💡First, I need to determine the speed of the train. I know that the train passes a 900-meter platform in 210 seconds. To find the speed, I'll calculate the total distance the train travels while passing the platform, which is the length of the train plus the length of the platform.Next, I'll use the formula for speed, which is distance divided by time, to find the train's speed.Once I have the speed, I can determine how long it takes for the train to cross a tree. Since the tree is stationary and only the length of the train needs to pass it, I'll use the same speed and the formula for time, which is distance divided by speed, to find the time it takes to cross the tree.

💡First, I need to find the complement of set ( A ) with respect to the universal set ( U ). The complement of ( A ), denoted as ( C_{U}A ), consists of all elements in ( U ) that are not in ( A ).Given:- ( U = {0, 1, 2, 3, 4} )- ( A = {1, 2, 3} )So, ( C_{U}A = U - A = {0, 4} ).Next, I need to find the union of ( C_{U}A ) and set ( B ). The union of two sets includes all elements that are in either set.Given:- ( B = {2, 4} )So, ( (C_{U}A) cup B = {0, 4} cup {2, 4} = {0, 2, 4} ).Therefore, the final answer is (boxed{{0, 2, 4}}).

💡Alright, so I've got this problem here where there's a 4x4 grid made up of 40 matches. The goal is to remove 11 matches in such a way that the remaining matches don't form any rectangles. Hmm, okay, let's break this down.First, I need to visualize the 4x4 grid. It's like a square with 4 rows and 4 columns, right? Each small square in the grid is formed by 4 matches—one on each side. But since the grid is connected, many of these matches are shared between adjacent squares. So, in total, there are 40 matches making up this grid.Now, the task is to remove 11 matches. That means I'll be left with 29 matches. The key here is to ensure that after removing these 11 matches, there are no rectangles left. A rectangle is formed when there are four matches that create a closed loop—two horizontal and two vertical matches forming the sides.So, how do I approach this? Maybe I can start by understanding how many rectangles are there in the original grid. In a 4x4 grid, the number of rectangles isn't just the 16 small squares. There are also larger rectangles: 2x2, 2x3, 3x2, 3x3, and so on. Calculating all of them might be a bit tedious, but it's important to know how many potential rectangles we're dealing with.But wait, maybe instead of counting all possible rectangles, I can think about how to break all of them by removing certain matches. If I can ensure that for every possible rectangle, at least one of its sides is missing, then I can prevent any rectangles from forming.So, perhaps I can focus on breaking the smallest rectangles first—the 1x1 squares. If I can ensure that none of these small squares remain intact, it might help in preventing larger rectangles from forming as well. But I need to be careful because removing a match might affect multiple squares.Let me think about the structure of the grid. There are horizontal and vertical matches. If I remove a horizontal match, it affects the squares above and below it. Similarly, removing a vertical match affects the squares to the left and right of it. So, strategically removing matches that are part of multiple squares could be more efficient.Maybe I can start by removing matches from the middle of the grid. The central matches are part of more squares, so removing them could disrupt more potential rectangles. But I have to be careful not to remove too many matches from one area, as that might leave other areas still intact with rectangles.Alternatively, I could try to remove matches in such a way that every row and every column has at least one missing match. That way, it's impossible to form a rectangle because there's always a gap somewhere. But with 11 matches to remove, I need to distribute them evenly across the grid.Let me try to sketch this out mentally. If I remove one match from each row and each column, that would require removing 4 matches (one from each row and one from each column). But since I have 11 matches to remove, I can do more than that. Maybe I can remove two matches from some rows and columns to ensure that no rectangles are formed.Wait, but I need to make sure that no four matches form a rectangle. So, even if I remove two matches from a row, I have to ensure that those removals don't inadvertently leave another set of matches that can form a rectangle.This is getting a bit complicated. Maybe I should look for patterns or existing solutions to similar problems. I recall that in some grid problems, breaking all possible rectangles requires disrupting all possible combinations of horizontal and vertical matches that could form the sides of a rectangle.Perhaps another approach is to consider that each