answer：Alright, so I have this problem here about an arithmetic sequence. Let me try to understand it step by step. The sequence is given as {a_n}, and they tell me that a₁ is -8 and a₂ is -6. Okay, so arithmetic sequences have a common difference between consecutive terms. That means I can find the common difference, d, by subtracting a₁ from a₂. So, d = a₂ - a₁ = (-6) - (-8) = 2. Got it, the common difference is 2. That means each term increases by 2 from the previous one. So, a₁ is -8, a₂ is -6, a₃ would be -4, a₄ would be -2, a₅ would be 0, and so on. Let me write that down:a₁ = -8 a₂ = -6 a₃ = -4 a₄ = -2 a₅ = 0 ...and so on.Now, the problem says that the same number is added to a₁, a₄, and a₅, and the resulting three numbers form a geometric sequence in order. Hmm, okay. So, let's denote this number as x. That means we're going to add x to each of these terms:- For a₁, which is -8, adding x gives us (-8 + x).- For a₄, which is -2, adding x gives us (-2 + x).- For a₅, which is 0, adding x gives us (0 + x) = x.So now, we have three new numbers: (-8 + x), (-2 + x), and x. And these three numbers form a geometric sequence. Alright, now I need to recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, r. So, if I have three terms, say b, c, d, in a geometric sequence, then c/b = d/c, which means c² = b*d. Applying that to our three terms: (-8 + x), (-2 + x), and x. So, the middle term squared should equal the product of the first and the third terms. Let me write that equation out:(-2 + x)² = (-8 + x) * xOkay, now I need to solve this equation for x. Let's expand both sides step by step.First, the left side: (-2 + x)². That's equal to (-2 + x) multiplied by (-2 + x). Let me compute that:(-2 + x) * (-2 + x) = (-2)*(-2) + (-2)*x + x*(-2) + x*x = 4 - 2x - 2x + x² = 4 - 4x + x²So, the left side simplifies to x² - 4x + 4.Now, the right side: (-8 + x) * x. Let's compute that:(-8 + x) * x = -8*x + x*x = -8x + x²So, the right side simplifies to x² - 8x.Now, set the left side equal to the right side:x² - 4x + 4 = x² - 8xHmm, okay. Let me subtract x² from both sides to simplify:x² - 4x + 4 - x² = x² - 8x - x² Simplifies to: -4x + 4 = -8xNow, let's solve for x. I'll add 8x to both sides:-4x + 4 + 8x = -8x + 8x Simplifies to: 4x + 4 = 0Now, subtract 4 from both sides:4x + 4 - 4 = 0 - 4 Simplifies to: 4x = -4Finally, divide both sides by 4:4x / 4 = -4 / 4 Simplifies to: x = -1So, the added number is -1. Let me just double-check to make sure I didn't make any mistakes.If x is -1, then:- a₁ + x = -8 + (-1) = -9 - a₄ + x = -2 + (-1) = -3 - a₅ + x = 0 + (-1) = -1So, the three numbers are -9, -3, -1. Let's check if this is a geometric sequence.The ratio between the second and first term: (-3)/(-9) = 1/3 The ratio between the third and second term: (-1)/(-3) = 1/3Yes, both ratios are equal to 1/3, so it is indeed a geometric sequence. That checks out.Wait, but just to be thorough, let me plug x = -1 back into the original equation to ensure it satisfies the equation.Left side: (-2 + x)² = (-2 + (-1))² = (-3)² = 9 Right side: (-8 + x)*x = (-8 + (-1))*(-1) = (-9)*(-1) = 9Yes, both sides equal 9, so x = -1 is definitely the correct solution.I think that's solid. I don't see any mistakes in my calculations, and the check confirms that it works. So, the added number is -1.**Final Answer**The added number is boxed{-1}.

