answer：Okay, so I have this problem about an arithmetic sequence. Let me try to figure it out step by step. First, I know that in an arithmetic sequence, each term increases by a constant difference. Let's call this common difference "d". The nth term of an arithmetic sequence can be written as a_n = a_1 + (n-1)d, where a_1 is the first term.The problem gives me two pieces of information: 1. The sum of the first n terms is denoted as s_n. So, s_n is the sum of the sequence up to the nth term.2. Specifically, a_2 + a_3 = 5, and s_5 = 20.I need to find a_5. Let me write down what I know:- a_2 is the second term, which would be a_1 + d.- a_3 is the third term, which would be a_1 + 2d.- So, a_2 + a_3 = (a_1 + d) + (a_1 + 2d) = 2a_1 + 3d. And this equals 5.So, equation one is: 2a_1 + 3d = 5.Next, the sum of the first 5 terms, s_5, is 20. The formula for the sum of the first n terms of an arithmetic sequence is s_n = n/2 * (2a_1 + (n-1)d). So, plugging in n=5, we get:s_5 = 5/2 * (2a_1 + 4d) = 20.Let me compute that:5/2 * (2a_1 + 4d) = 20.Multiplying both sides by 2 to eliminate the fraction:5*(2a_1 + 4d) = 40.Divide both sides by 5:2a_1 + 4d = 8.So, equation two is: 2a_1 + 4d = 8.Now, I have two equations:1. 2a_1 + 3d = 52. 2a_1 + 4d = 8I can solve these simultaneously. Let me subtract equation one from equation two to eliminate a_1.(2a_1 + 4d) - (2a_1 + 3d) = 8 - 5Simplify:2a_1 + 4d - 2a_1 - 3d = 3Which simplifies to:d = 3.Okay, so the common difference d is 3.Now, plug d = 3 back into equation one to find a_1.2a_1 + 3*3 = 52a_1 + 9 = 5Subtract 9 from both sides:2a_1 = 5 - 92a_1 = -4Divide both sides by 2:a_1 = -2.So, the first term is -2, and the common difference is 3.Now, I need to find a_5, which is the fifth term.Using the formula for the nth term:a_5 = a_1 + (5-1)d = a_1 + 4d.Plug in the values:a_5 = -2 + 4*3 = -2 + 12 = 10.So, a_5 is 10.Let me double-check my calculations to make sure I didn't make any mistakes.First, a_1 = -2 and d = 3.Compute a_2 and a_3:a_2 = a_1 + d = -2 + 3 = 1a_3 = a_1 + 2d = -2 + 6 = 4a_2 + a_3 = 1 + 4 = 5. That matches the given information.Now, compute s_5:s_5 = (-2) + 1 + 4 + 7 + 10.Let's add them up:-2 + 1 = -1-1 + 4 = 33 + 7 = 1010 + 10 = 20.That also matches the given information.So, my calculations seem correct. Therefore, a_5 is indeed 10.**Final Answer**boxed{C}

answer：Okay, so I've got this math problem here: (8000)(6000) equals (some number) times 10 to the fifth power. I need to find out what that "some number" is. Alright, let's break this down step by step.First, I need to understand what the problem is asking. It says that when you multiply 8000 by 6000, it's equal to some unknown number multiplied by 10^5. So, in equation form, that would be:8000 * 6000 = x * 10^5My goal is to find the value of x. That seems straightforward, but let's make sure I do this correctly.Alright, let's start by calculating 8000 multiplied by 6000. I think the easiest way to do this is to multiply the numbers as they are and then adjust for the zeros. So, 8 times 6 is 48, and then I have four zeros from 8000 and three zeros from 6000. Wait, that's actually four zeros in 8000 and three zeros in 6000, so that's a total of seven zeros. So, 48 followed by seven zeros would be 48,000,000.Let me double-check that multiplication to make sure I didn't make a mistake. 8000 times 6000... Yeah, 8 times 6 is 48, and then adding the seven zeros gives me 48,000,000. Okay, that seems right.So now, I have:48,000,000 = x * 10^5I need to solve for x. To do that, I should isolate x on one side of the equation. Since x is multiplied by 10^5, I can divide both sides of the equation by 10^5 to get x by itself.So, dividing both sides by 10^5:x = 48,000,000 / 10^5Now, 10^5 is 100,000. So, I'm dividing 48,000,000 by 100,000. Let's see, dividing by 100,000 is the same as moving the decimal point five places to the left.Starting with 48,000,000, which is 48,000,000.0. Moving the decimal five places to the left would give me 480.00000. So, x equals 480.Wait, let me make sure I did that correctly. 48,000,000 divided by 100,000. Another way to think about it is that 100,000 goes into 48,000,000 how many times? Well, 100,000 times 480 is 48,000,000, because 100,000 times 100 is 10,000,000, so 100,000 times 400 is 40,000,000, and 100,000 times 80 is 8,000,000. Adding those together gives 48,000,000. So, yes, x is indeed 480.Just to recap, I multiplied 8000 by 6000 to get 48,000,000, then I divided that by 10^5 (which is 100,000) to find that x equals 480. That makes sense, and I don't see any mistakes in my calculations.So, the value of the first number in the equation is 480.

