think I've thought this through thoroughly. I considered different coordinate systems, computed volumes directly, used the scalar triple product, and even thought about linear transformations and determinants. All approaches led me to the same conclusion. So, I'm pretty sure the answer is 9.

answer：Okay, so I need to figure out how much money I should invest now at an annual interest rate of 4% so that in 15 years, I'll have 600,000. Hmm, this sounds like a problem involving compound interest. I remember that compound interest means that the interest is calculated on the initial principal and also on the accumulated interest from previous periods. Since it's compounded annually, the interest is added once each year.First, I think I need to recall the formula for compound interest. I believe it's something like A equals P times (1 plus r) to the power of n, where A is the amount of money accumulated after n years, including interest. P is the principal amount, which is the initial amount of money invested. R is the annual interest rate, and n is the number of years the money is invested for.So, the formula should be:[ A = P times (1 + r)^n ]In this case, I know that A is 600,000, r is 4% or 0.04, and n is 15 years. I need to find P, the principal amount I need to invest now.So, rearranging the formula to solve for P, I get:[ P = frac{A}{(1 + r)^n} ]Plugging in the values I have:[ P = frac{600,000}{(1 + 0.04)^{15}} ]Now, I need to calculate (1 + 0.04) to the power of 15. Let me see, 1.04 raised to the 15th power. I think I can use a calculator for this part. Let me try calculating that.First, 1.04 to the power of 1 is 1.04.1.04 squared is 1.0816.1.04 cubed is approximately 1.124864.Continuing this way up to the 15th power would take too long manually, so I'll use a calculator.Using a calculator, 1.04 raised to the 15th power is approximately 1.8009435.So, now I can substitute that back into the equation:[ P = frac{600,000}{1.8009435} ]Now, dividing 600,000 by 1.8009435:[ P ≈ 333,088.86 ]So, I would need to invest approximately 333,088.86 today at an annual interest rate of 4% compounded annually to have 600,000 in 15 years.Wait, let me double-check my calculations to make sure I didn't make any mistakes. I'll recalculate (1.04)^15 to confirm it's approximately 1.8009435. Yes, that seems correct. Then, dividing 600,000 by 1.8009435 does give approximately 333,088.86.Just to be thorough, I'll consider if there are any other factors I might have missed, like taxes or fees, but the problem doesn't mention any, so I think it's safe to ignore those for now.Therefore, my final answer is that I need to invest approximately 333,088.86 today.

answer：Alright, so I've got this problem about earthwork transportation in a city renovation project. It's divided into two parts. Let me try to figure out how to tackle each part step by step.**Part 1: Finding the Earthwork Capacity of Each Vehicle**Okay, the problem says that 3 large vehicles and 4 small vehicles can transport 44 tons at once. Similarly, 4 large and 6 small can transport 62 tons. I need to find out how much each large and small vehicle can carry individually.Hmm, this sounds like a system of equations problem. Let me denote:- Let ( x ) be the capacity of one large vehicle in tons.- Let ( y ) be the capacity of one small vehicle in tons.So, from the first statement: ( 3x + 4y = 44 ).From the second statement: ( 4x + 6y = 62 ).Now, I have two equations:1. ( 3x + 4y = 44 )2. ( 4x + 6y = 62 )I need to solve this system to find ( x ) and ( y ).I think the best way is to use the elimination method. Let me try to eliminate one of the variables. Maybe I can eliminate ( y ) by making the coefficients of ( y ) the same in both equations.Looking at the coefficients:- In the first equation, the coefficient of ( y ) is 4.- In the second equation, it's 6.The least common multiple of 4 and 6 is 12. So, I can multiply the first equation by 3 and the second equation by 2 to make the coefficients of ( y ) equal to 12.Multiplying the first equation by 3:( 3*(3x + 4y) = 3*44 )Which gives:( 9x + 12y = 132 ) ... (3)Multiplying the second equation by 2:( 2*(4x + 6y) = 2*62 )Which gives:( 8x + 12y = 124 ) ... (4)Now, I have:3. ( 9x + 12y = 132 )4. ( 8x + 12y = 124 )Now, subtract equation (4) from equation (3):( (9x + 12y) - (8x + 12y) = 132 - 124 )Simplify:( 9x - 8x + 12y - 12y = 8 )Which simplifies to:( x = 8 )So, one large vehicle can carry 8 tons. Now, I need to find ( y ).Let me substitute ( x = 8 ) back into one of the original equations. Let's use the first one:( 3x + 4y = 44 )Substitute ( x = 8 ):( 3*8 + 4y = 44 )Calculate:( 24 + 4y = 44 )Subtract 24 from both sides:( 4y = 20 )Divide by 4:( y = 5 )So, one small vehicle can carry 5 tons.Alright, that seems straightforward. I think that's the answer for part (1).**Part 2: Finding Possible Dispatch Plans**Now, the company is dispatching a total of 12 vehicles, both large and small. The total earthwork transported each time should be at least 78 tons, and at least 4 small vehicles must be dispatched.Let me denote:- Let ( m ) be the number of small vehicles dispatched.