answer：Alright, so I have this problem where I need to figure out the relationship between four numbers: a, b, c, and d. The options are given as A, B, C, D, each suggesting a different order from smallest to largest. Let me try to break this down step by step.First, let's write down what each variable represents:- a = 0.2^{3.5}- b = 0.2^{4.1}- c = e^{1.1}- d = log_{0.2}3I need to compare these four numbers and determine their order. Let me tackle each one individually.Starting with a and b since they both involve the same base, 0.2, raised to different exponents. I remember that when the base is between 0 and 1, the function is decreasing. That means as the exponent increases, the value of the function decreases. So, since 3.5 is less than 4.1, a should be greater than b. So, a > b.Next, let's think about c, which is e^{1.1}. I know that e is approximately 2.718, so e^{1} is about 2.718. Since 1.1 is slightly more than 1, e^{1.1} should be a bit more than 2.718. Maybe around 3 or so? I can calculate it more precisely if needed, but for now, I'll just note that c is greater than 1 because e^{0} is 1, and since the exponent is positive, it's definitely larger than 1.Now, d is log base 0.2 of 3. Hmm, logarithms with bases between 0 and 1 are tricky because they are decreasing functions. So, log_{0.2}3 is asking, "0.2 raised to what power gives me 3?" Since 0.2 is less than 1, to get a number greater than 1 (which 3 is), the exponent must be negative. For example, log_{0.2}1 is 0 because 0.2^0 = 1. And since 3 is greater than 1, log_{0.2}3 will be negative. So, d is less than 0.Putting this together:- a and b are both positive numbers less than 1 because 0.2 raised to any positive power will be less than 1. But since a has a smaller exponent, a > b.- c is greater than 1.- d is negative.So, in order from smallest to largest, it should be d < b < a < c.Let me double-check to make sure I didn't make a mistake. For a and b, since the base is less than 1, higher exponents lead to smaller numbers. So, 0.2^{4.1} is indeed smaller than 0.2^{3.5}, meaning b < a. That seems right.For c, e^{1.1} is definitely greater than e^1, which is about 2.718, so c is more than 2.718. And d is negative because log base less than 1 of a number greater than 1 is negative. So, d is the smallest.Therefore, the order is d < b < a < c, which corresponds to option C.**Final Answer**boxed{C}

answer：Okay, so I have this problem about a regular tetrahedron (ABCD), and I need to find the dihedral angle (C-FG-E). The points (E), (F), and (G) are midpoints of edges (AB), (BC), and (CD) respectively. Hmm, dihedral angles can be tricky, but let me try to visualize this.First, I remember that a regular tetrahedron has all edges equal, so every edge is the same length. Let's assume each edge has length 2 for simplicity. That might make calculations easier later on.Now, (E), (F), and (G) are midpoints, so each of them divides their respective edges into two equal parts. That means each segment like (AE), (EB), (BF), (FC), (CG), and (GD) is 1 unit long if the total edge length is 2.I need to find the dihedral angle (C-FG-E). Dihedral angle is the angle between two planes. In this case, the planes are probably the ones containing the edges (FG) and the points (C) and (E). So, one plane would be the one containing (C), (F), and (G), and the other plane would be the one containing (E), (F), and (G). Is that right? Wait, actually, dihedral angle is between two planes that share a common line, which here is (FG). So, the two planes are (CFG) and (EFG). So, the dihedral angle is between these two planes along the line (FG).To find this dihedral angle, I might need to find the angle between the two planes. One way to do this is to find the angle between two