answer：Okay, so I have this problem about a regular dodecahedron, and I need to find the probability that if I randomly choose two vertices, they will be endpoints of an edge. Hmm, let me try to break this down step by step.First, I should recall what a regular dodecahedron is. From what I remember, it's one of the Platonic solids, right? It has 12 faces, and each face is a regular pentagon. But wait, actually, no, that's the regular dodecahedron. Wait, no, the regular dodecahedron has 12 faces, each of which is a regular pentagon, and it has 20 vertices. Wait, is that right? Or is it the other way around? Let me think. No, actually, the regular dodecahedron has 12 faces, 30 edges, and 20 vertices. Yeah, that sounds more familiar. So, 20 vertices, 30 edges, 12 faces.Okay, so the problem is about choosing two vertices at random. So, the total number of possible pairs of vertices is the combination of 20 vertices taken 2 at a time. The formula for combinations is n choose k, which is n! / (k! (n - k)!). So, in this case, it would be 20 choose 2, which is (20 × 19) / (2 × 1) = 190. So, there are 190 possible pairs of vertices.Now, out of these 190 pairs, how many of them are connected by an edge? Well, since the dodecahedron has 30 edges, each edge connects two vertices, so there are 30 such pairs that are endpoints of an edge.Therefore, the probability that two randomly chosen vertices are connected by an edge is the number of favorable outcomes (which is 30) divided by the total number of possible outcomes (which is 190). So, the probability is 30/190. Simplifying that fraction, both numerator and denominator are divisible by 10, so it becomes 3/19.Wait, but hold on a second. I think I might have made a mistake here. Earlier, I thought the dodecahedron has 12 vertices, but now I'm recalling it has 20 vertices. Let me double-check that. A regular dodecahedron has 12 faces, each face being a pentagon, so each face has 5 edges. But each edge is shared by two faces, so the total number of edges is (12 × 5) / 2 = 30. Okay, that matches up. Now, using Euler's formula for polyhedrons, which is V - E + F = 2, where V is vertices, E is edges, and F is faces. Plugging in the numbers, V - 30 + 12 = 2, so V = 20. Yes, so it has 20 vertices.So, going back, the total number of vertex pairs is 20 choose 2, which is 190, and the number of edges is 30. So, the probability is 30/190, which simplifies to 3/19.But wait, I think I might have confused the dodecahedron with another Platonic solid. Let me make sure. The regular dodecahedron has 12 faces, 30 edges, and 20 vertices. The regular icosahedron, on the other hand, has 20 faces, 30 edges, and 12 vertices. So, yes, I was correct earlier; the dodecahedron has 20 vertices.Therefore, my calculation seems correct: 20 vertices, 30 edges, so 30 favorable pairs out of 190 possible pairs, leading to a probability of 3/19.But just to be thorough, let me think about it another way. Each vertex in a regular dodecahedron is connected to how many other vertices? Since each face is a pentagon, and each vertex is part of multiple faces. Wait, actually, in a regular dodecahedron, each vertex is where three pentagonal faces meet. So, each vertex is connected to three other vertices by edges. Therefore, each vertex has a degree of 3.So, if each of the 20 vertices has 3 edges, then the total number of edges would be (20 × 3) / 2 = 30, which matches what we had before. So, that's consistent.Therefore, the number of edges is indeed 30, and the total number of vertex pairs is 190. So, the probability is 30/190, which simplifies to 3/19.Wait, but in the initial problem statement, it says "two of the vertices of a regular dodecahedron are to be chosen at random." So, I think my reasoning is correct.But just to make sure, let me think about another Platonic solid, like the cube. A cube has 8 vertices, 12 edges. The probability of choosing two vertices that are connected by an edge would be 12 / (8 choose 2) = 12 / 28 = 3/7. That makes sense.Similarly, for a tetrahedron, which has 4 vertices and 6 edges, the probability would be 6 / (4 choose 2) = 6 / 6 = 1, which makes sense because every pair of vertices is connected by an edge.So, applying the same logic to the dodecahedron, which has 20 vertices and 30 edges, the probability is 30 / (20 choose 2) = 30 / 190 = 3/19.Therefore, I think my answer is correct.

answer：To solve the given problem, we start by understanding the conditions for the equation ((a-2)x^{|a|-1} = -2) to be linear in (x). For an equation to be linear in (x), the power of (x) must be 1. Therefore, the absolute value expression (|a| - 1) must equal 1. We also note that (a - 2) cannot be zero because if (a - 2 = 0), the equation would not be valid as it would imply division by zero or the equation would become (0 = -2), which is not true. Hence, we have two conditions:1. (|a| - 1 = 1)2. (a - 2 neq 0)Let's solve these step by step:**Step 1: Solve (|a| - 1 = 1)**[|a| - 1 = 1 implies |a| = 2]This means (a) can be (2) or (-2). However, we must also consider the second condition.**Step 2: Consider (a - 2 neq 0)**Given (a - 2 neq 0), (a) cannot be (2) because that would make (a - 2 = 0), which is not allowed. Therefore, the only solution that satisfies both conditions is (a = -2).Thus, the value of (a) that makes ((a - 2)x^{|a| - 1} = -2) a linear equation in (x) is (boxed{-2}).