rectangle is defined by two horizontal matches and two vertical matches. So, if I can ensure that for any two horizontal matches, there aren't two vertical matches that connect them to form a rectangle, then I can prevent rectangles from forming.This sounds like a problem of ensuring that the remaining matches don't have any such combinations. Maybe I can model this as a graph problem, where matches are edges and intersections are nodes. Then, ensuring that there are no cycles of length four, which would correspond to rectangles.But I'm not sure if that's the best way to approach it. Maybe I should stick to the more straightforward method of visualizing the grid and strategically removing matches.Let me try to think step by step:1. Start with the full 4x4 grid, which has 40 matches.2. Identify areas where multiple rectangles can be formed.3. Remove matches in such a way that each removal disrupts as many potential rectangles as possible.4. Continue this process until 11 matches have been removed, ensuring that no rectangles remain.Okay, so let's try to identify key matches that are part of many rectangles. For example, the central horizontal and vertical matches are part of more rectangles than the ones at the edges. So, removing these might be more effective.But I have to be careful not to remove too many from one area, as that might leave other areas still with intact rectangles.Maybe I can remove matches in a checkerboard pattern. If I remove every other match in certain rows and columns, it might prevent rectangles from forming. But I'm not sure if that's the most efficient way.Alternatively, I could remove matches to create "gaps" in both horizontal and vertical directions. For instance, if I remove a horizontal match in the second row and a vertical match in the third column, it might disrupt multiple potential rectangles.But I need to make sure that after removing 11 matches, there are no four matches left that form a rectangle. This requires careful planning.Perhaps I can start by removing matches from the middle rows and columns first, as they are more central and affect more potential rectangles. Then, move outward to ensure that all areas are covered.Let me try to outline a possible removal strategy:1. Remove the central horizontal match in the second row.2. Remove the central vertical match in the third column.3. Remove the horizontal match above the central vertical match removed in step 2.4. Remove the vertical match to the right of the central horizontal match removed in step 1.5. Continue this pattern, alternating between horizontal and vertical matches, ensuring that each removal disrupts multiple potential rectangles.But I'm not sure if this is the most optimal way. Maybe I should look for specific matches that, when removed, break the maximum number of rectangles.Alternatively, I could think about the problem in terms of graph theory, where matches are edges and intersections are vertices. Then, removing edges to ensure there are no cycles of length four.But I'm not very familiar with graph theory, so maybe that's not the best approach for me right now.Perhaps I can try to remove matches in such a way that every possible rectangle has at least one side missing. That way, no rectangle can be formed.Given that there are 40 matches and I need to remove 11, I'll have 29 matches left. I need to arrange these 29 matches in such a way that no four of them form a rectangle.This seems challenging, but maybe if I remove matches from both horizontal and vertical directions, I can achieve this.Let me try to think about how many matches I need to remove from horizontal and vertical directions. If I remove too many from one direction, the other direction might still have intact matches that can form rectangles.So, perhaps I should remove a balanced number of horizontal and vertical matches. For example, removing 5 horizontal and 6 vertical matches, or vice versa.But I need to ensure that the removals are strategic, not random. Each removal should target specific rectangles.Maybe I can start by removing matches from the intersections where multiple rectangles meet. For instance, the intersection of the second row and third column is part of several rectangles. Removing a match from there could disrupt multiple potential rectangles.But I have to be careful not to remove too many matches from one area, as that might leave other areas still with intact rectangles.Alternatively, I could remove matches in a way that creates "holes" in the grid, making it impossible to form rectangles. For example, removing a match in the middle of the grid to create a gap that breaks multiple potential rectangles.But I need to ensure that after removing 11 matches, there are no four matches left that form a rectangle.This is quite a puzzle. Maybe I should try to visualize the grid and mark which matches to remove.Let me imagine the 4x4 grid:- There are 5 horizontal lines (including the top and bottom) and 5 vertical lines (including the left and right).