answer：Okay, so I have this problem about a sequence of positive numbers {x_n} where the sum S_n = x_1 + x_2 + ... + x_n satisfies S_n ≥ 2 S_{n-1} for n = 2, 3, and so on. I need to prove that there's a constant C > 0 such that x_n ≥ C * 2^n for all n = 1, 2, 3, etc.Hmm, let me try to unpack this step by step. First, the condition S_n ≥ 2 S_{n-1} tells me that each partial sum is at least twice the previous one. That seems like a pretty strong condition. It suggests that the sequence {x_n} is growing exponentially because each term is contributing enough to double the sum.Let me write down what I know:1. S_n = x_1 + x_2 + ... + x_n2. S_n ≥ 2 S_{n-1} for n ≥ 2From the second point, I can write S_n ≥ 2 S_{n-1}. If I think recursively, maybe I can express S_n in terms of S_1. Let's see:For n = 2: S_2 ≥ 2 S_1For n = 3: S_3 ≥ 2 S_2 ≥ 2*(2 S_1) = 4 S_1For n = 4: S_4 ≥ 2 S_3 ≥ 2*(4 S_1) = 8 S_1Wait a minute, this looks like S_n ≥ 2^{n-1} S_1. That seems to be a pattern. Let me test it for n = 1, 2, 3, 4:- n=1: S_1 = x_1- n=2: S_2 ≥ 2 S_1- n=3: S_3 ≥ 4 S_1- n=4: S_4 ≥ 8 S_1Yes, so in general, S_n ≥ 2^{n-1} S_1. That makes sense because each step doubles the previous sum.Now, I need to relate this back to x_n. Since S_n = S_{n-1} + x_n, and S_n ≥ 2 S_{n-1}, then:S_n = S_{n-1} + x_n ≥ 2 S_{n-1}Subtracting S_{n-1} from both sides gives:x_n ≥ S_{n-1}So, each term x_n is at least as big as the previous sum S_{n-1}. That's interesting because it means x_n is growing at least as fast as the sum of all previous terms.But I need to show that x_n is at least C * 2^n. So, maybe I can use the inequality S_{n-1} ≥ 2^{n-2} S_1 from the earlier result.Since x_n ≥ S_{n-1} and S_{n-1} ≥ 2^{n-2} S_1, then:x_n ≥ 2^{n-2} S_1Let me write that as:x_n ≥ (S_1 / 4) * 2^nBecause 2^{n-2} is equal to (1/4) * 2^n. So, if I let C = S_1 / 4, then:x_n ≥ C * 2^nThat seems to work for n ≥ 2. But what about n = 1?For n = 1, x_1 = S_1. If I plug n = 1 into the inequality x_n ≥ C * 2^n, I get:x_1 ≥ C * 2^1 = 2CBut since C = S_1 / 4, then 2C = S_1 / 2. So, x_1 = S_1 ≥ S_1 / 2, which is true because S_1 is positive. So, the inequality holds for n = 1 as well.Wait, but I defined C as S_1 / 4. Is that the best possible constant? Maybe not, but the problem just asks for the existence of some constant C > 0. So, as long as I can find such a C, it's fine.Let me recap:1. From S_n ≥ 2 S_{n-1}, we derived that S_n ≥ 2^{n-1} S_1.2. Then, since x_n = S_n - S_{n-1} ≥ S_{n-1}, we substituted