vectors, each lying on one of the planes and perpendicular to the line (FG). Alternatively, I can use the coordinates of the points to compute the angle.Maybe assigning coordinates to the tetrahedron would help. Let me place the tetrahedron in a coordinate system. Let me recall that a regular tetrahedron can be embedded in 3D space with vertices at ((1,1,1)), ((1,-1,-1)), ((-1,1,-1)), and ((-1,-1,1)), scaled appropriately. But maybe that's too complicated. Alternatively, I can place one vertex at the origin, but that might complicate things because of symmetry.Wait, perhaps it's better to use a regular tetrahedron with edge length 2, so that the midpoints are at integer coordinates. Let me try to assign coordinates such that calculations are easier.Let me set point (A) at ((0,0,0)), point (B) at ((2,0,0)), point (C) at ((1, sqrt{3}, 0)), and point (D) at ((1, sqrt{3}/3, sqrt{6}/3 * 2)). Wait, maybe that's too hasty. Let me think.Actually, a regular tetrahedron can be placed with one vertex at the origin, and the others in such a way that all edges are equal. Let me recall that the coordinates can be set as follows:Let me choose the coordinates so that the centroid is at the origin, but maybe that's overcomplicating. Alternatively, place one vertex at the top, one along the x-axis, one in the xy-plane, and the last one somewhere in 3D space.Alternatively, perhaps it's better to use vectors and midpoints without assigning specific coordinates. Let me think.Since all edges are equal, the midpoints will form smaller segments. Maybe I can use vectors to find the normals of the planes (CFG) and (EFG), and then find the angle between these normals.Yes, that sounds feasible. So, if I can find the normal vectors to the planes (CFG) and (EFG), then the dihedral angle between the planes is equal to the angle between these normals.To find the normal vectors, I can take the cross product of two vectors lying on each plane.First, let me define the coordinates of the points. Let me assign coordinates to the tetrahedron (ABCD) such that calculations are manageable.Let me set point (A) at ((0,0,0)), point (B) at ((2,0,0)), point (C) at ((1, sqrt{3}, 0)), and point (D) at ((1, sqrt{3}/3, h)), where (h) is the height from the base to the apex (D). Since it's a regular tetrahedron, all edges are equal, so the distance from (D) to each of (A), (B), and (C) should be 2.Let me compute (h). The distance from (D) to (A) is:[sqrt{(1-0)^2 + left(frac{sqrt{3}}{3} - 0right)^2 + (h - 0)^2} = 2]Simplify:[1 + frac{1}{3} + h^2 = 4][frac{4}{3} + h^2 = 4][h^2 = frac{8}{3}][h = frac{2sqrt{6}}{3}]So, point (D) is at ((1, sqrt{3}/3, 2sqrt{6}/3)).Now, let's find the coordinates of midpoints (E), (F), and (G).- (E) is the midpoint of (AB): so average the coordinates of (A) and (B). (A = (0,0,0)), (B = (2,0,0)), so (E = left(frac{0+2}{2}, frac{0+0}{2}, frac{0+0}{2}right) = (1,0,0)).- (F) is the midpoint of (BC): (B = (2,0,0)), (C = (1, sqrt{3}, 0)), so (F = left(frac{2+1}{2}, frac{0 + sqrt{3}}{2}, frac{0+0}{2}right) = left(frac{3}{2}, frac{sqrt{3}}{2}, 0right)).- (G) is the midpoint of (CD): (C = (1, sqrt{3}, 0)), (D = (1, sqrt{3}/3, 2sqrt{6}/3)), so (G = left(frac{1+1}{2}, frac{sqrt{3} + sqrt{3}/3}{2}, frac{0 + 2sqrt{6}/3}{2}right)).Simplify (G):- x-coordinate: (frac{2}{2} = 1)- y-coordinate: (frac{sqrt{3} + sqrt{3}/3}{2} = frac{(3sqrt{3} + sqrt{3})/3}{2} = frac{4sqrt{3}/3}{2} = frac{2sqrt{3}}{3})- z-coordinate: (frac{2sqrt{6}/3}{2} = frac{sqrt{6}}{3})So, (G = left(1, frac{2sqrt{3}}{3}, frac{sqrt{6}}{3}right)).Now, I have coordinates for points (C), (F), (G), and (E). Let me write them down:- (C = (1, sqrt{3}, 0))- (F = left(frac{3}{2}, frac{sqrt{3}}{2}, 0right))- (G = left(1, frac{2sqrt{3}}{3}, frac{sqrt{6}}{3}right))- (E = (1,0,0))Now, I need to