💡First, identify the original number of blue balloons Joan had, which is 9.Next, determine how many blue balloons Joan has now, which is 7.To find out how many balloons Joan lost, subtract the current number of balloons from the original number.So, 9 minus 7 equals 2.Therefore, Joan lost 2 blue balloons.

💡Alright, so I have this problem here: "If four times the larger of two numbers is nine times the smaller and the difference between the numbers is 12, find the larger of the two numbers." The options are given as fractions, which is a bit unusual, but I guess it's because the numbers might not be whole numbers. Let me try to break this down step by step.First, I need to identify what the problem is asking. There are two numbers, one larger and one smaller. The problem gives me two pieces of information:1. Four times the larger number is equal to nine times the smaller number.2. The difference between the two numbers is 12.I need to find the larger number. Okay, so let's assign some variables to make this easier. Let me call the smaller number "x" and the larger number "y." That seems logical because "y" is usually associated with being larger, but I could have used any letters. So, x is smaller, y is larger.Now, translating the first piece of information into an equation: "Four times the larger number is nine times the smaller." So, four times y equals nine times x. In math terms, that would be:4y = 9xGot that down. Now, the second piece of information: "The difference between the numbers is 12." Since y is larger and x is smaller, the difference would be y minus x equals 12. So:y - x = 12Alright, so now I have two equations:1. 4y = 9x2. y - x = 12I need to solve this system of equations to find the value of y, the larger number. Let's see how to approach this. I can use substitution or elimination. Since the second equation is already solved for y in terms of x, substitution might be the easiest here.From the second equation, y = x + 12. So, I can substitute this expression for y into the first equation.So, replacing y in the first equation with (x + 12):4(x + 12) = 9xLet me simplify this step by step. First, distribute the 4 on the left side:4x + 48 = 9xNow, I want to get all the x terms on one side and the constants on the other. Let me subtract 4x from both sides to move the x terms to the right:48 = 9x - 4xSimplifying the right side:48 = 5xNow, to solve for x, I can divide both sides by 5:x = 48 / 5Okay, so x is 48/5. That's a fraction, which makes sense given the answer choices. Now, I need to find y, the larger number. I can use the second equation again, which is y = x + 12. So, substituting x = 48/5 into this:y = (48/5) + 12Hmm, adding a fraction and a whole number. To add them, I need to express 12 as a fraction with the same denominator as 48/5. Since 12 is the same as 60/5, because 12 times 5 is 60, right?So, y = (48/5) + (60/5) = (48 + 60)/5 = 108/5Alright, so y is 108/5. Let me check if that makes sense. If y is 108/5, then x is 48/5, and the difference between y and x is 108/5 - 48/5 = 60/5 = 12, which matches the second condition. Also, four times y is 4*(108/5) = 432/5, and nine times x is 9*(48/5) = 432/5, so that also checks out.So, both conditions are satisfied with y = 108/5. Looking back at the answer choices:(A) 96/5(B) 100/5(C) 104/5(D) 108/5(E) 112/5108/5 is option D, so that must be the correct answer.Wait, just to make sure I didn't make any calculation errors. Let me go through the steps again.Starting with the two equations:4y = 9xy - x = 12From the second equation, y = x + 12. Substituting into the first equation:4(x + 12) = 9x4x + 48 = 9x48 = 5xx = 48/5Then, y = x + 12 = 48/5 + 60/5 = 108/5Yes, that seems consistent. So, I think I did everything correctly. The larger number is 108/5, which is option D.

