answer：Okay, so I have this problem about triangle ABC where I'm given the cosines of two angles, A and B, and I need to find the cosine of the third angle, C. Let me try to figure this out step by step.First, I remember that in any triangle, the sum of the angles is 180 degrees. So, angle C must be equal to 180 degrees minus the sum of angles A and B. That is, C = 180° - (A + B). Now, I need to find cos C. Since C is 180° minus (A + B), I can use the cosine of a supplementary angle identity. I recall that cos(180° - θ) = -cos θ. So, cos C = cos(180° - (A + B)) = -cos(A + B). Therefore, if I can find cos(A + B), I can find cos C by just taking the negative of that value.To find cos(A + B), I can use the cosine addition formula, which is:cos(A + B) = cos A cos B - sin A sin B.I already know cos A and cos B, which are given as 4/5 and 7/25 respectively. But I don't know sin A and sin B. I need to find those.I remember that sin²θ + cos²θ = 1, so I can find sin A and sin B using this identity.Starting with angle A:cos A = 4/5, so sin²A = 1 - cos²A = 1 - (4/5)² = 1 - 16/25 = 9/25. Therefore, sin A = √(9/25) = 3/5.Similarly, for angle B:cos B = 7/25, so sin²B = 1 - cos²B = 1 - (7/25)² = 1 - 49/625 = 576/625. Therefore, sin B = √(576/625) = 24/25.Now that I have sin A and sin B, I can plug all the values into the cosine addition formula:cos(A + B) = cos A cos B - sin A sin B= (4/5)(7/25) - (3/5)(24/25)= (28/125) - (72/125)= (28 - 72)/125= (-44)/125.So, cos(A + B) = -44/125.But earlier, I established that cos C = -cos(A + B). Therefore,cos C = -(-44/125) = 44/125.So, the value of d, which is cos C, is 44/125.Let me just double-check my steps to make sure I didn't make any mistakes. I found sin A and sin B correctly using the Pythagorean identity. Then, I applied the cosine addition formula accurately, multiplied the fractions correctly, and subtracted them. Finally, I applied the supplementary angle identity correctly to find cos C. Everything seems to check out.Yeah, I think that's the right answer.

answer：Okay, so I have this problem where I need to find the maximum value of the expression ((a - b)^2 + (b - c)^2 + (c - a)^2) given that (a^2 + b^2 + c^2 = 9). Hmm, let's see. I remember that when dealing with expressions involving squares, sometimes expanding them can help simplify or find relationships with other terms.First, I'll try expanding each squared term:[(a - b)^2 = a^2 - 2ab + b^2][(b - c)^2 = b^2 - 2bc + c^2][(c - a)^2 = c^2 - 2ca + a^2]Now, if I add all these together:[(a - b)^2 + (b - c)^2 + (c - a)^2 = (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ca + a^2)]Let me combine like terms:- The (a^2) terms: (a^2 + a^2 = 2a^2)- The (b^2) terms: (b^2 + b^2 = 2b^2)- The (c^2) terms: (c^2 + c^2 = 2c^2)- The cross terms: (-2ab - 2bc - 2ca)So putting it all together:[2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca]I can factor out a 2:[2(a^2 + b^2 + c^2 - ab - bc - ca)]Hmm, that looks a bit cleaner. I know that (a^2 + b^2 + c^2 = 9), so maybe I can substitute that in:[2(9 - ab - bc - ca)]So the expression simplifies to (18 - 2(ab + bc + ca)). Therefore, to maximize the original expression, I need to minimize the value of (ab + bc + ca).Wait, how do I find the minimum value of (ab + bc + ca) given that (a^2 + b^2 + c^2 = 9)? I think there might be a relationship involving the square of the sum of the variables.Recall that:[(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)]We know (a^2 + b^2 + c^2 = 9), so:[(a + b + c)^2 = 9 + 2(ab + bc + ca)]Let me solve