- Then, the number of large vehicles dispatched will be ( 12 - m ).Given that at least 4 small vehicles are dispatched, ( m geq 4 ).Also, the total earthwork transported should be at least 78 tons. So, the total capacity should satisfy:( 8*(12 - m) + 5*m geq 78 )Let me write that out:( 8*(12 - m) + 5m geq 78 )First, expand the terms:( 96 - 8m + 5m geq 78 )Combine like terms:( 96 - 3m geq 78 )Subtract 96 from both sides:( -3m geq -18 )Now, divide both sides by -3. Remember, when you divide or multiply both sides of an inequality by a negative number, the inequality sign flips.So,( m leq 6 )So, ( m leq 6 ).But we also have ( m geq 4 ) from the condition that at least 4 small vehicles are dispatched.Therefore, ( m ) can be 4, 5, or 6.So, the possible dispatch plans are:1. ( m = 4 ): 4 small vehicles and ( 12 - 4 = 8 ) large vehicles.2. ( m = 5 ): 5 small vehicles and ( 12 - 5 = 7 ) large vehicles.3. ( m = 6 ): 6 small vehicles and ( 12 - 6 = 6 ) large vehicles.Let me verify if these satisfy the total earthwork requirement.1. For ( m = 4 ):Total capacity = ( 8*8 + 5*4 = 64 + 20 = 84 ) tons. Which is more than 78 tons.2. For ( m = 5 ):Total capacity = ( 8*7 + 5*5 = 56 + 25 = 81 ) tons. Also more than 78 tons.3. For ( m = 6 ):Total capacity = ( 8*6 + 5*6 = 48 + 30 = 78 ) tons. Exactly 78 tons.So, all three plans satisfy the condition of at least 78 tons.Therefore, the possible dispatch plans are:- 8 large and 4 small vehicles,- 7 large and 5 small vehicles,- 6 large and 6 small vehicles.I think that's all the possible plans given the constraints.**Summary**So, after solving the system of equations for part (1), I found that one large vehicle can transport 8 tons and one small vehicle can transport 5 tons. For part (2), considering the constraints of total vehicles, minimum small vehicles, and the total earthwork requirement, I determined that there are three possible dispatch plans: 8 large and 4 small, 7 large and 5 small, and 6 large and 6 small vehicles.I should double-check my calculations to make sure I didn't make any mistakes.For part (1):- ( 3x + 4y = 44 )- ( 4x + 6y = 62 )Solving these, I got ( x = 8 ) and ( y = 5 ). Plugging back in:- ( 3*8 + 4*5 = 24 + 20 = 44 ) ✔️- ( 4*8 + 6*5 = 32 + 30 = 62 ) ✔️Good.For part (2):- Total vehicles: 12- Minimum small vehicles: 4- Total capacity: at least 78 tonsCalculations:- ( 8*(12 - m) + 5m geq 78 )- Simplified to ( m leq 6 )- So, ( m = 4, 5, 6 )Total capacities:- ( m = 4 ): 84 tons ✔️- ( m = 5 ): 81 tons ✔️- ( m = 6 ): 78 tons ✔️All good. I think I've covered all the necessary steps and verified my answers.

answer：Okay, so I have this problem here where I need to find the sum of g(n) from n equals 1 to infinity. The function g(n) is defined as the sum from k equals 1 to infinity of 1 divided by k raised to the power of n. That is, g(n) is 1/1^n + 1/2^n + 1/3^n + 1/4^n and so on. First, I remember that this kind of series is called a p-series, where p is the exponent in the denominator. The general form is the sum of 1/k^p from k=1 to infinity. I also recall that a p-series converges if p is greater than 1 and diverges if p is less than or equal to 1. So, for g(n), when n is 1, it becomes the harmonic series, which is known to diverge. For n greater than 1, like 2, 3, and so on, the series converges. That means g(n) is finite for n ≥ 2 but infinite when n=1. Now, the problem asks for the sum of g(n) from n=1 to infinity. So, I need to compute the sum of these series for each n starting from 1. That is, I have to add up g(1) + g(2) + g(3) + ... But wait, g(1) is the harmonic series, which diverges. So, if I'm adding up g(n) starting from n=1, and one of the terms in that sum is already infinite, does that mean the entire sum diverges? Let me think about it more carefully. The sum is an infinite series where each term is itself an infinite series. So, it's a double sum: sum over n from 1 to infinity of sum over k from 1 to infinity of 1/(k^n). I wonder if I can switch the order of summation. Maybe I can write it as sum over k from 1 to infinity of sum over n from 1 to infinity of 1/(k^n). If that's allowed, then for each k, I have a geometric series in n. Yes, for each fixed k, the sum over n from 1 to infinity of 1/(k^n) is a geometric series with first term 1/k and common ratio 1/k. The sum of a geometric series is a/(1 - r), where a is the first term and r is the common ratio, provided that |r| < 1. So, for each k, the inner sum becomes (1/k) / (1 - 1/k) = 1/(k - 1). Therefore, the entire double sum becomes the sum over k from 1 to infinity of 1/(k - 1). Wait a minute, when k=1, 1/(k - 1) is 1/0, which is undefined. So, actually, the first term when k=1 is problematic. Let me check my steps again. When k=1, the inner sum is sum over n from 1 to infinity of 1/(1^n) = sum over n from 1 to infinity of 1, which is clearly divergent. So, even if I switch the order of summation, the term when k=1 is still divergent. Therefore, the entire double sum diverges because one of the terms in the outer sum is divergent. So, putting it all together, the sum from n=1 to infinity of g(n) is divergent because it includes the harmonic series as one of its terms, which itself diverges. I think that's the conclusion. The sum doesn't converge; it goes to infinity.