the lower bound for S_{n-1} to get x_n ≥ 2^{n-2} S_1.3. By setting C = S_1 / 4, we rewrote this as x_n ≥ C * 2^n.4. Checked that this works for n = 1 as well.So, it looks like I've found the required constant C. Therefore, such a constant exists, and the proof is complete.But just to make sure I didn't make any mistakes, let me test this with an example. Suppose S_1 = x_1 = 4. Then C = 4 / 4 = 1.For n = 1: x_1 = 4 ≥ 1 * 2^1 = 2 ✔️For n = 2: S_2 ≥ 2 S_1 = 8, so x_2 = S_2 - S_1 ≥ 8 - 4 = 4 ≥ 1 * 2^2 = 4 ✔️For n = 3: S_3 ≥ 2 S_2 ≥ 16, so x_3 = S_3 - S_2 ≥ 16 - 8 = 8 ≥ 1 * 2^3 = 8 ✔️And so on. It seems to hold.Another example: Let S_1 = x_1 = 2. Then C = 2 / 4 = 0.5.For n = 1: x_1 = 2 ≥ 0.5 * 2 = 1 ✔️For n = 2: S_2 ≥ 4, so x_2 ≥ 4 - 2 = 2 ≥ 0.5 * 4 = 2 ✔️For n = 3: S_3 ≥ 8, so x_3 ≥ 8 - 4 = 4 ≥ 0.5 * 8 = 4 ✔️Good, it still works.What if S_1 is very small, say S_1 = 1. Then C = 1 / 4 = 0.25.For n = 1: x_1 = 1 ≥ 0.25 * 2 = 0.5 ✔️For n = 2: S_2 ≥ 2, so x_2 ≥ 2 - 1 = 1 ≥ 0.25 * 4 = 1 ✔️For n = 3: S_3 ≥ 4, so x_3 ≥ 4 - 2 = 2 ≥ 0.25 * 8 = 2 ✔️Still holds.Okay, so in all these examples, the inequality x_n ≥ C * 2^n holds with C = S_1 / 4. That gives me more confidence that the proof is correct.I think I've covered all the necessary steps and checked with examples. So, I'm ready to write up the formal proof.

answer：First, I need to determine the total number of dancers at the junior prom. Since a fourth of the 140 kids were dancers, I'll calculate this by dividing 140 by 4.Next, I know that out of these dancers, 10 did not participate in the slow dance. To find out how many dancers did the slow dance, I'll subtract the 10 dancers who did not slow dance from the total number of dancers.Finally, this subtraction will give me the number of dancers who danced the slow dance.

answer：To solve for the conjugate of the complex number ( z ) given the equation ( z(1 - i) = 1 + i ), follow these steps:1. **Express ( z ) in terms of ( i ):** [ z = frac{1 + i}{1 - i} ]2. **Simplify the expression by multiplying the numerator and the denominator by the conjugate of the denominator:** [ z = frac{(1 + i)(1 + i)}{(1 - i)(1 + i)} = frac{(1 + i)^2}{1^2 - i^2} ]3. **Expand and simplify the numerator and the denominator:** [ (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i ] [ 1^2 - i^2 = 1 - (-1) = 2 ] [ z = frac{2i}{2} = i ]4. **Find the conjugate of ( z ):** [ overline{z} = overline{i} = -i ]Therefore, the conjugate of the complex number ( z ) is (-i).