💡Alright, so I have this problem here: Given real numbers x, y, z that satisfy the equation x + 2y + 3z + 8 = 0, I need to prove that the expression (x - 1)^2 + (y + 2)^2 + (z - 3)^2 is greater than or equal to 14. Hmm, okay. Let me try to figure this out step by step.First, I know that expressions like (x - a)^2 + (y - b)^2 + (z - c)^2 usually represent the squared distance from the point (x, y, z) to the point (a, b, c) in three-dimensional space. So, in this case, the expression (x - 1)^2 + (y + 2)^2 + (z - 3)^2 is the squared distance from the point (x, y, z) to the point (1, -2, 3). Now, the condition given is x + 2y + 3z + 8 = 0. That looks like the equation of a plane in three-dimensional space. So, essentially, we're dealing with all points (x, y, z) that lie on this plane. The problem is asking us to prove that the minimum distance from any point on this plane to the point (1, -2, 3) is at least sqrt(14). Therefore, the squared distance should be at least 14.I remember that the distance from a point to a plane can be calculated using a specific formula. Let me recall that formula. I think it's something like the absolute value of (ax + by + cz + d) divided by the square root of (a^2 + b^2 + c^2), where ax + by + cz + d = 0 is the equation of the plane, and (x, y, z) is the point. Wait, let me write that down more clearly. The distance D from a point (x0, y0, z0) to the plane ax + by + cz + d = 0 is given by:D = |a*x0 + b*y0 + c*z0 + d| / sqrt(a^2 + b^2 + c^2)So, in our case, the plane is x + 2y + 3z + 8 = 0, so a = 1, b = 2, c = 3, and d = 8. The point is (1, -2, 3). Plugging these into the formula, we get:D = |1*1 + 2*(-2) + 3*3 + 8| / sqrt(1^2 + 2^2 + 3^2)Let me compute the numerator first:1*1 = 12*(-2) = -43*3 = 9Adding these together: 1 - 4 + 9 = 6Then, adding d: 6 + 8 = 14So, the absolute value is |14| = 14.Now, the denominator is sqrt(1 + 4 + 9) = sqrt(14).Therefore, D = 14 / sqrt(14) = sqrt(14).So, the distance from the point (1, -2, 3) to the plane x + 2y + 3z + 8 = 0 is sqrt(14). Therefore, the minimum distance is sqrt(14), which means the squared distance is 14. Hence, for any point (x, y, z) on the plane, the squared distance to (1, -2, 3) is at least 14. Therefore, (x - 1)^2 + (y + 2)^2 + (z - 3)^2 >= 14.Wait, but the problem didn't specify that we have to use the distance formula. Maybe there's another way to approach this, like using the Cauchy-Schwarz inequality or something else.Let me think about using Cauchy-Schwarz. The Cauchy-Schwarz inequality states that for any vectors u and v, (u ⋅ v)^2 <= (u ⋅ u)(v ⋅ v). Maybe I can set up the problem in such a way that I can apply this inequality.Let me denote vector u as (x - 1, y + 2, z - 3) and vector v as (1, 2, 3). Then, the dot product u ⋅ v would be (x - 1)*1 + (y + 2)*2 + (z - 3)*3.Let me compute that:(x - 1)*1 = x - 1(y + 2)*2 = 2y + 4(z - 3)*3 = 3z - 9Adding these together: x - 1 + 2y + 4 + 3z - 9 = x + 2y + 3z - 6But from the given condition, x + 2y + 3z + 8 = 0, so x + 2y + 3z = -8. Therefore, x + 2y + 3z - 6 = -8 - 6 = -14.So, u ⋅ v = -14.Now, applying the Cauchy-Schwarz inequality:(u ⋅ v)^2 <= (u ⋅ u)(v ⋅ v)Which is:(-14)^2 <= [(x - 1)^2 + (y + 2)^2 + (z - 3)^2] * (1^2 + 2^2 + 3^2)So, 196 <= [(x - 1)^2 + (y + 2)^2 + (z - 3)^2] * 14Dividing both sides by 14:14 <= (x - 1)^2 + (y + 2)^2 + (z - 3)^2Which is exactly what we needed to prove. So, that's another way to see it using Cauchy-Schwarz.Alternatively, maybe I can use Lagrange multipliers since this is an optimization problem with a