answer：Alright, so I have this function here: ( y = frac{1}{sqrt{log_{0.5}(4x - 3)}} ). I need to find its domain. Hmm, okay. Let me think about what steps I need to take.First, I remember that for a function to be defined, especially when dealing with fractions, square roots, and logarithms, there are certain conditions that need to be satisfied. Let me break it down step by step.Starting with the logarithm part: ( log_{0.5}(4x - 3) ). I know that the logarithm function is only defined when its argument is positive. So, ( 4x - 3 ) must be greater than 0. Let me write that down:( 4x - 3 > 0 )Solving for x:( 4x > 3 )( x > frac{3}{4} )Okay, so x has to be greater than 3/4. That's one condition.Next, since the logarithm is inside a square root, the expression inside the square root must be positive because we can't take the square root of a negative number, and also, we can't have zero in the denominator because division by zero is undefined. So, the expression inside the square root, which is ( log_{0.5}(4x - 3) ), must be greater than zero. Let me write that:( log_{0.5}(4x - 3) > 0 )Now, I need to solve this inequality. Remember that ( log_b(a) > 0 ) depends on the base b. If the base is between 0 and 1, the logarithm function is decreasing. So, for ( log_{0.5}(4x - 3) > 0 ), since the base is 0.5 (which is less than 1), the inequality flips when we remove the logarithm. Let me recall the rule: if ( log_b(a) > c ) and ( 0 < b < 1 ), then ( a < b^c ).Applying this to my inequality:( 4x - 3 < (0.5)^0 )Since ( (0.5)^0 = 1 ), this simplifies to:( 4x - 3 < 1 )Solving for x:( 4x < 4 )( x < 1 )So, combining this with the earlier condition, x must be greater than 3/4 and less than 1. So, the domain is ( (frac{3}{4}, 1) ).Wait, but let me double-check. Is there any other condition I might have missed? The denominator is ( sqrt{log_{0.5}(4x - 3)} ), so not only does the logarithm have to be positive, but also the entire denominator can't be zero. But since we already have ( log_{0.5}(4x - 3) > 0 ), that takes care of it because the square root of a positive number is also positive, so the denominator is never zero in this interval.Is there a possibility that ( 4x - 3 ) could be equal to 1? If ( 4x - 3 = 1 ), then ( x = 1 ). But at x = 1, ( log_{0.5}(1) = 0 ), which would make the denominator ( sqrt{0} = 0 ), leading to division by zero. So, x cannot be equal to 1. Similarly, at x = 3/4, ( 4x - 3 = 0 ), which is not allowed in the logarithm. So, x must be strictly between 3/4 and 1.Therefore, the domain is ( (frac{3}{4}, 1) ), which corresponds to option A.Wait, but hold on a second. Let me think about the logarithm base again. The base is 0.5, which is less than 1, so the logarithm function is indeed decreasing. So, when I have ( log_{0.5}(4x - 3) > 0 ), it translates to ( 4x - 3 < 1 ), which is correct. So, x < 1. And since ( 4x - 3 > 0 ), x > 3/4. So, yeah, the interval is ( (frac{3}{4}, 1) ).Is there any other aspect I need to consider? For example, does the function have any asymptotes or points where it's undefined within this interval? Well, within ( (frac{3}{4}, 1) ), the logarithm is positive, so the square root is defined, and the denominator is non-zero. So, the function is defined throughout this interval.Therefore, I'm confident that the domain is ( (frac{3}{4}, 1) ), which is option A.**Final Answer**boxed{A}