answer：Okay, so I have this problem about double-bottom sequences. Let me try to understand what a double-bottom sequence is. The definition says that a sequence {a_n} is a double-bottom sequence if two conditions are satisfied:1. There exist distinct natural numbers p and q such that a_p = a_q = c, where c is some constant.2. For any other natural number n (not equal to p or q), a_n is greater than c.So, essentially, the sequence has exactly two terms that are equal to the minimum value c, and all other terms are larger than c. Got it.Now, the problem has three parts. Let me tackle them one by one.**Part (1): Determine whether the given sequences are double-bottom sequences.**There are three sequences given:(i) a_n = n + 6/n(ii) a_n = sin(nπ/2)(iii) a_n = |(n - 3)(n - 5)|I need to check each of these to see if they satisfy the conditions of a double-bottom sequence.Starting with (i): a_n = n + 6/n.Hmm, this looks like a function that might have a minimum. Let me think about its behavior. For positive integers n, as n increases, n + 6/n will behave like n, which goes to infinity. But for small n, it might have a minimum.Let me compute some terms:n=1: 1 + 6/1 = 7n=2: 2 + 6/2 = 2 + 3 = 5n=3: 3 + 6/3 = 3 + 2 = 5n=4: 4 + 6/4 = 4 + 1.5 = 5.5n=5: 5 + 6/5 = 5 + 1.2 = 6.2n=6: 6 + 6/6 = 6 + 1 = 7So, the terms are 7, 5, 5, 5.5, 6.2, 7,...Wait, so at n=2 and n=3, the value is 5, which is the minimum. All other terms are greater than 5. So, this seems to satisfy the conditions. So, (i) is a double-bottom sequence.Moving on to (ii): a_n = sin(nπ/2).Let me compute the terms:n=1: sin(π/2) = 1n=2: sin(π) = 0n=3: sin(3π/2) = -1n=4: sin(2π) = 0n=5: sin(5π/2) = 1n=6: sin(3π) = 0n=7: sin(7π/2) = -1n=8: sin(4π) = 0So, the sequence goes: 1, 0, -1, 0, 1, 0, -1, 0,...Looking at this, the minimum value is -1, which occurs at n=3, 7, 11, etc. So, there are multiple terms where a_n = -1, not just two. Therefore, it doesn't satisfy condition (1) because there are more than two p and q where a_p = a_q = c. So, (ii) is not a double-bottom sequence.Now, (iii): a_n = |(n - 3)(n - 5)|Let me compute some terms:n=1: |(1-3)(1-5)| = |(-2)(-4)| = |8| = 8n=2: |(2-3)(2-5)| = |(-1)(-3)| = |3| = 3n=3: |(3-3)(3-5)| = |0*(-2)| = 0n=4: |(4-3)(4-5)| = |1*(-1)| = 1n=5: |(5-3)(5-5)| = |2*0| = 0n=6: |(6-3)(6-5)| = |3*1| = 3n=7: |(7-3)(7-5)| = |4*2| = 8So, the sequence is: 8, 3, 0, 1, 0, 3, 8,...Looking at this, the minimum value is 0, which occurs at n=3 and n=5. All other terms are greater than 0. So, this satisfies both conditions: two distinct terms equal to the minimum, and all others larger. So, (iii) is a double-bottom sequence.**Part (2): Given a piecewise sequence, find m and S_n.**The sequence is defined as:a_n = { 101 - 2n, for 1 ≤ n ≤ 50 { 2^{n - 50} + m, for n > 50We are told that this sequence is a double-bottom sequence. So, we need to find m and the sum S_n.First, since it's a double-bottom sequence, there must be exactly two terms equal to the minimum c, and all other terms must be greater than c.Looking at the sequence, for n from 1 to 50, it's an arithmetic sequence starting at 101 - 2*1 = 99, decreasing by 2 each time: 99, 97, 95,..., up to n=50: 101 - 2*50 = 101 - 100 = 1.Then, for n > 50, it's 2^{n - 50} + m. So, starting at n=51: 2^{1} + m = 2 + m, n=52: 4 + m, etc., which is an increasing sequence.Since the sequence is a double-bottom, the minimum value c must occur at two distinct positions. Let's see where the minimum could be.Looking at the first part (n ≤ 50), the sequence is decreasing from 99 to 1. The last term is 1. Then, for n=51, it's 2 + m. So, if 2 + m is less than 1, then the minimum would be at n=51, but we need two minima. Alternatively, if 2 + m is equal to 1, then n=50 and n=51 both have a_n = 1, which would make them the two minima.So, let's set a_{50} = a_{51}:a_{50} = 101 - 2*50 = 1a_{51} = 2^{51 - 50} + m = 2 + mSetting them equal: 1 = 2 + m => m = -1.So, m must be -1.Now, let's confirm that this makes the sequence a double-bottom. With m = -1, a_{51} = 2 - 1 = 1, same as a_{50}. All other terms:For n ≤ 50: a_n = 101 - 2n, which is decreasing from 99 to 1. So, all terms except a_{50} are