constraint. Let's try that approach.We need to minimize the function f(x, y, z) = (x - 1)^2 + (y + 2)^2 + (z - 3)^2 subject to the constraint g(x, y, z) = x + 2y + 3z + 8 = 0.The method of Lagrange multipliers tells us that at the minimum, the gradient of f is proportional to the gradient of g. So, ∇f = λ∇g for some λ.Computing the gradients:∇f = (2(x - 1), 2(y + 2), 2(z - 3))∇g = (1, 2, 3)So, setting up the equations:2(x - 1) = λ*12(y + 2) = λ*22(z - 3) = λ*3And the constraint equation: x + 2y + 3z + 8 = 0.Let me solve these equations step by step.From the first equation: 2(x - 1) = λ => x - 1 = λ/2 => x = 1 + λ/2From the second equation: 2(y + 2) = 2λ => y + 2 = λ => y = λ - 2From the third equation: 2(z - 3) = 3λ => z - 3 = (3λ)/2 => z = 3 + (3λ)/2Now, substitute x, y, z into the constraint equation:x + 2y + 3z + 8 = 0Substituting:(1 + λ/2) + 2*(λ - 2) + 3*(3 + (3λ)/2) + 8 = 0Let me compute each term:1 + λ/22*(λ - 2) = 2λ - 43*(3 + (3λ)/2) = 9 + (9λ)/2Adding all together:1 + λ/2 + 2λ - 4 + 9 + (9λ)/2 + 8 = 0Combine like terms:Constants: 1 - 4 + 9 + 8 = 14Lambda terms: λ/2 + 2λ + (9λ)/2Let me convert 2λ to (4λ)/2 to have a common denominator:λ/2 + (4λ)/2 + (9λ)/2 = (14λ)/2 = 7λSo, the equation becomes:14 + 7λ = 0 => 7λ = -14 => λ = -2Now, substitute λ = -2 back into expressions for x, y, z:x = 1 + (-2)/2 = 1 - 1 = 0y = (-2) - 2 = -4z = 3 + (3*(-2))/2 = 3 - 3 = 0So, the point (x, y, z) that minimizes the distance is (0, -4, 0). Let's compute the squared distance:(0 - 1)^2 + (-4 + 2)^2 + (0 - 3)^2 = (-1)^2 + (-2)^2 + (-3)^2 = 1 + 4 + 9 = 14So, the minimum squared distance is indeed 14, which means that for any other point on the plane, the squared distance will be greater than or equal to 14.Therefore, we've proven that (x - 1)^2 + (y + 2)^2 + (z - 3)^2 >= 14.I think all these methods—distance formula, Cauchy-Schwarz, and Lagrange multipliers—lead to the same conclusion. It's reassuring to see that different approaches confirm the same result. So, I feel confident that the inequality holds true.**Final Answer**The minimum value is boxed{14}.

💡First, I identify the numbers provided: 25, 41, 13, and 32.Next, I determine the largest number among them, which is 41.Then, I find the smallest number, which is 13.Finally, I calculate the sum of the largest and smallest numbers: 41 + 13 = 54.

💡Alright, let's tackle this problem step by step. So, we have a set of consecutive positive integers starting from 3, and one number is erased. The average of the remaining numbers is given as 53 and 8/19. We need to figure out which number was erased. The options are 800, 805, 810, 815, and 820.First, let me understand what the problem is saying. We have a sequence of numbers starting at 3, like 3, 4, 5, ..., up to some number, let's call it 'n'. Then, someone erases one number from this sequence, and the average of the remaining numbers is 53 8/19. We need to find out which number was erased.Okay, so let's break this down. The average of the remaining numbers is 53 8/19. That's a mixed number, so I should probably convert that to an improper fraction to make calculations easier. 