for (ab + bc + ca):[ab + bc + ca = frac{(a + b + c)^2 - 9}{2}]So substituting back into our expression:[18 - 2left(frac{(a + b + c)^2 - 9}{2}right) = 18 - left((a + b + c)^2 - 9right)][= 18 - (a + b + c)^2 + 9][= 27 - (a + b + c)^2]So now, the expression we're trying to maximize is (27 - (a + b + c)^2). To maximize this, we need to minimize ((a + b + c)^2). The smallest possible value of a square is 0, so if we can make (a + b + c = 0), then the expression becomes (27 - 0 = 27).But wait, can (a + b + c) actually be zero given that (a^2 + b^2 + c^2 = 9)? Let me think. Yes, it's possible. For example, if (a = 3), (b = 0), and (c = -3), then (a + b + c = 0) and (a^2 + b^2 + c^2 = 9 + 0 + 9 = 18), which is more than 9. Hmm, that doesn't work. Maybe another set.Wait, perhaps (a = sqrt{3}), (b = sqrt{3}), (c = -sqrt{3}). Then (a + b + c = sqrt{3} + sqrt{3} - sqrt{3} = sqrt{3}), which isn't zero. Hmm, maybe another approach.Alternatively, maybe using vectors. If I consider (a), (b), and (c) as components of a vector, then (a + b + c) is the sum of the components. To minimize the square of this sum, perhaps we can set the vector such that it's orthogonal to the vector (1,1,1). That might give the minimum.Wait, I'm getting a bit confused. Let me step back. The expression (27 - (a + b + c)^2) will be maximized when ((a + b + c)^2) is minimized. The minimum value of ((a + b + c)^2) is 0, so if we can find values of (a), (b), and (c) such that (a + b + c = 0) and (a^2 + b^2 + c^2 = 9), then the expression will be 27.Is such a triplet possible? Let's see. For example, let me take (a = 3), (b = 0), (c = -3). Then (a + b + c = 0), but (a^2 + b^2 + c^2 = 9 + 0 + 9 = 18), which is more than 9. So that doesn't work.Wait, maybe smaller numbers. Let me take (a = sqrt{3}), (b = sqrt{3}), (c = -sqrt{3}). Then (a + b + c = sqrt{3} + sqrt{3} - sqrt{3} = sqrt{3}), which isn't zero. Hmm, not helpful.Alternatively, maybe (a = t), (b = t), (c = -2t). Then (a + b + c = t + t - 2t = 0). Now, let's compute (a^2 + b^2 + c^2 = t^2 + t^2 + (4t^2) = 6t^2). We need this equal to 9, so (6t^2 = 9) which gives (t^2 = 1.5), so (t = sqrt{1.5}) or (t = -sqrt{1.5}).So, for example, (a = sqrt{1.5}), (b = sqrt{1.5}), (c = -2sqrt{1.5}). Then (a + b + c = 0) and (a^2 + b^2 + c^2 = 1.5 + 1.5 + 6 = 9). Perfect! So such values exist.Therefore, the expression (27 - (a + b + c)^2) can indeed reach 27 when (a + b + c = 0). Therefore, the maximum value of the original expression is 27.Wait, just to make sure, let me compute the original expression with these values:[(a - b)^2 + (b - c)^2 + (c - a)^2]Given (a = sqrt{1.5}), (b = sqrt{1.5}), (c = -2sqrt{1.5}):First, (a - b = 0), so ((a - b)^2 = 0).Next, (b - c = sqrt{1.5} - (-2sqrt{1.5}) = 3sqrt{1.5}), so ((b - c)^2 = 9 times 1.5 = 13.5).Similarly, (c - a = -2sqrt{1.5} - sqrt{1.5} = -3sqrt{1.5}), so ((c - a)^2 = 13.5).Adding them up: (0 + 13.5 + 13.5 = 27). Yep, that checks out.So, I think I'm confident that the maximum value is 27.**Final Answer**boxed{27}

answer：First, I need to calculate the dot product of vectors **a** and **b**. The dot product is found by multiplying the corresponding components of the vectors and then adding those products together.Given:- Vector **a** = (1, 1)- Vector **b** = (-1, 2)I will multiply the first components of each vector:1 * (-1) = -1Next, I will multiply the second components of each vector:1 * 2 = 2Finally, I will add these two results together to find the dot product:-1 + 2 = 1Therefore, the dot product of vectors **a** and **b** is 1.

