💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that in the arithmetic sequence {a_n}, the sum of the 4th, 6th, 8th, 10th, and 12th terms is 120. I need to find the value of 2a_{10} - a_{12}.Alright, arithmetic sequences have a common difference, right? So each term is the previous term plus a constant difference, which we can call 'd'. The general formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)d, where a_1 is the first term.Let me write down the terms mentioned in the problem using this formula:- a_4 = a_1 + 3d- a_6 = a_1 + 5d- a_8 = a_1 + 7d- a_{10} = a_1 + 9d- a_{12} = a_1 + 11dThe problem says that the sum of these terms is 120. So, let me add them up:a_4 + a_6 + a_8 + a_{10} + a_{12} = (a_1 + 3d) + (a_1 + 5d) + (a_1 + 7d) + (a_1 + 9d) + (a_1 + 11d)Let me combine like terms. There are five a_1 terms and the coefficients of d are 3, 5, 7, 9, and 11.So, that's 5a_1 + (3 + 5 + 7 + 9 + 11)d = 120Let me compute the sum of the coefficients of d: 3 + 5 is 8, plus 7 is 15, plus 9 is 24, plus 11 is 35. So, the equation becomes:5a_1 + 35d = 120Hmm, I can factor out a 5 from the left side:5(a_1 + 7d) = 120Dividing both sides by 5:a_1 + 7d = 24Wait a minute, a_1 + 7d is actually the 8th term of the sequence, right? Because a_n = a_1 + (n-1)d, so a_8 = a_1 + 7d. So, a_8 = 24.Okay, so the 8th term is 24. Now, I need to find 2a_{10} - a_{12}.Let me express a_{10} and a_{12} in terms of a_8. Since a_{10} is two terms after a_8, and a_{12} is four terms after a_8.So, a_{10} = a_8 + 2dAnd a_{12} = a_8 + 4dSo, substituting these into the expression 2a_{10} - a_{12}:2(a_8 + 2d) - (a_8 + 4d)Let me expand this:2a_8 + 4d - a_8 - 4dSimplify:(2a_8 - a_8) + (4d - 4d) = a_8 + 0 = a_8So, 2a_{10} - a_{12} simplifies to a_8, which we already found to be 24.Therefore, the value of 2a_{10} - a_{12} is 24.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I wrote down the terms correctly using the arithmetic sequence formula.2. I summed them up and combined like terms correctly, getting 5a_1 + 35d = 120.3. Factored out 5, leading to a_1 + 7d = 24, which is a_8 = 24.4. Expressed a_{10} and a_{12} in terms of a_8 and d.5. Substituted into 2a_{10} - a_{12} and simplified, ending up with a_8.Everything seems to check out. I don't see any errors in my calculations or reasoning. So, I'm confident that the answer is 24.**Final Answer**The value of (2a_{10} - a_{12}) is boxed{24}.

💡Okay, so I have this equation: 5x minus 2m equals 3x minus 6m plus 1. And I need to find the integer values of m given that the solution for x is between -3 and 2, not including -3 but including 2. Hmm, let me try to figure this out step by step.First, I think I need to solve the equation for x in terms of m. That way, I can see how x depends on m and then use the given range for x to find the possible values of m. Alright, let's start by writing down the equation again:5x - 2m = 3x - 6m + 1I want to get all the x terms on one side and the m terms on the other. So, maybe I can subtract 3x from both sides to get rid of it on the right side. Let's try that:5x - 3x - 2m = -6m + 1Simplifying the left side, 5x minus 3x is 2x, so:2x - 2m = -6m + 1Now, I want to get all the m terms on one side. Let me add 6m to both sides to move the -6m to the left:2x - 2m + 6m = 1Simplifying that, -2m plus 6m is 4m, so:2x + 4m = 1Hmm, now I need to solve for x. Let me subtract 4m from both sides:2x = 1 - 4mAnd then divide both sides by 2 to get x by itself:x = (1 - 4m)/2Okay, so x is equal to (1 - 4m)/2. Now, the problem says that x is greater than -3 and less than or equal to 2. So, I can write that as:-3 < x ≤ 2But since x is (1 - 4m)/2, I can substitute that into the inequality:-3 < (1 - 4m)/2 ≤ 2Now, I need to solve this compound inequality for m. Let me break it