53 times 19 is 1007, plus 8 is 1015, so 53 8/19 is equal to 1015/19. That might be useful later.Now, let's think about the original sequence. It starts at 3 and goes up to some number 'n'. The number of terms in this sequence is (n - 3 + 1) = (n - 2). So, the total number of terms is n - 2.The sum of an arithmetic series is given by the formula: sum = (number of terms)/2 * (first term + last term). So, the sum of the original sequence is (n - 2)/2 * (3 + n).After one number is erased, the number of terms becomes (n - 3). The average of the remaining numbers is given as 1015/19. So, the sum of the remaining numbers is average * number of terms, which is (1015/19) * (n - 3).Therefore, the sum of the original sequence minus the erased number equals the sum of the remaining numbers. So, we can write the equation:Original sum - erased number = Remaining sumWhich translates to:[(n - 2)/2 * (3 + n)] - x = (1015/19) * (n - 3)Here, 'x' is the erased number.So, let's write that equation out:[(n - 2)(n + 3)/2] - x = (1015/19)(n - 3)Now, we need to solve for 'x' and 'n'. But we have two variables here, so we need another equation or some way to relate 'n' and 'x'. Since 'x' is one of the numbers in the sequence, it must be an integer between 3 and 'n'. Also, the options given for 'x' are all in the 800s, so 'n' must be at least 820, because the highest option is 820.Wait, that might not necessarily be true. Let me think. If the average after erasing is 53 8/19, which is approximately 53.42, that suggests that the numbers are around that average. But since the sequence starts at 3, it must go up to a number much higher than 53 to have an average around 53. So, 'n' is likely in the 100s or maybe higher.But the options for 'x' are all in the 800s, so 'n' must be at least 820. Hmm, that seems like a big jump. Maybe I'm missing something.Wait, let's think about the average. The average of the remaining numbers is 53 8/19. That average is calculated after removing one number. So, the original average would be slightly different. Let me see.The original average would be the sum of the original sequence divided by (n - 2). The remaining average is the sum of the original sequence minus 'x' divided by (n - 3). So, maybe I can set up an equation involving the original average and the new average.But I don't know the original average. Maybe I can express it in terms of 'n'.Alternatively, perhaps I can manipulate the equation I have to solve for 'n' first.Let me rewrite the equation:[(n - 2)(n + 3)/2] - x = (1015/19)(n - 3)Let me multiply both sides by 2 to eliminate the denominator:(n - 2)(n + 3) - 2x = (2030/19)(n - 3)Hmm, 2030 divided by 19 is approximately 106.842. But let me see if 2030 is divisible by 19.19 times 106 is 2014, and 2030 - 2014 is 16, so 2030/19 is 106 and 16/19. So, it's 106.842.But maybe I can keep it as a fraction to make calculations exact.So, 2030/19 is equal to 106 + 16/19.So, the equation becomes:(n - 2)(n + 3) - 2x = (106 + 16/19)(n - 3)Hmm, this is getting a bit messy. Maybe I can multiply both sides by 19 to eliminate the denominators.So, multiply both sides by 19:19*(n - 2)(n + 3) - 38x = 2030*(n - 3)Now, let's expand the left side:19*(n^2 + 3n - 2n - 6) - 38x = 2030n - 6090Simplify inside the brackets:19*(n^2 + n - 6) - 38x = 2030n - 6090Now, distribute the 19:19n^2 + 19n - 114 - 38x = 2030n - 6090Now, let's bring all terms to one side:19n^2 + 19n - 114 - 38x - 2030n + 6090 = 0Combine like terms:19n^2 + (19n - 2030n) + (-114 + 6090) - 38x = 0Which simplifies to:19n^2 - 2011n + 5976 - 38x = 0Hmm, this is a quadratic equation in terms of 'n', but we still have 'x' in there. Since 'x' is one of the options, maybe we can substitute the options into the equation and see which one makes 'n' an integer.Let's try that approach.Given that 'x' is one of 800, 805, 810, 815, 820, let's substitute each into the equation and see if we can find an integer 'n'.Starting with option (B) 805, since