answer：Okay, so I'm trying to figure out this game problem. It's on a grid with 2012 rows and more than 2 columns. The piece starts in the leftmost column, and players take turns moving it right, up, or down without revisiting any cell. The game ends when someone moves it to the rightmost column. But here's the twist: whether moving to the rightmost column wins or loses is only revealed when the piece is in the second-to-last column. Hmm, interesting.First, I need to understand the rules clearly. The board is a grid with 2012 rows and k columns, where k is greater than 2. The piece starts somewhere in the first column, and players alternate moving it right, up, or down. No revisiting cells, so each move has to be to an unvisited cell. The game ends when someone moves it to the rightmost column, but whether that move wins or loses is only known when the piece is in the second-to-last column. So, the outcome depends on some condition that's revealed just before the end.I think the key here is to figure out if one of the players can force a win regardless of what happens when the piece is in the second-to-last column. Since the outcome is revealed only then, it might depend on who is forced to make that move.Let me consider the structure of the game. The piece starts in the first column, and players move it towards the right, up, or down. Since the grid is 2012 rows tall, that's a lot of rows, but the number of columns is more than 2, so at least 3 columns.If the piece is in the second-to-last column, then the next move will be to the last column, which ends the game. But the problem is that whether that move wins or loses is only revealed at that point. So, the player who moves to the second-to-last column can choose to either win or lose based on some condition.Wait, no, actually, the problem says that whether moving to the rightmost column wins or loses is only revealed when the piece reaches the second-to-last column. So, it's not that the player can choose, but rather that the outcome is determined by some condition that becomes known only when the piece is in the second-to-last column.Hmm, maybe it's like a hidden rule that is only known when you get there. So, depending on the position in the second-to-last column, moving to the last column might win or lose. But since it's only revealed then, the players don't know in advance.But the question is, can one of the players guarantee a win? So, regardless of what the outcome is when reaching the second-to-last column, can one player force a win?I think the answer might be yes, because of the structure of the game. Since the grid is so large, with 2012 rows, the first player can control the parity of the moves, ensuring that they can force the game into a state where they can win.Let me think about parity. The game is essentially a path from the first column to the last column, with each move being a step right, up, or down. Since the grid is 2012 rows, which is an even number, that might play a role.If the first player can always move in such a way that they control the number of moves, they can force the second player into a position where they have to make the losing move.Alternatively, maybe the first player can mirror the second player's moves in some way, ensuring that they always have a move and can force the second player to be the one to move into the last column.Wait, but the grid is 2012 rows, which is even, so if the first player can pair up the rows, they can mirror the second player's moves vertically, ensuring that they always have a move.But I'm not sure if that's the right approach. Maybe I need to think about it as a game of parity, where the first player can always force the game to end on an odd or even move, depending on what's needed.Since the outcome is only revealed when the piece is in the second-to-last column, the first player can plan their moves so that they are the one to move into the second-to-last column, and then based on the revealed outcome, they can decide whether to move to the last column or force the second player to do it.But wait, the outcome is revealed when the piece is in the second-to-last column, so the player who is about to move from the second-to-last column knows whether moving to the last column wins or loses.Therefore, if the first player can force the game to reach the second-to-last column on the second player's turn, then the second player will have to make the move to the last column, and if that move is a losing move, the second player loses. If it's a winning move, the second player wins.But the first player doesn't know in advance whether moving to the last column wins or loses, so they can't rely on that. Instead, they need to ensure that regardless of what happens when the piece is