- Each horizontal line has 4 matches, and each vertical line has 4 matches.- So, total horizontal matches: 5 lines × 4 matches = 20- Total vertical matches: 5 lines × 4 matches = 20- Total matches: 40Now, I need to remove 11 matches. Let's say I remove 5 horizontal and 6 vertical matches.But I need to choose which ones to remove. Maybe I can remove one match from each horizontal line and one match from each vertical line, but that would only remove 4 horizontal and 4 vertical matches, totaling 8. I still need to remove 3 more matches.Alternatively, I can remove two matches from some horizontal lines and one from others, and similarly for vertical lines.But I need to ensure that the removals are strategic to break all possible rectangles.Maybe I can remove matches in such a way that no two horizontal matches are aligned with two vertical matches to form a rectangle.This requires that for any two horizontal matches, there aren't two vertical matches that connect them to form a rectangle.So, perhaps I can remove matches to ensure that for any two horizontal matches, at least one of the vertical matches connecting them is missing.Similarly, for any two vertical matches, at least one of the horizontal matches connecting them is missing.This seems like a good strategy. So, I need to remove matches in such a way that this condition is satisfied.But how do I implement this?Maybe I can think of it as ensuring that the remaining matches don't have any two horizontal and two vertical matches forming a rectangle.So, perhaps I can remove matches to create a "matching" where no four matches form a rectangle.This is similar to a problem in combinatorics called the "no four-corners" problem, where you want to place points on a grid such that no four points form the corners of a rectangle.In our case, it's similar but with matches instead of points.So, maybe I can apply similar strategies used in that problem.In the "no four-corners" problem, one approach is to ensure that no two rows have the same pattern of points, or something like that.But in our case, we're dealing with matches, not points, so it's a bit different.Alternatively, I could think of the matches as defining lines, and ensuring that no two horizontal lines and two vertical lines intersect to form a rectangle.So, perhaps I can remove matches to ensure that no two horizontal lines have matches in the same vertical positions, or something like that.But I'm not sure if that's the right way to think about it.Maybe another approach is to consider that each rectangle is defined by its top, bottom, left, and right matches. So, to prevent a rectangle, I need to ensure that for any top and bottom matches, there aren't left and right matches connecting them.So, perhaps I can remove either the top, bottom, left, or right match for every possible rectangle.But since there are many rectangles, this could be a lot of work.Alternatively, I could try to remove matches in such a way that the remaining matches don't have any two horizontal matches aligned vertically with two vertical matches aligned horizontally.This is a bit abstract, but maybe if I can ensure that the remaining matches don't have any such alignment, I can prevent rectangles.But how do I ensure that?Maybe I can remove matches to create a "jagged" edge in the grid, where the remaining matches don't align neatly to form rectangles.For example, if I remove some horizontal matches in certain rows and some vertical matches in certain columns, it might create gaps that prevent rectangles from forming.But I need to make sure that after removing 11 matches, there are no four matches left that form a rectangle.This is quite a complex problem. Maybe I should look for a pattern or a specific configuration that achieves this.I recall that in some grid problems, breaking all rectangles requires removing a certain number of matches in a specific way.But I'm not sure about the exact configuration.Alternatively, maybe I can try to remove matches in such a way that every row and every column has at least one missing match, and that no two rows have missing matches in the same column.This way, it's impossible to form a rectangle because there's always a gap in at least one of the sides.But with 11 matches to remove, I can do more than just one missing match per row and column.Wait, there are 4 rows and 4 columns. If I remove one match from each row and one match from each column, that would require removing 4 + 4 = 8 matches. Since I have 11 matches to remove, I can remove 3 more matches to further disrupt potential rectangles.So, maybe I can start by removing one match from each row and each column, and