greater than 1.For n > 50: a_n = 2^{n - 50} - 1. Since 2^{n - 50} is increasing, starting at 2 for n=51, so a_{51}=1, a_{52}=4 -1=3, a_{53}=8 -1=7, etc., all greater than 1.Thus, the minimum value is 1, achieved at n=50 and n=51, and all other terms are greater than 1. So, it's a double-bottom sequence.Now, we need to find S_n, the sum of the first n terms.We have two cases: n ≤ 50 and n > 50.Case 1: 1 ≤ n ≤ 50.The sequence is an arithmetic sequence with first term a_1 = 99, common difference d = -2.The sum S_n is given by:S_n = (n/2)(2a_1 + (n - 1)d)Plugging in:S_n = (n/2)(2*99 + (n - 1)*(-2)) = (n/2)(198 - 2n + 2) = (n/2)(200 - 2n) = n(100 - n) = 100n - n^2.Case 2: n > 50.We need to compute the sum of the first 50 terms plus the sum from n=51 to n.Sum of first 50 terms: S_{50} = 100*50 - 50^2 = 5000 - 2500 = 2500.Now, for n > 50, the terms from 51 to n are a geometric sequence starting at a_{51} = 1, with common ratio 2.Wait, hold on: a_n = 2^{n - 50} - 1 for n > 50.But wait, when n=51, a_{51}=2^{1} -1=1, n=52: 4 -1=3, n=53:8 -1=7, etc. So, the sequence is 1, 3, 7, 15,... which is 2^1 -1, 2^2 -1, 2^3 -1,...So, from n=51 onwards, a_n = 2^{n - 50} -1.So, the sum from n=51 to n=k is sum_{m=1}^{k - 50} (2^m -1) = sum_{m=1}^{k -50} 2^m - sum_{m=1}^{k -50} 1.Sum of 2^m from m=1 to t is 2^{t+1} - 2.Sum of 1 from m=1 to t is t.So, sum from n=51 to n=k is (2^{k -50 +1} - 2) - (k -50) = 2^{k -49} - 2 - k +50 = 2^{k -49} - k +48.Therefore, the total sum S_n for n >50 is S_{50} + sum from 51 to n.So, S_n = 2500 + 2^{n -49} - n +48 = 2^{n -49} -n +2548.Thus, summarizing:For 1 ≤ n ≤50: S_n = 100n -n^2For n >50: S_n =2^{n -49} -n +2548**Part (3): Determine if there's an integer k such that a_n = (kn +3)(9/10)^n is a double-bottom sequence.**So, we need to find integer k such that the sequence has exactly two terms equal to the minimum c, and all other terms are greater than c.First, let's analyze the behavior of a_n.a_n = (kn +3)(9/10)^nSince (9/10)^n is a decaying exponential, the term (kn +3) is linear in n.Depending on the sign of k, the behavior changes.Case 1: k >0Then, kn +3 is increasing, but multiplied by a decaying exponential. So, a_n might have a single maximum and then decay. But since it's a product of a linear increasing and exponential decay, it might have a single peak.But we need two minima. Wait, but if k>0, the term kn +3 is increasing, so (kn +3)(9/10)^n might first increase and then decrease, but the minimum would be at n=1 or somewhere else?Wait, let's compute the ratio a_{n+1}/a_n:a_{n+1}/a_n = [(k(n+1)+3)(9/10)^{n+1}]/[(kn +3)(9/10)^n] = [(kn +k +3)/(kn +3)]*(9/10)Set this ratio equal to 1 to find critical points:[(kn +k +3)/(kn +3)]*(9/10) =1Multiply both sides by (kn +3):(kn +k +3)*(9/10) = kn +3Multiply both sides by 10:(kn +k +3)*9 =10kn +30Expand left side:9kn +9k +27 =10kn +30Bring all terms to one side:9kn +9k +27 -10kn -30 =0Simplify:(-kn) +9k -3 =0So, -kn +9k -3=0Solve for n:kn =9k -3n= (9k -3)/k =9 - 3/kSince n must be a positive integer, 9 -3/k must be a positive integer.So, 3/k must be an integer subtracted from 9. So, 3/k must be rational, but k is integer.So, 3/k must be integer, which implies that k divides 3.Thus, possible k are divisors of 3: k=1,3,-1,-3.But earlier, we considered k>0. So, k=1 or 3.But let's check if n=9 -3/k is positive integer.For k=1: n=9 -3=6. So, n=6.For k=3: n=9 -1=8.So, for k=1, n=6 is the critical point where a_{n+1}/a_n=1, meaning a_n has a peak at n=6.Similarly, for k=3, peak at n=8.But we need two minima. Wait, but in this case, if k>0, the sequence increases to a peak and then decreases. So, the minimum would be at n=1 or somewhere else?Wait, actually, for k>0, the sequence starts at n=1: a_1=(k +3)(9/10). Then, increases to a peak at n=6 or 8, then decreases towards zero.So, the minimum would be at n=1 or n approaching infinity (which is zero). But since n is finite, the minimum would be at n=1 or somewhere else.Wait, actually, for k>0, the sequence starts at a certain value, increases to a peak, then decreases. So, the minimum would be at n=1 or n= some other point.But in our case, for k=1, the peak is at n=6. So, a_1=(1 +3)(9/10)=4*(9/10)=3.6a_2=(2 +3)(81/100)=5*0.81=4.05a_3=(3 +3)(729/1000)=6*0.729=4.374a_4=(4 +3)(6561/10000)=7*0.6561≈4.5927a_5=(5 +3)(59049/100000)=8*0.59049≈4.7239a_6=(6 +3)(531441/1000000)=9*0.531441≈4.782969a_7=(7 +3)(4782969/10000000)=10*0.4782969≈4.782969Wait, so at n=6 and n=7, the values are approximately equal. So, a_6 ≈ a_7.But wait, let's compute more accurately.a_6=(6 +3)(9/10)^6=9*(531441/1000000)=9*0.531441=4.782969a_7=(7 +3)(9/10)^7=10*(4782969/10000000)=10*0.4782969=4.782969So, a_6 = a_7 ≈4.782969So, for k=1, the sequence has a peak at n=6 and n=7, both equal. So, the maximum is achieved at two points. But we need a double-bottom, which is a minimum. So, this is the opposite.Wait, but for k=1, the sequence increases to n=6 and n=7, then decreases. So, the minimum would be at n=1 and as n approaches infinity (which is zero). But since n is finite, the minimum is at n=1.But for a double-bottom, we need two minima. So, unless the sequence has two minima, which would require the function to have two points where it reaches the minimum.But for k>0, the sequence increases to a peak and then decreases. So, the minimum is at n=1 and as n approaches infinity, but since n is finite, the minimum is at n=1. So, only one minimum. So, k=1 doesn't work.Similarly, for k=3, let's check.n=9 -3/3=9 -1=8.So, peak at n=8.Compute a_8 and a_9.a_8=(3*8 +3)(9/10)^8=27*(43046721/100000000)=27*0.43046721≈11.62261467a_9=(3*9 +3)(9/10)^9=30*(387420489/1000000000)=30*0.387420489≈11.62261467So, a_8 = a_9≈11.62261467Again, the maximum is achieved at two points, but the minimum is at n=1.So, for k=3, the sequence increases to n=8 and n=9, then decreases. So, the minimum is at n=1.Thus, for k>0, we only get one minimum at n=1, so it's not a double-bottom sequence.Case 2: k <0Now, let's consider k negative. Let k be a negative integer.So, kn +3 is a linear function with negative slope. So, as n increases, kn +3 decreases.But multiplied by (9/10)^n, which is also decreasing. So, the product might have a single minimum or something else.Wait, let's analyze the ratio a_{n+1}/a_n again.a_{n+1}/a_n = [(k(n+1)+3)(9/10)^{n+1}]/[(kn +3)(9/10)^n] = [(kn +k +3)/(kn +3)]*(9/10)Set this equal to 1:[(kn +k +3)/(kn +3)]*(9/10) =1Multiply both sides by (kn +3):(kn +k +3)*(9/10) = kn +3Multiply both sides by 10:(kn +k +3)*9 =10kn +30Expand:9kn +9k +27 =10kn +30Bring all terms to one side:9kn +9k +27 -10kn -30 =0Simplify:(-kn) +9k -3=0So, -kn +9k -3=0Solve for n:kn =9k -3n= (9k -3)/k =9 - 3/kAgain, n must be a positive integer.Since k is negative integer, 3/k is negative, so 9 -3/k is greater than 9.But n must be a positive integer, so 9 -3/k must be integer.Let me denote k as negative integer: k = -m, where m is positive integer.So, n=9 -3/(-m)=9 +3/mSo, n=9 +3/m must be integer.Thus, 3/m must be integer, so m divides 3.Thus, m=1,3So, k= -1, -3So, possible k are -1 and -3.Let's check for k=-1:n=9 +3/1=12So, n=12.Similarly, for k=-3:n=9 +3/3=9 +1=10So, n=10.Now, let's analyze the behavior for k=-1 and k=-3.First, k=-1:a_n = (-n +3)(9/10)^nWe need to find where a_n has two minima.But let's compute the ratio a_{n+1}/a_n:a_{n+1}/a_n = [(- (n+1) +3)(9/10)^{n+1}]/[(-n +3)(9/10)^n] = [(-n +2)/(-n +3)]*(9/10)Simplify:[(-n +2)/(-n +3)] = (n -2)/(n -3)So, a_{n+1}/a_n = (n -2)/(n -3)*(9/10)Set this equal to 1:(n -2)/(n -3)*(9/10)=1Multiply both sides by (n -3):(n -2)*(9/10)=n -3Multiply both sides by 10:9(n -2)=10(n -3)9n -18=10n -30Bring terms:-18 +30=10n -9n12=nSo, n=12.Thus, the sequence a_n for k=-1 has a critical point at n=12.Let's check the behavior around n=12.Compute a_{11}, a_{12}, a_{13}.a_{11}=(-11 +3)(9/10)^11=(-8)(approx 0.313810596)= -2.510484768a_{12}=(-12 +3)(9/10)^12=(-9)(approx 0.282429536)= -2.541865824a_{13}=(-13 +3)(9/10)^13=(-10)(approx 0.254186582)= -2.54186582Wait, so a_{12}=a_{13}≈-2.54186582Wait, that's interesting. So, a_{12}=a_{13}But wait, let's compute more accurately.Compute a_{12}=(-12 +3)(9/10)^12= (-9)*(9^12/10^12)= (-9)*(282429536481/1000000000000)= (-9)*(0.282429536481)= -2.541865828329a_{13}=(-13 +3)(9/10)^13= (-10)*(9^13/10^13)= (-10)*(2541865828329/10000000000000)= (-10)*(0.2541865828329)= -2.541865828329So, a_{12}=a_{13}≈-2.541865828329So, the