down into two separate inequalities:First, -3 < (1 - 4m)/2And second, (1 - 4m)/2 ≤ 2I'll solve each part separately and then find the intersection of the solutions.Starting with the first inequality:-3 < (1 - 4m)/2To eliminate the denominator, I can multiply both sides by 2:-6 < 1 - 4mNow, subtract 1 from both sides:-6 - 1 < -4mWhich simplifies to:-7 < -4mNow, I need to solve for m. To do that, I'll divide both sides by -4. But wait, when I divide or multiply both sides of an inequality by a negative number, the inequality sign flips. So, dividing both sides by -4:-7 / (-4) > mWhich simplifies to:7/4 > mOr, written another way:m < 7/4Okay, so m is less than 7/4, which is 1.75.Now, moving on to the second inequality:(1 - 4m)/2 ≤ 2Again, I'll eliminate the denominator by multiplying both sides by 2:1 - 4m ≤ 4Subtract 1 from both sides:-4m ≤ 3Now, divide both sides by -4, remembering to flip the inequality sign:m ≥ -3/4So, m is greater than or equal to -3/4, which is -0.75.Putting both inequalities together, we have:-0.75 ≤ m < 1.75So, m is between -0.75 and 1.75. But the problem asks for integer values of m. Let's list the integers within this range.Starting from -0.75, the next integer is 0, then 1. Since m must be less than 1.75, the integer values are 0 and 1.Let me double-check to make sure these values of m satisfy the original inequality for x.First, let's take m = 0:x = (1 - 4*0)/2 = 1/2 = 0.5Is 0.5 between -3 and 2? Yes, it is.Next, m = 1:x = (1 - 4*1)/2 = (1 - 4)/2 = (-3)/2 = -1.5Is -1.5 between -3 and 2? Yes, it is.What about m = -1? Wait, m = -1 is less than -0.75, so it's outside our range. Let's check it anyway:x = (1 - 4*(-1))/2 = (1 + 4)/2 = 5/2 = 2.5But 2.5 is greater than 2, which is outside the given range for x. So, m = -1 doesn't work.Similarly, m = 2:x = (1 - 4*2)/2 = (1 - 8)/2 = (-7)/2 = -3.5Which is less than -3, so it's also outside the range.Therefore, the only integer values of m that satisfy the condition are 0 and 1.

💡Okay, so I have this problem about three towns, A, B, and C, connected by straight roads. The problem gives me some distances from each town to the opposite road and mentions a point D that's equidistant from all three roads. I need to figure out the maximum volume of fuel someone would need to reach any of the roads from point D. First, let me try to visualize the problem. There's a triangle formed by the towns A, B, and C. Point D is inside this triangle and is equidistant from all three sides. That makes me think of the incenter of the triangle, which is the point where the angle bisectors meet and is equidistant from all sides. So, D is probably the incenter.The problem states that the distance from town A to the road BC is 100 km. So, if I imagine the triangle ABC, the height from A to BC is 100 km. Then, it says the sum of the distances from B to AC and from C to AB is 300 km. So, if I denote the height from B to AC as h_b and from C to AB as h_c, then h_b + h_c = 300 km.I need to find the maximum volume of fuel required to reach any of the roads from point D. Since fuel consumption is 1 liter per 10 km, I need to find the maximum distance from D to any of the roads and then divide that by 10 to get the liters.But wait, point D is equidistant from all three roads, so the distance from D to each road is the same. That distance is the inradius of the triangle. So, if I can find the inradius, that will give me the distance from D to each road, and then I can calculate the fuel needed.To find the inradius, I remember the formula for the inradius of a triangle: r = A/s, where A is the area of the triangle and s is the semi-perimeter. But I don't know the sides of the triangle or the semi-perimeter. Hmm, maybe I can express the area in terms of the given heights.The