it's the middle option, maybe that's the answer.So, let's set x = 805.Then, the equation becomes:19n^2 - 2011n + 5976 - 38*805 = 0Calculate 38*805:38*800 = 30,40038*5 = 190So, 30,400 + 190 = 30,590So, the equation is:19n^2 - 2011n + 5976 - 30,590 = 0Simplify:19n^2 - 2011n - 24,614 = 0Now, let's see if this quadratic equation has integer solutions.We can use the quadratic formula:n = [2011 ± sqrt(2011^2 - 4*19*(-24,614))]/(2*19)Calculate discriminant D:D = 2011^2 - 4*19*(-24,614)First, 2011^2:2011*2011. Let's calculate that.2000^2 = 4,000,0002*2000*11 = 44,00011^2 = 121So, (2000 + 11)^2 = 4,000,000 + 44,000 + 121 = 4,044,121Now, 4*19*24,614:First, 4*19 = 7676*24,614Let's calculate 76*24,614:24,614 * 70 = 1,722,98024,614 * 6 = 147,684So, total is 1,722,980 + 147,684 = 1,870,664But since it's -4*19*(-24,614), it becomes +1,870,664So, D = 4,044,121 + 1,870,664 = 5,914,785Now, sqrt(5,914,785). Let's see if this is a perfect square.Let me estimate sqrt(5,914,785). 2430^2 = 5,904,900. 2440^2 = 5,953,600. So, it's between 2430 and 2440.Let's try 2435^2:2435^2 = (2400 + 35)^2 = 2400^2 + 2*2400*35 + 35^2 = 5,760,000 + 168,000 + 1,225 = 5,929,225That's higher than 5,914,785.Let's try 2430^2 = 5,904,900Difference: 5,914,785 - 5,904,900 = 9,885So, 2430 + x squared is 5,914,785.(2430 + x)^2 = 2430^2 + 2*2430*x + x^2 = 5,904,900 + 4,860x + x^2 = 5,914,785So, 4,860x + x^2 = 9,885Assuming x is small, x^2 is negligible, so 4,860x ≈ 9,885x ≈ 9,885 / 4,860 ≈ 2.035So, x ≈ 2.035, so let's try x=2:(2430 + 2)^2 = 2432^2 = ?2432^2 = (2400 + 32)^2 = 2400^2 + 2*2400*32 + 32^2 = 5,760,000 + 153,600 + 1,024 = 5,914,624That's very close to 5,914,785.Difference: 5,914,785 - 5,914,624 = 161So, 2432^2 = 5,914,6242433^2 = 2432^2 + 2*2432 + 1 = 5,914,624 + 4,864 + 1 = 5,919,489Which is way higher. So, the discriminant is not a perfect square, which means 'n' is not an integer. Hmm, that's a problem.Wait, maybe I made a mistake in calculations. Let me double-check.Wait, when I calculated 38*805, I got 30,590. Let me confirm:805 * 38:800*38 = 30,4005*38 = 19030,400 + 190 = 30,590. That's correct.Then, 19n^2 - 2011n + 5976 - 30,590 = 0Which is 19n^2 - 2011n - 24,614 = 0Discriminant D = 2011^2 - 4*19*(-24,614) = 4,044,121 + 1,870,664 = 5,914,785sqrt(5,914,785) ≈ 2432.03, which is not an integer. So, 'n' is not an integer, which contradicts our assumption that 'n' is an integer because it's the last term in the sequence.Hmm, so maybe x is not 805. Let's try another option.Let's try option (A) 800.So, x = 800.Then, the equation becomes:19n^2 - 2011n + 5976 - 38*800 = 0Calculate 38*800 = 30,400So, the equation is:19n^2 - 2011n + 5976 - 30,400 = 0Simplify:19n^2 - 2011n - 24,424 = 0Again, use quadratic formula:n = [2011 ± sqrt(2011^2 - 4*19*(-24,424))]/(2*19)Calculate discriminant D:D = 4,044,121 + 4*19*24,424First, 4*19 = 7676*24,424Calculate 76*24,424:24,424 * 70 = 1,709,68024,424 * 6 = 146,544Total = 1,709,680 + 146,544 = 1,856,224So, D = 4,044,121 + 1,856,224 = 5,900,345Now, sqrt(5,900,345). Let's see.2430^2 = 5,904,900, which is higher than 5,900,345.2425^2 = ?2425^2 = (2400 + 25)^2 = 2400^2 + 2*2400*25 + 25^2 = 5,760,000 + 120,000 + 625 = 5,880,625Difference: 5,900,345 - 5,880,625 = 19,720So, 2425 + x squared is 5,900,345.