sequence reaches a minimum at n=12 and n=13, both equal.Now, let's check the behavior around n=12.Compute a_{11}=(-11 +3)(9/10)^11=(-8)*(31381059609/100000000000)= (-8)*(0.31381059609)= -2.51048476872a_{12}=≈-2.541865828329a_{13}=≈-2.541865828329a_{14}=(-14 +3)(9/10)^14=(-11)*(22876792454961/100000000000000)= (-11)*(0.22876792454961)=≈-2.5164471690457So, a_{14}≈-2.5164471690457So, the sequence decreases to n=12 and n=13, reaching the minimum, then starts increasing again (becoming less negative).Wait, but in terms of the actual values, the minimum is at n=12 and n=13, and all other terms are greater than this minimum (since the sequence is negative, but the minimum is the most negative, so all other terms are greater than this).Wait, but hold on: a_n is negative for n ≥4, since kn +3 for k=-1:At n=1: (-1 +3)=2, so a_1=2*(9/10)=1.8n=2: (-2 +3)=1, a_2=1*(81/100)=0.81n=3: (-3 +3)=0, a_3=0n=4: (-4 +3)=-1, a_4=-1*(729/1000)= -0.729n=5: (-5 +3)=-2, a_5=-2*(6561/10000)= -1.3122n=6: (-6 +3)=-3, a_6=-3*(59049/100000)= -1.77147n=7: (-7 +3)=-4, a_7=-4*(531441/1000000)= -2.125764n=8: (-8 +3)=-5, a_8=-5*(4782969/10000000)= -2.3914845n=9: (-9 +3)=-6, a_9=-6*(43046721/100000000)= -2.58280326n=10: (-10 +3)=-7, a_{10}=-7*(387420489/1000000000)= -2.711943423n=11: (-11 +3)=-8, a_{11}=-8*(3486784401/10000000000)= -2.7894275208n=12: (-12 +3)=-9, a_{12}=-9*(31381059609/100000000000)= -2.82429536481n=13: (-13 +3)=-10, a_{13}=-10*(282429536481/1000000000000)= -2.82429536481n=14: (-14 +3)=-11, a_{14}=-11*(2541865828329/10000000000000)= -2.7960524111619So, the sequence goes from positive at n=1,2,3, then becomes negative, reaching a minimum at n=12 and n=13, then starts increasing again (becoming less negative).Thus, the minimum value is at n=12 and n=13, both equal to approximately -2.82429536481.All other terms are greater than this minimum (since they are either positive or less negative).Therefore, for k=-1, the sequence is a double-bottom sequence.Similarly, let's check for k=-3.n=9 -3/k=9 -3/(-3)=9 +1=10.So, critical point at n=10.Compute a_{10}, a_{11}, a_{9}.a_n=( -3n +3)(9/10)^nCompute a_9=(-27 +3)(9/10)^9=(-24)*(387420489/1000000000)= (-24)*0.387420489≈-9.298091736a_{10}=(-30 +3)(9/10)^10=(-27)*(282429536481/100000000000)= (-27)*0.282429536481≈-7.625597485a_{11}=(-33 +3)(9/10)^11=(-30)*(2541865828329/10000000000000)= (-30)*0.2541865828329≈-7.625597485Wait, so a_{10}=a_{11}≈-7.625597485Wait, let's compute more accurately.a_{10}=(-30 +3)(9/10)^10=(-27)*(9^10/10^10)= (-27)*(3486784401/10000000000)= (-27)*(0.3486784401)= -9.4143178827Wait, that doesn't match my previous calculation. Wait, 9^10 is 3486784401, so 3486784401/10^10=0.3486784401So, a_{10}=(-27)*0.3486784401≈-9.4143178827Similarly, a_{11}=(-33 +3)(9/10)^11=(-30)*(9^11/10^11)= (-30)*(31381059609/100000000000)= (-30)*(0.31381059609)= -9.4143178827So, a_{10}=a_{11}≈-9.4143178827Now, let's check a_9 and a_{12}.a_9=(-27 +3)(9/10)^9=(-24)*(387420489/1000000000)= (-24)*0.387420489≈-9.298091736a_{12}=(-36 +3)(9/10)^12=(-33)*(282429536481/1000000000000)= (-33)*0.282429536481≈-9.320174603873So, the sequence for k=-3:a_1=( -3 +3)=0*(9/10)=0a_2=(-6 +3)=-3*(81/100)= -2.43a_3=(-9 +3)=-6*(729/1000)= -4.374a_4=(-12 +3)=-9*(6561/10000)= -5.9049a_5=(-15 +3)=-12*(59049/100000)= -7.08588a_6=(-18 +3)=-15*(531441/1000000)= -7.971615a_7=(-21 +3)=-18*(4782969/10000000)= -8.6093442a_8=(-24 +3)=-21*(43046721/100000000)= -9.03981141a_9=(-27 +3)=-24*(387420489/1000000000)= -9.298091736a_{10}=≈-9.4143178827a_{11}=≈-9.4143178827a_{12}=≈-9.320174603873So, the sequence decreases to n=10 and n=11, reaching the minimum, then starts increasing again.Thus, the minimum is at n=10 and n=11, both equal to≈-9.4143178827.All other terms are greater than this minimum.Therefore, for k=-3, the sequence is also a double-bottom sequence.Thus, the possible integer values of k are -1 and -3.**Final Answer**(1) (i) boxed{text{Yes}}, (ii) boxed{text{No}}, (iii) boxed{text{Yes}}; (2) m = boxed{-1}, S_n = boxed{100n - n^2} for 1 leqslant n leqslant 50, and S_n = boxed{2^{n-49} - n + 2548} for n geqslant 51; (3) k = boxed{-1} or boxed{-3}.