area of the triangle can be expressed using any of the heights. For example, using the height from A: Area = (1/2)*BC*100. Similarly, using the height from B: Area = (1/2)*AC*h_b, and from C: Area = (1/2)*AB*h_c.Since all these expressions equal the area, I can set them equal to each other. But I don't know the lengths of the sides, so maybe I can find a relationship between the heights.Given that h_b + h_c = 300, maybe I can express the area in terms of h_b and h_c. Let me denote the sides opposite to A, B, and C as a, b, and c respectively. So, a is BC, b is AC, and c is AB.From the area expressions:Area = (1/2)*a*100 = (1/2)*b*h_b = (1/2)*c*h_c.So, a*100 = b*h_b = c*h_c = 2*Area.Also, the semi-perimeter s = (a + b + c)/2.I need to find r = Area/s. But I don't have s or Area directly. Maybe I can find a relationship between a, b, c, h_b, and h_c.Since a*100 = b*h_b = c*h_c, let's denote this common value as 2*Area. So, 2*Area = a*100 = b*h_b = c*h_c.From this, I can express b and c in terms of a:b = (2*Area)/h_b = (a*100)/h_bc = (2*Area)/h_c = (a*100)/h_cSo, b = (a*100)/h_b and c = (a*100)/h_c.Now, the semi-perimeter s = (a + b + c)/2 = (a + (a*100)/h_b + (a*100)/h_c)/2.Let me factor out a/2:s = (a/2)*(1 + 100/h_b + 100/h_c).But I know that h_b + h_c = 300, so maybe I can express 1/h_b + 1/h_c in terms of h_b and h_c.Let me denote h_b = x, then h_c = 300 - x.So, 1/h_b + 1/h_c = 1/x + 1/(300 - x).I need to find the maximum value of r, which is Area/s. But Area = (1/2)*a*100, so Area = 50a.Thus, r = Area/s = (50a)/( (a/2)*(1 + 100/x + 100/(300 - x)) ) = (50a)/( (a/2)*(1 + 100/x + 100/(300 - x)) )Simplify this:r = (50a) / ( (a/2)*(1 + 100/x + 100/(300 - x)) ) = (50a) * (2/a) / (1 + 100/x + 100/(300 - x)) ) = 100 / (1 + 100/x + 100/(300 - x))So, r = 100 / (1 + 100/x + 100/(300 - x)).I need to find the maximum value of r, which would correspond to the minimum value of the denominator.So, to maximize r, I need to minimize the denominator: 1 + 100/x + 100/(300 - x).Let me denote f(x) = 1 + 100/x + 100/(300 - x). I need to find the minimum of f(x) for x in (0, 300).To find the minimum, I can take the derivative of f(x) with respect to x and set it to zero.f(x) = 1 + 100/x + 100/(300 - x)f'(x) = -100/x² + 100/(300 - x)²Set f'(x) = 0:-100/x² + 100/(300 - x)² = 0-100/x² = -100/(300 - x)²1/x² = 1/(300 - x)²Taking square roots:1/x = ±1/(300 - x)Since x is between 0 and 300, both denominators are positive, so:1/x = 1/(300 - x)Which implies:300 - x = x300 = 2xx = 150So, the minimum occurs at x = 150, which means h_b = 150 and h_c = 150.Plugging back into f(x):f(150) = 1 + 100/150 + 100/150 = 1 + 2/3 + 2/3 = 1 + 4/3 = 7/3So, the minimum value of the denominator is 7/3, which means the maximum r is:r = 100 / (7/3) = 100 * (3/7) = 300/7 ≈ 42.857 kmTherefore, the distance from D to any road is approximately 42.857 km. Since fuel consumption is 1 liter per 10 km, the fuel needed is 42.857 / 10 ≈ 4.2857 liters. But wait, the problem asks for the maximum volume of fuel needed to reach any of the roads. Since D is equidistant from all roads, the distance is the same, so the fuel needed is the same for all roads. Therefore, the maximum fuel needed is approximately 4.2857 liters, which is 300/7 liters.But wait, 300/7 is approximately 42.857, but divided by 10 gives 4.2857 liters. However, the problem might be asking for the maximum distance, which is 300/7 km, and then the fuel is 300/70 liters, which simplifies to 30/7 liters, approximately 4.2857 liters.Wait, but the problem says "the maximum volume of fuel a motorist would need to reach any of the roads connecting the other towns from point D." Since D is equidistant to all roads, the maximum distance is the same as the inradius, which is 300/7 km. Therefore, the fuel needed