in the second-to-last column, they can win.Wait, maybe the first player can control the parity of the number of moves required to reach the second-to-last column. Since the grid has 2012 rows, which is even, the first player can ensure that the number of moves to reach the second-to-last column is odd, meaning that the second player is the one to move into the second-to-last column, and then the first player can respond accordingly.But I'm not sure. Let me try to break it down.The game starts in the first column. Each move can go right, up, or down. The goal is to reach the last column. The outcome is determined when the piece is in the second-to-last column.So, the game is essentially a path from the first column to the last column, with the outcome determined at the penultimate step.Since the grid is 2012 rows, which is even, the first player can use this to their advantage. They can move in such a way that they control the number of moves, ensuring that they can always respond to the second player's moves.Alternatively, maybe the first player can always move to a position that leaves an even number of moves to the last column, forcing the second player to be the one to make the final move.But I'm not entirely sure. Maybe I need to think about smaller grids to see the pattern.Let's consider a smaller grid, say 2 rows and 3 columns. The piece starts in the first column. The first player can move right, up, or down. If they move right, they're in the second column. Then the second player can move right to the last column, but the outcome is only revealed when in the second column. Wait, no, in this case, the second column is the penultimate column, so the outcome is revealed when the piece is in the second column.So, if the first player moves right to the second column, then the second player knows whether moving to the last column wins or loses. If it's a winning move, the second player can win. If it's a losing move, the second player would prefer not to move there.But since the second player has to move, they have to make a move. So, if moving to the last column is a winning move, the second player can win. If it's a losing move, the second player has to make it and lose.But the first player could have moved up or down instead of right. If they move up or down, they stay in the first column, but change rows. Then the second player can move right, up, or down.Wait, in a 2-row grid, moving up or down from the first column would just switch rows, but since there are only 2 rows, moving up from row 1 would go to row 2, and moving down from row 2 would go to row 1.So, in this case, the first player could move up or down, forcing the second player to move right or up/down.But in a 2x3 grid, the first player can control the game by moving vertically, forcing the second player into a position where they have to move right, potentially into a losing position.But this is a very small grid, so maybe the pattern isn't clear. Let's try a slightly bigger grid, say 4 rows and 3 columns.The piece starts in the first column. The first player can move right, up, or down. If they move right, they're in the second column, and the outcome is revealed. If they move up or down, they stay in the first column but change rows.If the first player moves up or down, the second player can then move right or up/down. If the second player moves right, they're in the second column, and the outcome is revealed. If they move up or down, they stay in the first column.So, the first player can choose to either move right immediately or move vertically. If they move vertically, they can potentially control the game by forcing the second player to move right.But in this case, since the grid is 4 rows, which is even, the first player can pair up the rows and mirror the second player's moves, ensuring that they can always respond and control the game.Wait, maybe that's the key. If the number of rows is even, the first player can mirror the second player's vertical moves, ensuring that they can always respond and control the parity of the game.In the original problem, the grid has 2012 rows, which is even. So, the first player can use this to their advantage by mirroring the second player's vertical moves, ensuring that they can always respond and control the game.By doing so, the first player can force the second player into a position where they have to make the final move into the last column, regardless of whether that move wins or loses.Therefore, the first player can guarantee a win by controlling the game's parity and forcing the second player to make the final move.I think that's the reasoning. The key is the even number of rows, which allows the first player to mirror the second player's moves and control the game's outcome.