answer：First, I recognize that when dividing 100 by an unknown number, the quotient is 9 and the remainder is 1. To find the divisor, I'll use the division formula: Dividend = (Divisor × Quotient) + Remainder.Plugging in the known values:100 = (Divisor × 9) + 1Next, I'll isolate the term with the divisor by subtracting 1 from both sides:99 = Divisor × 9Finally, I'll solve for the divisor by dividing both sides by 9:Divisor = 99 / 9Divisor = 11Therefore, the divisor is 11.

answer：Okay, so I have this problem here where I need to find the value of 'a' given that the function f(x) = ln(x) - sin(x) reaches an extreme value at x = π/3. The options are A: π/6, B: π/2, C: 2π/3, D: 3π/2. Hmm, let's see how to approach this.First, I remember that extreme values of a function occur where the first derivative is zero or undefined, provided the function is defined there. So, I should probably start by finding the derivative of f(x).The function is f(x) = ln(x) - sin(x). Let me recall the derivatives of these functions. The derivative of ln(x) is 1/x, and the derivative of sin(x) is cos(x). So, putting that together, the derivative f'(x) should be 1/x - cos(x). Wait, but in the problem statement, it's written as f'(x) = a/x - cos(x). Hmm, that's interesting. So, maybe there was a typo or something, but in any case, I think the derivative is 1/x - cos(x). But since the problem mentions a/x, maybe 'a' is a constant that we need to find such that the derivative equals zero at x = π/3.So, the problem is telling me that at x = π/3, the function has an extreme value, which means that f'(π/3) = 0. So, plugging x = π/3 into the derivative, we get:f'(π/3) = a/(π/3) - cos(π/3) = 0Simplifying that, a/(π/3) is the same as 3a/π, and cos(π/3) is 0.5 or 1/2. So, the equation becomes:3a/π - 1/2 = 0Now, I need to solve for 'a'. Let's add 1/2 to both sides:3a/π = 1/2Then, multiply both sides by π:3a = π/2Now, divide both sides by 3:a = π/6So, the value of 'a' is π/6, which is option A.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from f'(x) = a/x - cos(x), plugging in x = π/3:a/(π/3) - cos(π/3) = 0Simplify a/(π/3) to 3a/π, and cos(π/3) is indeed 1/2. So, 3a/π - 1/2 = 0. Solving for 'a' gives 3a/π = 1/2, so a = (1/2)*(π/3) = π/6. Yep, that seems correct.I think I got it right. So, the answer should be A: π/6.

answer：Okay, so I have this problem where I need to evaluate the sum of two complex numbers in exponential form and express the result as ( r e^{i theta} ). The given expression is ( 15 e^{3 pi i/13} + 15 e^{24 pi i/26} ). Hmm, let me break this down step by step.First, I notice that both terms have the same magnitude, which is 15. That might be helpful because when adding complex numbers with the same magnitude, there might be some symmetry or simplification I can use. Looking at the exponents, the first term is ( e^{3 pi i/13} ) and the second term is ( e^{24 pi i/26} ). I should probably simplify these angles to see if they can be expressed in a more comparable form. Let me convert ( 24 pi /26 ) to a simpler fraction. Dividing numerator and denominator by 2, that becomes ( 12 pi /13 ). So, the second term is ( e^{12 pi i/13} ).Now, I have ( 15 e^{3 pi i/13} + 15 e^{12 pi i/13} ). Hmm, both angles are in terms of ( pi /13 ), which is good. Let me write them as:1. ( 3 pi /13 )2. ( 12 pi /13 )I wonder if these angles are related in some way. Let me compute the difference between them. ( 12 pi /13 - 3 pi /13 = 9 pi /13 ). So, the angles are ( 3 pi /13 ) and ( 12 pi /13 ), which are ( 9 pi /13 ) apart. Wait, ( 12 pi /13 ) is actually ( pi - pi /13 ), right? Because ( pi = 13 pi /13 ), so subtracting ( pi /13 ) gives ( 12 pi /13 ). So, ( 12 pi /13 = pi - pi /13 ). That might be useful because it relates to the concept of supplementary angles in trigonometry.So, if I think of these two complex numbers on the complex plane, they have the same magnitude (15) and their angles are ( 3 pi /13 ) and ( pi - pi /13 ). That means they are symmetric with respect to the imaginary axis? Or maybe not exactly, but there's some symmetry here.Let me recall that when adding two complex numbers in