(2425 + x)^2 = 5,880,625 + 2*2425*x + x^2 = 5,880,625 + 4,850x + x^2 = 5,900,345So, 4,850x + x^2 = 19,720Assuming x is small, x^2 is negligible, so 4,850x ≈ 19,720x ≈ 19,720 / 4,850 ≈ 4.066So, x ≈ 4.066, so let's try x=4:2425 + 4 = 24292429^2 = ?2429^2 = (2400 + 29)^2 = 2400^2 + 2*2400*29 + 29^2 = 5,760,000 + 139,200 + 841 = 5,899,041Difference: 5,900,345 - 5,899,041 = 1,304So, 2429^2 = 5,899,0412430^2 = 5,904,900So, sqrt(5,900,345) is between 2429 and 2430, not an integer. So, 'n' is not an integer here either.Hmm, okay, let's try option (C) 810.x = 810Equation becomes:19n^2 - 2011n + 5976 - 38*810 = 0Calculate 38*810:38*800 = 30,40038*10 = 380Total = 30,400 + 380 = 30,780So, equation:19n^2 - 2011n + 5976 - 30,780 = 0Simplify:19n^2 - 2011n - 24,804 = 0Quadratic formula:n = [2011 ± sqrt(2011^2 - 4*19*(-24,804))]/(2*19)Calculate discriminant D:D = 4,044,121 + 4*19*24,8044*19 = 7676*24,804Calculate 76*24,804:24,804 * 70 = 1,736,28024,804 * 6 = 148,824Total = 1,736,280 + 148,824 = 1,885,104So, D = 4,044,121 + 1,885,104 = 5,929,225Now, sqrt(5,929,225). Let's see.2435^2 = 5,929,225. Yes, because earlier I calculated 2435^2 as 5,929,225.So, sqrt(D) = 2435Therefore, n = [2011 ± 2435]/(38)Calculate both possibilities:First, n = (2011 + 2435)/38 = (4446)/38 = 117Second, n = (2011 - 2435)/38 = (-424)/38 = -11.157Since 'n' must be positive and greater than 3, we take n=117.So, n=117.Now, let's verify if this makes sense.Original sequence is from 3 to 117.Number of terms: 117 - 3 + 1 = 115Sum of original sequence: (115)/2 * (3 + 117) = (115)/2 * 120 = 115*60 = 6,900After erasing x=810, the sum becomes 6,900 - 810 = 6,090Number of remaining terms: 115 - 1 = 114Average of remaining numbers: 6,090 / 114Calculate 6,090 ÷ 114:114*53 = 6,0426,090 - 6,042 = 48So, 53 + 48/114 = 53 + 16/38 = 53 + 8/19Which is exactly 53 8/19.So, this works!Therefore, the erased number is 810.Wait, but in the options, 810 is option (C). But earlier, when I tried x=805, it didn't work, but x=810 works.Wait, but in the initial problem, the options are (A) 800, (B) 805, (C) 810, (D) 815, (E) 820.So, according to this, the answer should be (C) 810.But wait, in my earlier calculation, when I tried x=805, the discriminant was not a perfect square, but when I tried x=810, it worked.Wait, but in the initial problem, the average after erasing is 53 8/19, which is approximately 53.42. If the original average was higher, then erasing a higher number would bring the average down. So, if the original average was higher than 53.42, erasing a higher number would make the average lower.But in our case, the original average was 6,900 / 115 ≈ 59.565, which is higher than 53.42. So, erasing a higher number (810) would bring the average down to 53.42.Wait, but 810 is much higher than the original average. So, erasing 810 would significantly lower the average.Wait, let me check the calculations again.Original sum: 6,900After erasing 810: 6,090Number of terms: 114Average: 6,090 / 114 = 53.42, which is 53 8/19.Yes, that's correct.But wait, in the options, 810 is option (C), but the initial problem statement says the options are (A) 800, (B) 805, (C) 810, (D) 815, (E) 820.So, according to my calculation, the answer is (C) 810.But in the initial problem, the user wrote:"A set of consecutive positive integers beginning with 3 is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is 53 8/19. Based on this new series, determine which number was erased.