answer：Okay, so I have this problem about an equilateral triangle ABC. Let me try to visualize it first. Since it's equilateral, all sides are equal, and all angles are 60 degrees. Point K is the midpoint of side AB, so that splits AB into two equal parts. Point M is on BC such that BM to MC is 1:3. So if BC is, say, 4 units, BM would be 1 unit and MC would be 3 units. Now, I need to find a point P on AC such that the perimeter of triangle PKM is minimized. Hmm, interesting.I remember that in problems where you need to minimize the perimeter or distance, reflections can be useful. Maybe I can use some reflection technique here. Let me think. If I reflect point M over AC, I can get a symmetric point M'. Then, the perimeter of PKM would be the same as the distance from K to M' via P on AC. So, the minimal perimeter would correspond to the straight line from K to M', intersecting AC at P.Wait, is that right? Let me make sure. If I reflect M over AC to get M', then the path K-P-M would be equivalent to K-P-M', but since M' is the reflection, the distance from P to M is the same as from P to M'. So, minimizing the perimeter of PKM is equivalent to minimizing the distance from K to M via a point P on AC, which is just the straight line from K to M'. Therefore, P is the intersection point of AC and the line KM'.Okay, that makes sense. So, I need to find where the line from K to M' intersects AC. Once I have that point P, I can determine the ratio AP:PC.Let me set up a coordinate system to make this more concrete. Let's place the triangle ABC with point A at (0, 0), point B at (2, 0), and since it's equilateral, point C will be at (1, √3). That way, the side length is 2 units, which should make calculations easier.Point K is the midpoint of AB, so its coordinates are ((0+2)/2, (0+0)/2) = (1, 0). Point M is on BC such that BM:MC = 1:3. Since BC goes from (2, 0) to (1, √3), let me find the coordinates of M.The vector from B to C is (-1, √3). Since BM:MC = 1:3, M divides BC in the ratio 1:3. So, using the section formula, the coordinates of M are:M_x = (3*2 + 1*1)/(1+3) = (6 + 1)/4 = 7/4M_y = (3*0 + 1*√3)/4 = √3/4Wait, that doesn't seem right. Let me double-check. The section formula is (mx2 + nx1)/(m + n), where m:n is the ratio. Since BM:MC = 1:3, M is closer to B, so m = 1, n = 3.So, M_x = (1*1 + 3*2)/(1+3) = (1 + 6)/4 = 7/4M_y = (1*√3 + 3*0)/4 = √3/4Yes, that's correct. So, M is at (7/4, √3/4).Now, I need to find the reflection of M over AC to get M'. To reflect a point over a line, I can use the formula for reflection over a line. The line AC goes from (0, 0) to (1, √3). Let me find the equation of line AC first.The slope of AC is (√3 - 0)/(1 - 0) = √3. So, the equation is y = √3 x.To reflect point M(7/4, √3/4) over the line y = √3 x, I can use the reflection formula. The formula for reflecting a point (x, y) over the line ax + by + c = 0 is:x' = x - 2a(ax + by + c)/(a² + b²)y' = y - 2b(ax + by + c)/(a² + b²)But in this case, the line is y - √3 x = 0, so a = -√3, b = 1, c = 0.So, plugging into the formula:x' = x - 2*(-√3)*(-√3 x + y)/((-√3)² + 1²)y' = y - 2*(1)*(-√3 x + y)/((-√3)² + 1²)Simplify the denominator: (-√3)² + 1² = 3 + 1 = 4.So,x' = x - 2*(-√3)*(-√3 x + y)/4x' = x - (2√3)(-√3 x + y)/4x' = x - (√3/2)(-√3 x + y)Similarly,y' = y - 2*(1)*(-√3 x + y)/4y' = y - (2/4)*(-√3 x + y)y' = y - (1/2)*(-√3 x + y)Let me compute x' and y' step by step.First, compute (-√3 x + y):For point M(7/4, √3/4):(-√3*(7/4) + √3/4) = (-7√3/4 + √3/4) = (-6√3)/4 = (-3√3)/2Now, compute x':x' = x - (√3/2)*(-3√3/2)x' = 7/4 - (√3/2)*(-3√3/2)x' = 7/4 - ( (√3)*(-3√3) )/(2*2)x' = 7/4 - ( (-3*3) )/4x' = 7/4 - (-9/4)x' = 7/4 + 9/4x' = 16/4x' = 4Wait, that can't be right. If I reflect M over AC, which is a line inside the triangle, getting x' = 4 would be outside the triangle. Maybe I made a mistake in the calculation.Let me double-check the reflection formula. Alternatively, maybe I should use a different approach.Another way to find the reflection is to find the