is (300/7)/10 = 30/7 liters, which is approximately 4.2857 liters.But the problem might be expecting the answer in liters, so 30/7 liters is approximately 4.2857 liters, which is about 4.29 liters. However, since the problem mentions "maximum volume," and 30/7 is an exact value, I should present it as 30/7 liters.Wait, but earlier I thought the inradius was 300/7 km, which is approximately 42.857 km, and then divided by 10 gives 4.2857 liters. But 300/7 divided by 10 is 30/7, which is correct.So, the maximum fuel needed is 30/7 liters, which is approximately 4.2857 liters. But the problem might expect the answer in a box, so I should write it as boxed{dfrac{30}{7}} liters.Wait, but let me double-check. The inradius r is 300/7 km, so the distance from D to any road is 300/7 km. Since fuel consumption is 1 liter per 10 km, the fuel needed is (300/7)/10 = 30/7 liters. Yes, that's correct.So, the maximum volume of fuel needed is 30/7 liters.

💡Okay, so I have this problem here: If (x + 3y = 6) and (xy = -12), what is the value of (x^2 + 9y^2)? Hmm, let me see how to approach this. First, I remember that when you have equations involving variables and their products, sometimes squaring one of the equations can help. Maybe I can square the first equation (x + 3y = 6) to see if that relates to (x^2 + 9y^2). Let me try that.So, if I square both sides of (x + 3y = 6), I get:[(x + 3y)^2 = 6^2]Which simplifies to:[x^2 + 6xy + 9y^2 = 36]Alright, that's interesting. Now, I notice that (x^2 + 9y^2) is part of this equation. If I can isolate that, maybe I can find its value. Let me see.From the equation above:[x^2 + 6xy + 9y^2 = 36]I can subtract (6xy) from both sides to get:[x^2 + 9y^2 = 36 - 6xy]Now, I know that (xy = -12), so I can substitute that into the equation:[x^2 + 9y^2 = 36 - 6(-12)]Calculating the right side:[36 - 6(-12) = 36 + 72 = 108]So, (x^2 + 9y^2 = 108). That seems straightforward. Let me double-check my steps to make sure I didn't make any mistakes.1. I started by squaring the first equation (x + 3y = 6), which gave me (x^2 + 6xy + 9y^2 = 36).2. Then, I isolated (x^2 + 9y^2) by subtracting (6xy), resulting in (x^2 + 9y^2 = 36 - 6xy).3. I substituted (xy = -12) into the equation, leading to (36 - 6(-12) = 36 + 72 = 108).Everything checks out. I don't see any errors in my calculations or reasoning. So, I'm confident that the value of (x^2 + 9y^2) is indeed 108.

💡First, I need to determine the number of students in each grade level. Given that 23% are juniors, 70% are not sophomores, and there are 160 seniors, I can calculate the number of juniors, seniors, sophomores, and freshmen.Next, I'll find the number of male students in each grade by applying the given gender distribution percentages.Then, I'll calculate the number of students in each club based on the provided percentages for each grade level.After that, I'll determine the number of male students in the specific clubs by applying the gender distribution percentages to the club members.Finally, I'll find the difference between the number of male freshmen in the Science Club and male sophomores in the Drama Club to answer the question.

💡First, calculate the total number of permutations of the digits 1, 2, 3, and 4, which is (4! = 24).Next, group the permutations by their starting digit:- Numbers starting with 1: (3! = 6) permutations- Numbers starting with 2: (3! = 6) permutationsThis accounts for the first 12 numbers.Now, consider numbers starting with 3:- The remaining digits are 1, 2, and 4.- The permutations starting with 3 are: - 3124 - 3142 - 3214 - 3241 - 3412 - 3421Since the first 12 numbers are accounted for, the numbers starting with 3 begin at position 13. The number 3142 is the second number in this group, placing it at position 14.