polar form, it's often helpful to convert them to rectangular form (a + bi), add them, and then convert back to polar form. Maybe that's a straightforward approach here.So, let's try that. Let me denote the first complex number as ( z_1 = 15 e^{3 pi i/13} ) and the second as ( z_2 = 15 e^{12 pi i/13} ).Converting ( z_1 ) to rectangular form:( z_1 = 15 left( cos(3 pi /13) + i sin(3 pi /13) right) ).Similarly, converting ( z_2 ):( z_2 = 15 left( cos(12 pi /13) + i sin(12 pi /13) right) ).Now, adding ( z_1 ) and ( z_2 ):( z_1 + z_2 = 15 cos(3 pi /13) + 15 cos(12 pi /13) + i [15 sin(3 pi /13) + 15 sin(12 pi /13)] ).Factor out the 15:( z_1 + z_2 = 15 [ cos(3 pi /13) + cos(12 pi /13) ] + i 15 [ sin(3 pi /13) + sin(12 pi /13) ] ).Hmm, now I need to compute these cosine and sine terms. Let me see if I can simplify them using trigonometric identities.I remember that ( cos(pi - x) = -cos(x) ) and ( sin(pi - x) = sin(x) ). So, ( cos(12 pi /13) = cos(pi - pi /13) = -cos(pi /13) ). Similarly, ( sin(12 pi /13) = sin(pi - pi /13) = sin(pi /13) ).So, substituting these into the expression:( z_1 + z_2 = 15 [ cos(3 pi /13) - cos(pi /13) ] + i 15 [ sin(3 pi /13) + sin(pi /13) ] ).Now, let's compute the real and imaginary parts separately.First, the real part:( 15 [ cos(3 pi /13) - cos(pi /13) ] ).And the imaginary part:( 15 [ sin(3 pi /13) + sin(pi /13) ] ).Hmm, these expressions might be simplified using sum-to-product identities. Let me recall those.For the real part, ( cos A - cos B = -2 sin left( frac{A + B}{2} right) sin left( frac{A - B}{2} right) ).For the imaginary part, ( sin A + sin B = 2 sin left( frac{A + B}{2} right) cos left( frac{A - B}{2} right) ).Let me apply these identities.Starting with the real part:Let ( A = 3 pi /13 ) and ( B = pi /13 ).So,( cos(3 pi /13) - cos(pi /13) = -2 sin left( frac{3 pi /13 + pi /13}{2} right) sin left( frac{3 pi /13 - pi /13}{2} right) ).Simplify the arguments:( frac{4 pi /13}{2} = 2 pi /13 ),( frac{2 pi /13}{2} = pi /13 ).So,( cos(3 pi /13) - cos(pi /13) = -2 sin(2 pi /13) sin(pi /13) ).Therefore, the real part becomes:( 15 [ -2 sin(2 pi /13) sin(pi /13) ] = -30 sin(2 pi /13) sin(pi /13) ).Now, the imaginary part:( sin(3 pi /13) + sin(pi /13) = 2 sin left( frac{3 pi /13 + pi /13}{2} right) cos left( frac{3 pi /13 - pi /13}{2} right) ).Simplify the arguments:( frac{4 pi /13}{2} = 2 pi /13 ),( frac{2 pi /13}{2} = pi /13 ).So,( sin(3 pi /13) + sin(pi /13) = 2 sin(2 pi /13) cos(pi /13) ).Therefore, the imaginary part becomes:( 15 [ 2 sin(2 pi /13) cos(pi /13) ] = 30 sin(2 pi /13) cos(pi /13) ).So, putting it all together, the sum ( z_1 + z_2 ) is:( -30 sin(2 pi /13) sin(pi /13) + i 30 sin(2 pi /13) cos(pi /13) ).Hmm, I can factor out ( 30 sin(2 pi /13) ):( 30 sin(2 pi /13) [ -sin(pi /13) + i cos(pi /13) ] ).Now, let me look at the expression inside the brackets: ( -sin(pi /13) + i cos(pi /13) ). That looks similar to the exponential form ( e^{i theta} ), but with a negative sine and positive cosine. Let me see.Recall that ( e^{i theta} = cos theta + i sin theta ). So, if I have ( cos theta - i sin theta ), that would be ( e^{-i theta} ). But in our case, it's ( -sin(pi /13) + i cos(pi /13) ). Hmm, not exactly matching.Wait, perhaps I can factor out an ( i ) from the expression:( -sin(pi /13) + i cos(pi /13) = i [ cos(pi /13) + i sin(pi /13) ] ).Wait, let me check:( i [ cos(pi /13) + i sin(pi /13) ] = i cos(pi /13) + i^2 sin(pi /13) = i cos(pi /13) - sin(pi /13) ).Yes, that's exactly the expression we have. So,( -sin(pi /13) + i cos(pi /13) = i [ cos(pi /13) + i sin(pi /13) ] = i e^{i pi /13} ).Therefore, substituting back into the sum:( z_1 + z_2 = 30 sin(2 pi /13) cdot i e^{i pi /13} ).Simplify this:( z_1 + z_2 = 30 i sin(2 pi /13) e^{i pi /13} ).But ( i = e^{i pi /2} ), so:( z_1 + z_2 = 30 e^{i pi /2} sin(2 pi /13) e^{i pi /13} ).Combine the exponents:( e^{i pi /2} cdot e^{i pi /13} = e^{i (pi /2 + pi /13)} ).Compute ( pi /2 + pi /13 ):Convert to a common denominator, which is 26:( pi /2 = 13 pi /26 ),( pi /13 = 2 pi /26 ).So,( 13 pi /26 + 2 pi /26 = 15 pi /26 ).Therefore,( z_1 + z_2 = 30 sin(2 pi /13) e^{i 15 pi /26} ).Wait, so this is