find the dihedral angle between the planes (CFG) and (EFG) along the line (FG). To find this, I can find the normal vectors to each plane and then compute the angle between these normals.First, let's find two vectors in each plane.For plane (CFG):Vectors can be ( overrightarrow{FG} ) and ( overrightarrow{CG} ).Compute ( overrightarrow{FG} ):(G - F = left(1 - frac{3}{2}, frac{2sqrt{3}}{3} - frac{sqrt{3}}{2}, frac{sqrt{6}}{3} - 0right) )Simplify:x: ( -frac{1}{2} )y: ( frac{2sqrt{3}}{3} - frac{sqrt{3}}{2} = frac{4sqrt{3} - 3sqrt{3}}{6} = frac{sqrt{3}}{6} )z: ( frac{sqrt{6}}{3} )So, ( overrightarrow{FG} = left(-frac{1}{2}, frac{sqrt{3}}{6}, frac{sqrt{6}}{3}right) )Compute ( overrightarrow{CG} ):(G - C = left(1 - 1, frac{2sqrt{3}}{3} - sqrt{3}, frac{sqrt{6}}{3} - 0right) )Simplify:x: 0y: ( frac{2sqrt{3}}{3} - sqrt{3} = -frac{sqrt{3}}{3} )z: ( frac{sqrt{6}}{3} )So, ( overrightarrow{CG} = left(0, -frac{sqrt{3}}{3}, frac{sqrt{6}}{3}right) )Now, the normal vector to plane (CFG) is the cross product of ( overrightarrow{FG} ) and ( overrightarrow{CG} ).Let me compute that:Let ( mathbf{a} = overrightarrow{FG} = left(-frac{1}{2}, frac{sqrt{3}}{6}, frac{sqrt{6}}{3}right) )Let ( mathbf{b} = overrightarrow{CG} = left(0, -frac{sqrt{3}}{3}, frac{sqrt{6}}{3}right) )Cross product ( mathbf{a} times mathbf{b} ) is:[begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} -frac{1}{2} & frac{sqrt{3}}{6} & frac{sqrt{6}}{3} 0 & -frac{sqrt{3}}{3} & frac{sqrt{6}}{3} end{vmatrix}]Compute determinant:- i component: ( left(frac{sqrt{3}}{6} cdot frac{sqrt{6}}{3}right) - left(frac{sqrt{6}}{3} cdot -frac{sqrt{3}}{3}right) )- j component: ( -left( left(-frac{1}{2} cdot frac{sqrt{6}}{3}right) - left(frac{sqrt{6}}{3} cdot 0right) right) )- k component: ( left(-frac{1}{2} cdot -frac{sqrt{3}}{3}right) - left(frac{sqrt{3}}{6} cdot 0right) )Compute each component:i component:( frac{sqrt{3} cdot sqrt{6}}{18} - left(-frac{sqrt{6} cdot sqrt{3}}{9}right) = frac{sqrt{18}}{18} + frac{sqrt{18}}{9} = frac{3sqrt{2}}{18} + frac{3sqrt{2}}{9} = frac{sqrt{2}}{6} + frac{sqrt{2}}{3} = frac{sqrt{2}}{6} + frac{2sqrt{2}}{6} = frac{3sqrt{2}}{6} = frac{sqrt{2}}{2} )j component:( -left( -frac{sqrt{6}}{6} - 0 right) = -left( -frac{sqrt{6}}{6} right) = frac{sqrt{6}}{6} )k component:( frac{sqrt{3}}{6} - 0 = frac{sqrt{3}}{6} )So, the normal vector ( mathbf{n}_1 = left( frac{sqrt{2}}{2}, frac{sqrt{6}}{6}, frac{sqrt{3}}{6} right) )Now, let's find the normal vector for plane (EFG).For plane (EFG):Points (E), (F), (G). Let's find vectors ( overrightarrow{FG} ) and ( overrightarrow{EG} ).Wait, we already have ( overrightarrow{FG} ) from earlier: ( left(-frac{1}{2}, frac{sqrt{3}}{6}, frac{sqrt{6}}{3}right) )Compute ( overrightarrow{EG} ):(G - E = left(1 - 1, frac{2sqrt{3}}{3} - 0, frac{sqrt{6}}{3} - 0right) = left(0, frac{2sqrt{3}}{3}, frac{sqrt{6}}{3}right) )So, ( overrightarrow{EG} = left(0, frac{2sqrt{3}}{3}, frac{sqrt{6}}{3}right) )Now, the normal vector to plane (EFG) is the cross product of ( overrightarrow{FG} ) and ( overrightarrow{EG} ).Let ( mathbf{c} = overrightarrow{FG} = left(-frac{1}{2}, frac{sqrt{3}}{6}, frac{sqrt{6}}{3}right) )Let ( mathbf{d} = overrightarrow{EG} = left(0, frac{2sqrt{3}}{3}, frac{sqrt{6}}{3}right) )Cross product ( mathbf{c} times mathbf{d} ):[begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} -frac{1}{2} & frac{sqrt{3}}{6} & frac{sqrt{6}}{3} 0 & frac{2sqrt{3}}{3} & frac{sqrt{6}}{3} end{vmatrix}]Compute determinant:- i component: ( left(frac{sqrt{3}}{6} cdot frac{sqrt{6}}{3}right) - left(frac{sqrt{6}}{3} cdot frac{2sqrt{3}}{3}right) )- j component: ( -left( left(-frac{1}{2} cdot frac{sqrt{6}}{3}right) - left(frac{sqrt{6}}{3} cdot 0right) right) )- k component: ( left(-frac{1}{2} cdot frac{2sqrt{3}}{3}right) - left(frac{sqrt{3}}{6} cdot 