answer：Okay, so I have this logarithmic equation to solve:[log _{2} log _{3}left(x^{2}-16right)-log _{1 / 2} log _{1 / 3} frac{1}{x^{2}-16}=2]Hmm, that looks a bit complicated with all the different bases and nested logarithms. Let me try to break it down step by step.First, I need to figure out the domain of this function because logarithms have specific requirements. The arguments inside the logarithms must be positive. So, let's look at the innermost parts:1. For (log_{3}(x^2 - 16)) to be defined, (x^2 - 16) must be greater than 0. That means (x^2 > 16), so (x > 4) or (x < -4).2. Also, the argument of the outer logarithm, (log_{3}(x^2 - 16)), must be positive. So, (log_{3}(x^2 - 16) > 0). Since (log_{3}(y) > 0) when (y > 1), this implies (x^2 - 16 > 1), so (x^2 > 17), which means (x > sqrt{17}) or (x < -sqrt{17}). That's approximately (x > 4.123) or (x < -4.123).3. Now, looking at the second logarithmic term: (log_{1/3}(1/(x^2 - 16))). Since the base is (1/3), which is less than 1, the logarithm is defined when its argument is positive. So, (1/(x^2 - 16) > 0). That means (x^2 - 16 > 0), which is the same as the first condition, so nothing new here.4. The outer logarithm with base (1/2) requires that its argument, which is (log_{1/3}(1/(x^2 - 16))), must be positive. Let's see: (log_{1/3}(1/(x^2 - 16)) > 0). Since the base is (1/3), which is less than 1, the logarithm is positive when its argument is between 0 and 1. So, (0 < 1/(x^2 - 16) < 1). - (1/(x^2 - 16) < 1) implies (x^2 - 16 > 1), so (x^2 > 17), which is the same as before. - (1/(x^2 - 16) > 0) is already satisfied as long as (x^2 - 16 > 0).So, combining all these, the domain is (x > sqrt{17}) or (x < -sqrt{17}).Okay, so now I know the domain. Next, I need to solve the equation:[log _{2} log _{3}left(x^{2}-16right)-log _{1 / 2} log _{1 / 3} frac{1}{x^{2}-16}=2]Let me try to simplify each term. I remember that (log_{1/a}(b) = -log_{a}(b)). So, maybe I can rewrite the second logarithm term using this property.First, let's rewrite (log_{1/2}) and (log_{1/3}):- (log_{1/2}(y) = -log_{2}(y))- (log_{1/3}(y) = -log_{3}(y))So, let's apply this to the second term:[log_{1/2} log_{1/3} frac{1}{x^{2}-16} = log_{1/2} left( -log_{3} frac{1}{x^{2}-16} right)]Wait, that might complicate things. Let me think differently. Maybe I can express everything in terms of base 2 and base 3 logarithms.Alternatively, let me consider substituting (y = x^2 - 16). Then, the equation becomes:[log_{2} log_{3}(y) - log_{1/2} log_{1/3} left( frac{1}{y} right) = 2]That might make it easier. So, let (y = x^2 - 16). Then, the equation is:[log_{2} log_{3}(y) - log_{1/2} log_{1/3} left( frac{1}{y} right) = 2]Now, let's simplify each logarithmic term.First term: (log_{2} log_{3}(y)) remains as it is.Second term: (log_{1/2} log_{1/3} left( frac{1}{y} right)). Let's break this down.First, compute (log_{1/3} left( frac{1}{y} right)). Using the property (log_{a}(1/b) = -log_{a}(b)), this becomes:[log_{1/3} left( frac{1}{y} right) = -log_{1/3}(y)]But (log_{1/3}(y)) can be rewritten as (-log_{3}(y)), so:[-log_{1/3}(y) = -(-log_{3}(y)) = log_{3}(y)]So, (log_{1/3} left( frac{1}{y} right) = log_{3}(y)).Now, the second term is (log_{1/2}) of that, which is:[log_{1/2} left( log_{3}(y) right)]Again, using the property (log_{1/a}(b) = -log_{a}(b)), this becomes:[log_{1/2} left( log_{3}(y) right) = -log_{2} left( log_{3}(y) right)]So, putting it all together, the original equation becomes:[log_{2} log_{3}(y) - left( -log_{2} log_{3}(y) right) = 2]Simplify the equation:[log_{2} log_{3}(y) + log_{2} log_{3}(y) = 2]Combine like terms:[2 log_{2} log_{3}(y) = 2]Divide both sides by 2:[log_{2} log_{3}(y) = 1]Now, solve for (log_{3}(y)). Remember that (log_{2}(a) = 1) implies (a = 2^1 = 2). So:[log_{3}(y) = 2]Solve for (y). Since (log_{3}(y) = 2), this means (y = 3^2 = 9).But (y = x^2 - 16), so:[x^2 - 16 = 9]Solve for (x^2):[x^2 = 9 + 16 = 25]Take the square root of both sides:[x = pm 5]Now, check if these solutions are within the domain we found earlier, which was (x > sqrt{17}) or (x < -sqrt{17}). Since (sqrt{17} approx 4.123), and 5 is greater than that, both (x = 5) and (x = -5) are valid solutions.Wait, let me double-check the steps to make sure I didn't make any mistakes.1. I substituted (y = x^2 - 16), which seems fine.2. I converted the logarithms with bases (1/2) and (1/3) into negative logarithms with bases 2 and 3, respectively. That seems correct.3. I simplified the second term step by step, which led me to express it in terms of (log_{2} log_{3}(y)), which then combined with the first term.4. After simplifying, I ended up with an equation that led me to (y = 9), and then (x = pm 5).Everything seems to check out. So, the solutions are (x = 5) and (x = -5).**Final Answer**The solutions are boxed{-5} and boxed{5}.

answer：Okay, so I have this problem where the sum of the first ( n ) terms of a sequence ( {a_n} ) is given by ( S_n = n^2 - 5n + 2 ). I need to find the sum of the first 10 terms of the sequence ( {|a_n|} ). The options are A: 56, B: 58, C: 62, D: 60. Hmm, let me try to figure this out step by step.First, I remember that the sum of the first ( n ) terms ( S_n ) is related to the individual terms ( a_n ) by the formula ( a_n = S_n - S_{n-1} ) for ( n geq 2 ), and ( a_1 = S_1 ). So, maybe I can find each term ( a_n ) and then take their absolute values to sum up.Let me start by calculating ( S_n ) for ( n = 1 ) to ( n = 10 ). That might help me see the pattern or identify when the terms change sign, which is important because we're dealing with absolute values.Calculating ( S_n ):- ( S_1 = 1^2 - 5(1) + 2 = 1 - 5 + 2 = -2 )- ( S_2 = 2^2 - 5(2) + 2 = 4 - 10 + 2 = -4 )- ( S_3 = 3^2 - 5(3) + 2 = 9 - 15 + 2 = -4 )- ( S_4 = 4^2 - 5(4) + 2 = 16 - 20 + 2 = -2 )- ( S_5 = 5^2 - 5(5) + 2 = 25 - 25 + 2 = 2 )- ( S_6 = 6^2 - 5(6) + 2 = 36 - 30 + 2 = 8 )- ( S_7 = 7^2 - 5(7) + 2 = 49 - 35 + 2 = 16 )- ( S_8 = 8^2 - 5(8) + 2 = 64 - 40 + 2 = 26 )- ( S_9 = 9^2 - 5(9) + 2 = 81 - 45 + 2 = 38 )- ( S_{10} = 10^2 - 5(10) + 2 = 100 - 50 + 2 = 52 )Okay, so the sums ( S_n ) start negative, reach a minimum, and then become positive. Now, let me find each ( a_n ) using ( a_n = S_n - S_{n-1} ) for ( n geq 2 ) and ( a_1 = S_1 ).Calculating ( a_n ):- ( a_1 = S_1 = -2 )- ( a_2 = S_2 - S_1 = (-4) - (-2) = -2 )- ( a_3 = S_3 - S_2 = (-4) - (-4) = 0 )- ( a_4 = S_4 - S_3 = (-2) - (-4) = 2 )- ( a_5 = S_5 - S_4 = 2 - (-2) = 4 )- ( a_6 = S_6 - S_5 = 8 - 2 = 6 )- ( a_7 = S_7 - S_6 = 16 - 8 = 8 )- ( a_8 = S_8 - S_7 = 26 - 16 = 10 )- ( a_9 = S_9 - S_8 = 38 - 26 = 12 )- ( a_{10} = S_{10} - S_9 = 52 - 38 = 14 )So, the terms ( a_n ) are:- ( a_1 = -2 )- ( a_2 = -2 )- ( a_3 = 0 )- ( a_4 = 2 )- ( a_5 = 4 )- ( a_6 = 6 )- ( a_7 = 8 )- ( a_8 = 10 )- ( a_9 = 12 )- ( a_{10} = 14 )Now, I need the absolute values of these terms. Let's compute ( |a_n| ):- ( |a_1| = |-2| = 2 )- ( |a_2| = |-2| = 2 )- ( |a_3| = |0| = 0 )- ( |a_4| = |2| = 2 )- ( |a_5| = |4| = 4 )- ( |a_6| = |6| = 6 )- ( |a_7| = |8| = 8 )- ( |a_8| = |10| = 10 )- ( |a_9| = |12| = 12 )- ( |a_{10}| = |14| = 14 )Now, summing these absolute values:( 2 + 2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 )Let me add them step by step:- Start with 2 (from ( |a_1| ))- Add 2: total 4- Add 0: total 4- Add 2: total 6- Add 4: total 10- Add 6: total 16- Add 8: total 24- Add 10: total 34- Add 12: total 46- Add 14: total 60So, the sum of the first 10 terms of ( {|a_n|} ) is 60. Looking at the options, D is 60, so that should be the answer.Wait, just to make sure I didn't make any calculation mistakes. Let me double-check the sum:2 (a1) + 2 (a2) + 0 (a3) + 2 (a4) + 4 (a5) + 6 (a6) + 8 (a7) + 10 (a8) + 12 (a9) + 14 (a10)Adding them:2 + 2 = 44 + 0 = 44 + 2 = 66 + 4 = 1010 + 6 = 1616 + 8 = 2424 + 10 = 3434 + 12 = 4646 + 14 = 60Yes, that's correct. So, 60 is the right answer, which is option D.Alternatively, another approach is to realize that the sum of absolute values can be calculated by considering when the terms change from negative to positive. From the ( a_n ) terms, we saw that ( a_1 ) and ( a_2 ) are negative, ( a_3 ) is zero, and from ( a_4 ) onwards, they are positive. So, the sum of absolute values would be the sum of the first 10 terms minus twice the sum of the negative terms (since we're taking absolute values, which flips their sign).Let me try this method as a cross-check.First, the sum of the first 10 terms ( S_{10} = 52 ).Now, the negative terms are ( a_1 = -2 ) and ( a_2 = -2 ). Their sum is ( -4 ). So, the absolute sum would be ( S_{10} - 2 times text{sum of negative terms} ).Wait, actually, since we're taking absolute values, the total sum would be ( |a_1| + |a_2| + |a_3| + dots + |a_{10}| ). Since ( a_3 ) is zero, it doesn't affect the sum. So, the sum is ( |a_1| + |a_2| + a_4 + a_5 + dots + a_{10} ).But ( |a_1| + |a_2| = 2 + 2 = 4 ). Then, ( a_4 ) to ( a_{10} ) are all positive, so their sum is ( S_{10} - S_3 ). Because ( S_{10} ) is the sum up to 10, and ( S_3 ) is the sum up to 3, so subtracting gives the sum from 4 to 10.Calculating ( S_{10} - S_3 = 52 - (-4) = 52 + 4 = 56 ).Then, adding the absolute values of the first two terms: 4 + 56 = 60. So, same result.Therefore, both methods give me 60, which is option D. I think that's solid.**Final Answer**boxed{D}