almost in the form ( r e^{i theta} ), except for the ( sin(2 pi /13) ) term. Let me compute ( sin(2 pi /13) ) numerically to see if it's a constant or if it can be expressed in terms of another trigonometric function.But before that, let me think if there's another way to express ( sin(2 pi /13) ). Alternatively, maybe I made a miscalculation earlier.Wait, let me recap:We had ( z_1 + z_2 = 30 sin(2 pi /13) e^{i 15 pi /26} ). So, the magnitude ( r ) is ( 30 sin(2 pi /13) ) and the angle ( theta ) is ( 15 pi /26 ).But in the initial problem statement, the answer is expected to be in the form ( r e^{i theta} ), so this seems correct. However, let me verify if ( sin(2 pi /13) ) can be simplified or expressed differently.Alternatively, perhaps I can express the sum in terms of cosine instead of sine. Let me recall that ( sin(x) = cos(pi /2 - x) ). So,( sin(2 pi /13) = cos(pi /2 - 2 pi /13) = cos(13 pi /26 - 4 pi /26) = cos(9 pi /26) ).Yes, that's correct because ( pi /2 = 13 pi /26 ), so subtracting ( 4 pi /26 ) gives ( 9 pi /26 ).Therefore,( sin(2 pi /13) = cos(9 pi /26) ).So, substituting back, we have:( z_1 + z_2 = 30 cos(9 pi /26) e^{i 15 pi /26} ).Therefore, the magnitude ( r ) is ( 30 cos(9 pi /26) ) and the angle ( theta ) is ( 15 pi /26 ).Let me double-check this result. Another approach is to consider the average angle and the difference in angles.Given two complex numbers with the same magnitude, ( 15 e^{i alpha} ) and ( 15 e^{i beta} ), their sum can be expressed as ( 15 e^{i (alpha + beta)/2} cdot 2 cos( (beta - alpha)/2 ) ).Let me apply this formula.Here, ( alpha = 3 pi /13 ) and ( beta = 12 pi /13 ).Compute the average angle:( (alpha + beta)/2 = (3 pi /13 + 12 pi /13)/2 = (15 pi /13)/2 = 15 pi /26 ).Compute the difference:( (beta - alpha)/2 = (12 pi /13 - 3 pi /13)/2 = (9 pi /13)/2 = 9 pi /26 ).Therefore, the sum is:( 15 cdot 2 cos(9 pi /26) e^{i 15 pi /26} = 30 cos(9 pi /26) e^{i 15 pi /26} ).Yes, this matches the result I obtained earlier. So, this confirms that the magnitude is ( 30 cos(9 pi /26) ) and the angle is ( 15 pi /26 ).Just to be thorough, let me compute the numerical values to ensure consistency.First, compute ( 9 pi /26 ):( 9 pi /26 approx 9 * 3.1416 /26 ≈ 28.2744 /26 ≈ 1.0875 ) radians.Compute ( cos(1.0875) ):Using a calculator, ( cos(1.0875) ≈ 0.4695 ).So, ( 30 * 0.4695 ≈ 14.085 ).Now, compute ( 15 pi /26 ):( 15 pi /26 ≈ 15 * 3.1416 /26 ≈ 47.124 /26 ≈ 1.8125 ) radians.Now, let me compute the original sum numerically to check.Compute ( z_1 = 15 e^{3 pi i /13} ):( 3 pi /13 ≈ 0.7236 ) radians.( cos(0.7236) ≈ 0.748 ), ( sin(0.7236) ≈ 0.663 ).So, ( z_1 ≈ 15 (0.748 + 0.663i) ≈ 11.22 + 9.945i ).Compute ( z_2 = 15 e^{12 pi i /13} ):( 12 pi /13 ≈ 2.8798 ) radians.( cos(2.8798) ≈ -0.970 ), ( sin(2.8798) ≈ 0.242 ).So, ( z_2 ≈ 15 (-0.970 + 0.242i) ≈ -14.55 + 3.63i ).Adding ( z_1 + z_2 ):Real part: ( 11.22 - 14.55 ≈ -3.33 ).Imaginary part: ( 9.945 + 3.63 ≈ 13.575 ).So, the sum is approximately ( -3.33 + 13.575i ).Now, let's compute ( r = 30 cos(9 pi /26) ≈ 30 * 0.4695 ≈ 14.085 ).Compute ( theta = 15 pi /26 ≈ 1.8125 ) radians.Convert ( theta ) to degrees to get a better sense: ( 1.8125 * (180/π) ≈ 103.8 degrees ).Now, let's compute ( r e^{i theta} ):( r cos theta ≈ 14.085 * cos(1.8125) ≈ 14.085 * (-0.222) ≈ -3.13 ).( r sin theta ≈ 14.085 * sin(1.8125) ≈ 14.085 * 0.975 ≈ 13.73 ).So, the rectangular form is approximately ( -3.13 + 13.73i ), which is close to our earlier calculation of ( -3.33 + 13.575i ). The slight discrepancy is likely due to rounding errors in the approximations.Therefore, this numerical check supports that our analytical result is correct.In summary, by converting both complex numbers to rectangular form, adding them, and then converting back to polar form, we found that the sum is ( 30 cos(9 pi /26) e^{i 15 pi /26} ). This matches the result obtained using the formula for the sum of two complex numbers with the same magnitude.So, the ordered pair ( (r, theta) ) is ( left(30 cos left(frac{9 pi}{26}right), frac{15 pi}{26}right) ).**Final Answer**The ordered pair is boxed{left(30 cos left(frac{9 pi}{26}right), frac{15 pi}{26}right)}.