0right) )Compute each component:i component:( frac{sqrt{18}}{18} - frac{2sqrt{18}}{9} = frac{3sqrt{2}}{18} - frac{6sqrt{2}}{9} = frac{sqrt{2}}{6} - frac{2sqrt{2}}{3} = frac{sqrt{2}}{6} - frac{4sqrt{2}}{6} = -frac{3sqrt{2}}{6} = -frac{sqrt{2}}{2} )j component:( -left( -frac{sqrt{6}}{6} - 0 right) = -left( -frac{sqrt{6}}{6} right) = frac{sqrt{6}}{6} )k component:( -frac{2sqrt{3}}{6} - 0 = -frac{sqrt{3}}{3} )So, the normal vector ( mathbf{n}_2 = left( -frac{sqrt{2}}{2}, frac{sqrt{6}}{6}, -frac{sqrt{3}}{3} right) )Now, we have two normal vectors:( mathbf{n}_1 = left( frac{sqrt{2}}{2}, frac{sqrt{6}}{6}, frac{sqrt{3}}{6} right) )( mathbf{n}_2 = left( -frac{sqrt{2}}{2}, frac{sqrt{6}}{6}, -frac{sqrt{3}}{3} right) )To find the dihedral angle between the two planes, we can find the angle between these two normal vectors. The dihedral angle is equal to the angle between the normals or its supplement, depending on the orientation.The formula for the angle ( theta ) between two vectors is:[cos theta = frac{mathbf{n}_1 cdot mathbf{n}_2}{|mathbf{n}_1| |mathbf{n}_2|}]First, compute the dot product ( mathbf{n}_1 cdot mathbf{n}_2 ):[left( frac{sqrt{2}}{2} times -frac{sqrt{2}}{2} right) + left( frac{sqrt{6}}{6} times frac{sqrt{6}}{6} right) + left( frac{sqrt{3}}{6} times -frac{sqrt{3}}{3} right)]Compute each term:1. ( frac{sqrt{2}}{2} times -frac{sqrt{2}}{2} = -frac{2}{4} = -frac{1}{2} )2. ( frac{sqrt{6}}{6} times frac{sqrt{6}}{6} = frac{6}{36} = frac{1}{6} )3. ( frac{sqrt{3}}{6} times -frac{sqrt{3}}{3} = -frac{3}{18} = -frac{1}{6} )Add them up:( -frac{1}{2} + frac{1}{6} - frac{1}{6} = -frac{1}{2} )So, ( mathbf{n}_1 cdot mathbf{n}_2 = -frac{1}{2} )Now, compute the magnitudes of ( mathbf{n}_1 ) and ( mathbf{n}_2 ):First, ( |mathbf{n}_1| ):[sqrt{left( frac{sqrt{2}}{2} right)^2 + left( frac{sqrt{6}}{6} right)^2 + left( frac{sqrt{3}}{6} right)^2}]Compute each term:1. ( left( frac{sqrt{2}}{2} right)^2 = frac{2}{4} = frac{1}{2} )2. ( left( frac{sqrt{6}}{6} right)^2 = frac{6}{36} = frac{1}{6} )3. ( left( frac{sqrt{3}}{6} right)^2 = frac{3}{36} = frac{1}{12} )Add them up:( frac{1}{2} + frac{1}{6} + frac{1}{12} = frac{6}{12} + frac{2}{12} + frac{1}{12} = frac{9}{12} = frac{3}{4} )So, ( |mathbf{n}_1| = sqrt{frac{3}{4}} = frac{sqrt{3}}{2} )Similarly, compute ( |mathbf{n}_2| ):[sqrt{left( -frac{sqrt{2}}{2} right)^2 + left( frac{sqrt{6}}{6} right)^2 + left( -frac{sqrt{3}}{3} right)^2}]Compute each term:1. ( left( -frac{sqrt{2}}{2} right)^2 = frac{2}{4} = frac{1}{2} )2. ( left( frac{sqrt{6}}{6} right)^2 = frac{6}{36} = frac{1}{6} )3. ( left( -frac{sqrt{3}}{3} right)^2 = frac{3}{9} = frac{1}{3} )Add them up:( frac{1}{2} + frac{1}{6} + frac{1}{3} = frac{3}{6} + frac{1}{6} + frac{2}{6} = frac{6}{6} = 1 )So, ( |mathbf{n}_2| = sqrt{1} = 1 )Now, plug into the cosine formula:[cos theta = frac{ -frac{1}{2} }{ frac{sqrt{3}}{2} times 1 } = frac{ -frac{1}{2} }{ frac{sqrt{3}}{2} } = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3}]So, ( cos theta = -frac{sqrt{3}}{3} ). Therefore, ( theta = arccos left( -frac{sqrt{3}}{3} right) ).But dihedral angles are typically measured between 0 and π, so this is acceptable. However, looking at the answer choices, none of them directly match ( arccos left( -frac{sqrt{3}}{3} right) ). Let me see.Wait, the options are:A. ( arcsin frac{sqrt{6}}{3} )B. ( frac{pi}{2} + arccos frac{sqrt{3}}{3} )C. ( frac{pi}{2} - arctan sqrt{2} )D. ( pi - operatorname{arccot} frac{sqrt{2}}{2} )Hmm, my result is ( arccos left( -frac{sqrt{3}}{3} right) ). Let me see if this can be expressed differently.Note that ( arccos(-x) = pi - arccos(x) ). So,( arccos left( -frac{sqrt{3}}{3} right) = pi - arccos left( frac{sqrt{3}}{3} right) )But looking at the options, option B is ( frac{pi}{2} + arccos frac{sqrt{3}}{3} ), and option D is ( pi - operatorname{arccot} frac{sqrt{2}}{2} ).Wait, perhaps I can relate ( arccos left( frac{sqrt{3}}{3} right) ) to ( operatorname{arccot} frac{sqrt{2}}{2} ).Let me compute ( operatorname{arccot} frac{sqrt{2}}{2} ). Let ( phi = operatorname{arccot} frac{sqrt{2}}{2} ), which means ( cot phi = frac{sqrt{2}}{2} ). So, ( tan phi = frac{2}{sqrt{2}} = sqrt{2} ). Therefore, ( phi = arctan sqrt{2} ).So, ( operatorname{arccot} frac{sqrt{2}}{2} = arctan sqrt{2} ). Therefore, option D is ( pi - arctan sqrt{2} ).Now, let's see if ( pi - arctan sqrt{2} ) is equal to ( arccos left( -frac{sqrt{3}}{3} right) ).Compute ( arccos left( -frac{sqrt{3}}{3} right) ). Let me denote ( theta = arccos left( -frac{sqrt{3}}{3} right) ). Then, ( cos theta = -frac{sqrt{3}}{3} ), so ( theta ) is in the second quadrant.Alternatively, ( theta = pi - arccos left( frac{sqrt{3}}{3} right) ). Let me compute ( arccos left( frac{sqrt{3}}{3} right) ).Let ( alpha = arccos left( frac{sqrt{3}}{3} right) ). Then, ( cos alpha = frac{sqrt{3}}{3} ). Let's find ( sin alpha ):( sin alpha = sqrt{1 - cos^2 alpha} = sqrt{1 - frac{3}{9}} = sqrt{frac{6}{9}} = frac{sqrt{6}}{3} ).So, ( tan alpha = frac{sin alpha}{cos alpha} = frac{sqrt{6}/3}{sqrt{3}/3} = sqrt{2} ). Therefore, ( alpha = arctan sqrt{2} ).Therefore, ( arccos left( frac{sqrt{3}}{3} right) = arctan sqrt{2} ).Thus, ( theta = pi - arctan sqrt{2} ), which is exactly option D.Wait, but earlier I had ( theta = arccos left( -frac{sqrt{3}}{3} right) = pi - arccos left( frac{sqrt{3}}{3} right) = pi - arctan sqrt{2} ). So, yes, that's option D.But just to make sure, let me check if the dihedral angle is indeed ( pi - arctan sqrt{2} ) or if it's the acute angle instead.Wait, the dihedral angle is the angle between two planes, which can be acute or obtuse. In our case, since the normals form an angle of ( pi - arctan sqrt{2} ), which is obtuse, but sometimes dihedral angles are considered as the smaller angle between the two planes. However, in the context of the problem, since the dihedral angle is specified as ( C-FG-E ), which likely refers to the angle between the two planes as they meet along ( FG ), and given the options, it's probably the obtuse angle.Alternatively, perhaps I made a miscalculation in the direction of the normals. Let me verify.Wait, the normal vectors ( mathbf{n}_1 ) and ( mathbf{n}_2 ) were computed as:( mathbf{n}_1 = left( frac{sqrt{2}}{2}, frac{sqrt{6}}{6}, frac{sqrt{3}}{6} right) )( mathbf{n}_2 = left( -frac{sqrt{2}}{2}, frac{sqrt{6}}{6}, -frac{sqrt{3}}{3} right) )The dot product was ( -frac{1}{2} ), and the magnitudes were ( frac{sqrt{3}}{2} ) and ( 1 ), so ( cos theta = -frac{sqrt{3}}{3} ), leading to ( theta = arccos(-sqrt{3}/3) approx 150^circ ), which is obtuse.But in the options, option D is ( pi - operatorname{arccot} frac{sqrt{2}}{2} ), which is ( pi - arctan sqrt{2} approx pi - 54.7^circ approx 125.3^circ ). Wait, that's not matching with 150 degrees. Wait, perhaps I made a mistake in the calculation.Wait, let me compute ( arctan sqrt{2} ). ( tan theta = sqrt{2} ), so ( theta approx 54.7^circ ). Therefore, ( pi - 54.7^circ approx 125.3^circ ), but earlier, ( arccos(-sqrt{3}/3) approx 150^circ ). Hmm, discrepancy here.Wait, perhaps I messed up the angle. Let me compute ( arccos(-sqrt{3}/3) ).Compute ( arccos(-sqrt{3}/3) ). Let me compute it numerically.