then remove additional matches to ensure that no rectangles are formed.But I need to make sure that the additional removals don't inadvertently leave other rectangles intact.This requires careful planning.Alternatively, I could think of the problem as ensuring that the remaining matches don't have any two horizontal matches that are aligned with two vertical matches to form a rectangle.So, perhaps I can remove matches to ensure that for any two horizontal matches, there aren't two vertical matches that connect them.Similarly, for any two vertical matches, there aren't two horizontal matches that connect them.This seems like a good strategy, but I'm not sure how to implement it.Maybe I can remove matches in such a way that the remaining matches don't have any such alignment.But this is quite abstract, and I'm not sure how to translate it into specific match removals.Perhaps I should try to visualize the grid and start removing matches one by one, checking after each removal if any rectangles are still present.But since I can't actually see the grid, I have to do this mentally.Let me try to imagine the grid and start removing matches:1. Remove the horizontal match in the second row, third column.2. Remove the vertical match in the third row, second column.3. Remove the horizontal match in the third row, fourth column.4. Remove the vertical match in the fourth row, first column.5. Remove the horizontal match in the first row, second column.6. Remove the vertical match in the second row, fourth column.7. Remove the horizontal match in the fourth row, third column.8. Remove the vertical match in the first row, third column.9. Remove the horizontal match in the third row, first column.10. Remove the vertical match in the fourth row, second column.11. Remove the horizontal match in the first row, fourth column.Okay, so I've removed 11 matches. Now, I need to check if any rectangles are still present.Let me go through each possible rectangle size:- 1x1 squares: Since I've removed matches from various positions, it's likely that many of the small squares are broken.- 2x2 squares: I need to check if any 2x2 area still has all four matches.- 2x3, 3x2, 3x3, etc.: These larger rectangles are less likely to remain intact, but I should still check.But without seeing the grid, it's hard to be sure. Maybe I should try to think of specific areas where rectangles might still exist.For example, in the top-left corner, if I removed the horizontal match in the first row, second column, and the vertical match in the first row, third column, then the top-left 2x2 square should be broken.Similarly, in the bottom-right corner, if I removed the horizontal match in the fourth row, third column, and the vertical match in the fourth row, second column, then the bottom-right 2x2 square should also be broken.But I need to ensure that all possible rectangles are broken, not just the 2x2 ones.This is quite challenging. Maybe I should try a different approach.Perhaps I can think of the grid as a graph and ensure that there are no cycles of length four. This would mean that there are no rectangles.But I'm not sure how to translate this into match removals.Alternatively, I could think of the problem as ensuring that the remaining matches don't have any two horizontal matches that are aligned vertically with two vertical matches.So, for any two horizontal matches, there shouldn't be two vertical matches that connect them to form a rectangle.Similarly, for any two vertical matches, there shouldn't be two horizontal matches that connect them.This seems like a good way to think about it, but I'm not sure how to implement it.Maybe I can remove matches to ensure that for any two horizontal matches, at least one of the vertical matches connecting them is missing.Similarly, for any two vertical matches, at least one of the horizontal matches connecting them is missing.This would prevent any rectangles from forming.But how do I ensure this with only 11 match removals?It seems like a lot to ask, but perhaps it's possible.Maybe I can start by removing matches in such a way that every pair of horizontal matches has at least one missing vertical match between them.Similarly, for every pair of vertical matches, ensure that at least one horizontal match is missing between them.This would require careful planning.But with 11 matches to remove, I think it's feasible.Alternatively, I could think of the problem as ensuring that the remaining matches don't have any two horizontal matches that are "aligned" with two vertical matches.This is similar to ensuring that the remaining matches don't have any two horizontal and two vertical matches forming a rectangle.But again, I'm not sure how to translate this into specific match removals.Maybe I should try to remove matches in a way that creates a "checkerboard" pattern, where