perpendicular from M to AC, find the midpoint, and then extend it to get M'.So, let's try that.First, find the equation of AC: y = √3 x.The slope of AC is √3, so the slope of the perpendicular is -1/√3.So, the equation of the perpendicular from M(7/4, √3/4) is:y - √3/4 = (-1/√3)(x - 7/4)Let me find the intersection point of this perpendicular with AC.Set y = √3 x in the above equation:√3 x - √3/4 = (-1/√3)(x - 7/4)Multiply both sides by √3 to eliminate the denominator:3x - 3/4 = - (x - 7/4)Simplify:3x - 3/4 = -x + 7/4Bring all terms to one side:3x + x = 7/4 + 3/44x = 10/4x = (10/4)/4 = 10/16 = 5/8Wait, that doesn't seem right. Let me check the algebra again.Starting from:√3 x - √3/4 = (-1/√3)(x - 7/4)Multiply both sides by √3:3x - 3/4 = - (x - 7/4)So,3x - 3/4 = -x + 7/4Bring -x to the left and -3/4 to the right:3x + x = 7/4 + 3/44x = 10/4x = (10/4)/4 = 10/16 = 5/8Wait, that seems correct. So, x = 5/8. Then y = √3*(5/8) = 5√3/8.So, the intersection point is (5/8, 5√3/8). This is the midpoint between M and M'.So, to find M', we can use the midpoint formula. Let me denote M' as (x', y').Midpoint between M(7/4, √3/4) and M'(x', y') is (5/8, 5√3/8).So,( (7/4 + x')/2 , (√3/4 + y')/2 ) = (5/8, 5√3/8)Therefore,(7/4 + x')/2 = 5/8 => 7/4 + x' = 10/8 = 5/4 => x' = 5/4 - 7/4 = -2/4 = -1/2Similarly,(√3/4 + y')/2 = 5√3/8 => √3/4 + y' = 5√3/4 => y' = 5√3/4 - √3/4 = 4√3/4 = √3So, M' is at (-1/2, √3).Wait, that seems correct. So, M' is at (-1/2, √3). Let me plot this mentally. Since AC is from (0,0) to (1, √3), reflecting M over AC would place M' outside the triangle on the other side of AC.Now, I need to find the line from K(1, 0) to M'(-1/2, √3). Let me find the equation of this line.The slope of KM' is (√3 - 0)/(-1/2 - 1) = √3 / (-3/2) = -2√3/3.So, the equation is y - 0 = (-2√3/3)(x - 1)Simplify: y = (-2√3/3)x + 2√3/3Now, find where this line intersects AC. AC is y = √3 x.Set √3 x = (-2√3/3)x + 2√3/3Bring all terms to one side:√3 x + (2√3/3)x - 2√3/3 = 0Factor out √3:√3 [x + (2/3)x - 2/3] = 0Simplify inside the brackets:(5/3)x - 2/3 = 0Multiply both sides by 3:5x - 2 = 0 => x = 2/5So, x = 2/5. Then y = √3*(2/5) = 2√3/5.Therefore, point P is at (2/5, 2√3/5).Now, I need to find the ratio AP:PC. Since AC is from (0,0) to (1, √3), which is length 2 units in my coordinate system, but actually, in coordinates, the length from (0,0) to (1, √3) is 2 units because the distance formula gives √[(1)^2 + (√3)^2] = √(1 + 3) = 2.But in terms of the coordinates, AP is from (0,0) to (2/5, 2√3/5), and PC is from (2/5, 2√3/5) to (1, √3).So, the length of AP is the distance from (0,0) to (2/5, 2√3/5):AP = √[(2/5)^2 + (2√3/5)^2] = √[(4/25) + (12/25)] = √[16/25] = 4/5Similarly, PC is the distance from (2/5, 2√3/5) to (1, √3):PC = √[(1 - 2/5)^2 + (√3 - 2√3/5)^2] = √[(3/5)^2 + (3√3/5)^2] = √[(9/25) + (27/25)] = √[36/25] = 6/5So, AP = 4/5 and PC = 6/5. Therefore, the ratio AP:PC is (4/5):(6/5) = 4:6 = 2:3.Wait, that's interesting. So, the ratio is 2:3. Let me just confirm that with the coordinates.Alternatively, since AC is divided by P at (2/5, 2√3/5), the ratio can be found by the x-coordinates or y-coordinates. Since AC is parameterized from (0,0) to (1, √3), the parameter t can be found such that P = (t, t√3). So, t = 2/5. Therefore, AP = t * AC length, and PC = (1 - t) * AC length. Since AC length is 2, AP = 2*(2/5) = 4/5, PC = 2*(3/5) = 6/5, so ratio 4/5 : 6/5 = 2:3.Yes, that confirms it. So, the ratio is 2:3.I think that's the answer. Let me just recap to make sure I didn't make any mistakes.1. I set up the coordinate system with A at (0,0), B at (2,0), and C at (1, √3).2. Found coordinates of K (midpoint of AB) as (1,0).3. Found coordinates of M on BC with BM:MC = 1:3, which gave M at (7/4, √3/4).4. Reflected M over AC to get M' at (-1/2, √3).5. Found the equation of line KM' and found its intersection with AC, which gave P at (2/5, 2√3/5).6. Calculated the lengths AP and PC, resulting in a ratio of 2:3.Everything seems to check out. I think the ratio is indeed 2:3.