( sqrt{3} approx 1.732 ), so ( sqrt{3}/3 approx 0.577 ). Therefore, ( arccos(-0.577) approx 125.26^circ ), which is approximately ( 2.186 ) radians.Wait, but ( pi - arctan sqrt{2} approx 3.1416 - 0.9553 approx 2.1863 ) radians, which is approximately 125.26 degrees. So, yes, it's consistent.Wait, so my initial thought that it was 150 degrees was incorrect because I miscalculated. The actual value is approximately 125.26 degrees, which is ( pi - arctan sqrt{2} ).Therefore, the dihedral angle is ( pi - operatorname{arccot} frac{sqrt{2}}{2} ), which is option D.But just to make sure, let me see if there's another way to compute this angle, maybe using geometric properties instead of coordinates.Another approach is to recognize that in a regular tetrahedron, the midpoints form smaller tetrahedrons and octahedrons. The figure formed by connecting midpoints is an octahedron, and the dihedral angles of an octahedron are known.Wait, a regular octahedron has dihedral angles of ( arccos(-1/3) approx 109.47^circ ). But that's different from what we have here.Alternatively, perhaps the dihedral angle in question is related to the angle between face normals in the octahedron.Wait, but in our case, the dihedral angle is between two specific planes, not necessarily the regular octahedron's dihedral angle.Alternatively, maybe using vector projections or something else.But since the coordinate method gave me a clear answer matching option D, and the numerical value matches as well, I think that's the correct choice.Therefore, the answer is D.**Final Answer**boxed{D}

answer：Okay, so I have this problem about a cubic function, and I need to find the sum of f evaluated at a bunch of points. Let me try to understand what's going on here.First, the function is given as f(x) = (1/3)x³ - (1/2)x² + 3x - 5/12. I remember that for cubic functions, the second derivative can tell us about points of inflection, which are centers of symmetry for the graph. The problem mentions that if f''(x₀) = 0, then (x₀, f(x₀)) is the center of symmetry. So, I think I need to find x₀ where the second derivative is zero.Let me compute the first and second derivatives. The first derivative, f'(x), should be the derivative of f(x). Taking the derivative term by term:- The derivative of (1/3)x³ is x².- The derivative of -(1/2)x² is -x.- The derivative of 3x is 3.- The derivative of a constant, -5/12, is 0.So, f'(x) = x² - x + 3.Now, the second derivative, f''(x), is the derivative of f'(x). Let's compute that:- The derivative of x² is 2x.- The derivative of -x is -1.- The derivative of 3 is 0.So, f''(x) = 2x - 1.To find where f''(x) = 0, set 2x - 1 = 0. Solving for x:2x - 1 = 0 2x = 1 x = 1/2.Okay, so the point of inflection is at x = 1/2. That means the center of symmetry is at (1/2, f(1/2)). I need to find f(1/2) to get the full coordinates.Let me compute f(1/2):f(1/2) = (1/3)(1/2)³ - (1/2)(1/2)² + 3*(1/2) - 5/12 First, compute each term:- (1/3)(1/2)³ = (1/3)(1/8) = 1/24- -(1/2)(1/2)² = -(1/2)(1/4) = -1/8- 3*(1/2) = 3/2- -5/12 remains as is.Now, add them all together:1/24 - 1/8 + 3/2 - 5/12.To add these fractions, I need a common denominator. The denominators are 24, 8, 2, and 12. The least common denominator is 24.Convert each term:- 1/24 stays as 1/24.- -1/8 = -3/24.- 3/2 = 36/24.- -5/12 = -10/24.Now, add them:1/24 - 3/24 + 36/24 - 10/24 Combine the numerators:1 - 3 + 36 - 10 = (1 - 3) + (36 - 10) = (-2) + 26 = 24.So, 24/24 = 1.Therefore, f(1/2) = 1. So, the center of symmetry is at (1/2, 1).Now, the problem is asking for the sum f(1/2017) + f(2/2017) + f(3/2017) + ... + f(2016/2017). That's a lot of terms! 2016 terms, to be exact.I remember that if a function is symmetric about a point (a, b), then for any x, f(a + h) + f(a - h) = 2b. In this case, the center of symmetry is (1/2, 1), so for any x, f(1/2 + h) + f(1/2 - h) = 2*1 = 2.So, if I take pairs of terms in the sum that are symmetric around 1/2, each pair should add up to 2.Let me see how the terms are arranged. The terms are f(k/2017) where k goes from 1 to 2016. So, each term is f(k/2017). Let me pair the first term with the last term, the second term with the second last term, and so on.For example, pair f(1/2017) with f(2016/2017). Let's