(A) 800 (B) 805 (C) 810 (D) 815 (E) 820"So, according to my calculation, the answer is (C) 810.But wait, in the initial problem, the user wrote the final answer as (B) 805, but according to my calculation, it's (C) 810.Hmm, maybe I made a mistake somewhere.Wait, let's double-check the calculation when x=805.When x=805, we had:19n^2 - 2011n - 24,614 = 0Discriminant D = 2011^2 - 4*19*(-24,614) = 4,044,121 + 1,870,664 = 5,914,785sqrt(5,914,785) ≈ 2432.03, not an integer.But when x=810, we had:19n^2 - 2011n - 24,804 = 0Discriminant D = 4,044,121 + 1,885,104 = 5,929,225sqrt(5,929,225) = 2435, which is an integer.So, n=117.So, the original sequence is from 3 to 117, which is 115 numbers.Sum is 6,900.After erasing 810, sum is 6,090, which divided by 114 gives 53 8/19.So, that works.But why did the initial problem say the answer is (B) 805? Maybe I misread the problem.Wait, let me check the problem again."A set of consecutive positive integers beginning with 3 is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is 53 8/19. Based on this new series, determine which number was erased.(A) 800 (B) 805 (C) 810 (D) 815 (E) 820"So, according to my calculation, the answer is (C) 810.But maybe I made a mistake in setting up the equation.Wait, let's go back to the beginning.Original sequence: 3, 4, 5, ..., nNumber of terms: n - 2Sum: S = (n - 2)(n + 3)/2After erasing x, number of terms: n - 3Sum: S - xAverage: (S - x)/(n - 3) = 1015/19So, equation:[(n - 2)(n + 3)/2 - x] / (n - 3) = 1015/19Multiply both sides by (n - 3):(n - 2)(n + 3)/2 - x = (1015/19)(n - 3)Multiply both sides by 2:(n - 2)(n + 3) - 2x = (2030/19)(n - 3)Multiply both sides by 19:19(n - 2)(n + 3) - 38x = 2030(n - 3)Expand left side:19(n^2 + n - 6) - 38x = 2030n - 6090Which is:19n^2 + 19n - 114 - 38x = 2030n - 6090Bring all terms to left:19n^2 + 19n - 114 - 38x - 2030n + 6090 = 0Simplify:19n^2 - 2011n + 5976 - 38x = 0Yes, that's correct.So, when x=810, we get n=117, which works.When x=805, n is not integer.So, the answer should be (C) 810.But in the initial problem, the user wrote the final answer as (B) 805. Maybe that was a mistake.Alternatively, perhaps the original sequence is longer, and 'n' is larger.Wait, let's try x=805 again, but with a larger 'n'.Wait, if n=117, x=810 is erased.But what if n is larger, say n=118.Wait, let's see.If n=118, number of terms is 116.Sum is (116)/2*(3 + 118) = 58*121 = 7,038If we erase x=805, sum becomes 7,038 - 805 = 6,233Number of terms: 115Average: 6,233 / 115 ≈ 54.2, which is higher than 53 8/19.So, not matching.Alternatively, if n=116.Sum: (116 - 2)/2*(3 + 116) = 114/2*119 = 57*119 = 6,783Erase x=805: 6,783 - 805 = 5,978Number of terms: 113Average: 5,978 / 113 ≈ 52.9, which is less than 53 8/19.So, not matching.Hmm, so n=117 gives us the correct average when x=810 is erased.Therefore, the answer should be (C) 810.But the initial problem's final answer was (B) 805. Maybe the initial problem had a different setup.Alternatively, perhaps I made a mistake in the calculation when x=805.Wait, let's try x=805 again.Equation: 19n^2 - 2011n - 24,614 = 0Discriminant D = 2011^2 + 4*19*24,614Wait, earlier I thought it was 2011^2 - 4*19*(-24,614), which is correct.But 4*19*24,614 = 76*24,614 = 1,870,664So, D = 4,044,121 + 1,870,664 = 5,914,785sqrt(5,914,785) ≈ 2432.03, not an integer.So, n is not integer.Therefore, x=805 is not the correct answer.Thus, the correct answer is (C) 810.

💡First, recall the formula for the curved surface area (CSA) of a cone:[ text{CSA} = pi r l ]Given:[ l = a sin(alpha) ][ r = a cos(beta) ]Substitute the expressions for ( l ) and ( r ) into the CSA formula:[ text{CSA} = pi times (a cos(beta)) times (a sin(alpha)) ]Simplify the expression:[ text{CSA} = pi a^2 sin(alpha) cos(beta) ]Therefore, the general formula for the curved surface area of the cone in terms of ( a ), ( alpha ), and ( beta ) is:[ text{CSA} = pi a^2 sin(alpha) cos(beta) ]