check if these are symmetric around 1/2.Compute 1/2017 + 2016/2017 = (1 + 2016)/2017 = 2017/2017 = 1. So, 1/2017 and 2016/2017 are symmetric around 1/2 because 1/2 is the midpoint between them.Similarly, 2/2017 and 2015/2017 add up to 1, so they are symmetric around 1/2. This pattern continues for all pairs.Since there are 2016 terms, the number of pairs is 2016 / 2 = 1008 pairs. Each pair sums to 2, so the total sum is 1008 * 2 = 2016.Wait, but let me double-check. Is each pair indeed symmetric around 1/2?Take k and 2017 - k. Then, k/2017 + (2017 - k)/2017 = (k + 2017 - k)/2017 = 2017/2017 = 1. So, yes, each pair adds up to 1, which is symmetric around 1/2.Therefore, each pair f(k/2017) + f((2017 - k)/2017) = 2. Since there are 1008 such pairs, the total sum is 1008 * 2 = 2016.So, the answer should be 2016, which is option D.But just to be thorough, let me consider if there's any term that isn't paired or if there's a middle term when the number of terms is odd. However, 2016 is even, so all terms can be paired without any leftover. Therefore, the reasoning holds.I think that's solid. The key was recognizing the symmetry of the cubic function about its inflection point and then using that to pair terms in the sum.**Final Answer**boxed{D}

answer：First, I'll denote the average of ( a ) and ( b ) as ( X ).The average of ( a ) and ( b ) is given by:[frac{a + b}{2} = X]The average of ( b ) and ( c ) is:[frac{b + c}{2} = 90]We are also given that:[c - a = 90]From this, we can express ( c ) in terms of ( a ):[c = a + 90]Substituting ( c = a + 90 ) into the equation for the average of ( b ) and ( c ):[frac{b + (a + 90)}{2} = 90][frac{b + a + 90}{2} = 90][b + a + 90 = 180][b + a = 90]Now, calculating the average of ( a ) and ( b ):[frac{a + b}{2} = frac{90}{2} = 45]Therefore, the average of ( a ) and ( b ) is 45.

answer：First, I need to identify the costs of each item Amy purchased at the fair. The items and their prices are as follows:- Stuffed toy: 2.00- Hot dog: 3.50- Candy apple: 1.50- Soda: 1.75- Ferris wheel ticket: 2.50Next, I need to determine the most expensive and the second most expensive items to apply the discounts correctly. The most expensive item is the hot dog at 3.50, and the second most expensive is the Ferris wheel ticket at 2.50.Before applying discounts, I should calculate the sales tax for each item. The sales tax rate is 10%, so I will calculate 10% of each item's price and add it to the original price.1. **Stuffed toy:** - Original price: 2.00 - Sales tax: 2.00 × 10% = 0.20 - Total with tax: 2.00 + 0.20 = 2.202. **Hot dog:** - Original price: 3.50 - Sales tax: 3.50 × 10% = 0.35 - Total with tax: 3.50 + 0.35 = 3.853. **Candy apple:** - Original price: 1.50 - Sales tax: 1.50 × 10% = 0.15 - Total with tax: 1.50 + 0.15 = 1.654. **Soda:** - Original price: 1.75 - Sales tax: 1.75 × 10% = 0.175 - Total with tax: 1.75 + 0.175 = 1.925 (rounded to 1.93)5. **Ferris wheel ticket:** - Original price: 2.50 - Sales tax: 2.50 × 10% = 0.25 - Total with tax: 2.50 + 0.25 = 2.75Now, I will apply the discounts to the most expensive and second most expensive items.1. **Hot dog with 50% discount:** - Total with tax: 3.85 - Discount: 3.85 × 50% = 1.925 - Price after discount: 3.85 - 1.925 = 1.925 (rounded to 1.93)2. **Ferris wheel ticket with 25% discount:** - Total with tax: 2.75 - Discount: 2.75 × 25% = 0.6875 - Price after discount: 2.75 - 0.6875 = 2.0625 (rounded to 2.06)Finally, I will sum up all the costs after discounts to find the total amount Amy spent.- Stuffed toy: 2.20- Discounted hot dog: 1.93- Candy apple: 1.65- Soda: 1.93- Discounted Ferris wheel ticket: 2.06Total cost after discounts: 2.20 + 1.93 + 1.65 + 1.93 + 2.06 = 9.77Amy started with 15.00, so the amount she has left is:15.00 - 9.77 = 5.23