matches are removed in alternating positions.But I'm not sure if that would prevent rectangles from forming.Alternatively, I could remove matches to create a "zigzag" pattern, disrupting the straight lines needed to form rectangles.But I'm not sure if that's the best approach.This problem is quite tricky. I think I need to approach it step by step, carefully removing matches and checking after each removal if any rectangles are still present.But since I can't see the grid, I have to do this mentally.Let me try to outline a specific removal strategy:1. Remove the horizontal match in the second row, third column.2. Remove the vertical match in the third row, second column.3. Remove the horizontal match in the third row, fourth column.4. Remove the vertical match in the fourth row, first column.5. Remove the horizontal match in the first row, second column.6. Remove the vertical match in the second row, fourth column.7. Remove the horizontal match in the fourth row, third column.8. Remove the vertical match in the first row, third column.9. Remove the horizontal match in the third row, first column.10. Remove the vertical match in the fourth row, second column.11. Remove the horizontal match in the first row, fourth column.Now, let's check if any rectangles are still present.Starting with the top-left 2x2 square:- The horizontal matches in the first and second rows are intact except for the second row, third column, which is removed.- The vertical matches in the first and second columns are intact except for the second row, fourth column, which is removed.- So, the top-left 2x2 square should be broken because the horizontal match in the second row, third column is missing.Similarly, the top-right 2x2 square:- The horizontal matches in the first and second rows are intact except for the first row, fourth column, which is removed.- The vertical matches in the third and fourth columns are intact except for the second row, fourth column, which is removed.- So, the top-right 2x2 square should also be broken.Moving to the bottom-left 2x2 square:- The horizontal matches in the third and fourth rows are intact except for the third row, first column, which is removed.- The vertical matches in the first and second columns are intact except for the fourth row, second column, which is removed.- So, the bottom-left 2x2 square should be broken.Finally, the bottom-right 2x2 square:- The horizontal matches in the third and fourth rows are intact except for the fourth row, third column, which is removed.- The vertical matches in the third and fourth columns are intact except for the fourth row, second column, which is removed.- So, the bottom-right 2x2 square should also be broken.Now, let's check for larger rectangles, like 2x3 or 3x2.For a 2x3 rectangle, I need to ensure that there aren't two horizontal matches and three vertical matches forming a rectangle.But since I've removed matches in various positions, it's likely that such rectangles are broken.Similarly, for 3x2 rectangles, the same logic applies.Overall, it seems like removing these 11 matches should prevent any rectangles from forming.Of course, without seeing the grid, I can't be 100% sure, but this seems like a reasonable approach.So, in conclusion, by strategically removing 11 matches from both horizontal and vertical directions, ensuring that no four matches form a rectangle, I can achieve the desired configuration.

💡Okay, so I have this geometry problem here. It says that line a is parallel to plane α, and line b is contained in plane α. I need to figure out the positional relationship between lines a and b. The options are: A: Parallel, B: Intersecting or skew, C: Skew, D: Parallel or skew.Hmm, let me visualize this. If line a is parallel to plane α, that means line a doesn't intersect plane α and lies entirely outside of it. Line b, on the other hand, is inside plane α. So, how can these two lines relate?Well, in three-dimensional space, two lines can be parallel, intersecting, or skew. Since line a is parallel to the entire plane, it can't intersect any line within that plane, right? So, they can't intersect. That rules out option B, which includes intersecting.Now, could they be parallel? If line a is parallel to plane α, it doesn't necessarily mean it's parallel to every line in the plane. It just means it doesn't intersect the plane. So, line a could be parallel to line b if they are in the same direction, but they could also be going in different directions.Wait, but if they aren't parallel, what else could they be? Since they don't intersect and aren't parallel, they must be skew. Skew lines are non-parallel, non-intersecting lines in different planes. In this case, line a is outside plane α, and line b is inside plane α, so they are in different planes. Therefore, they can't be parallel or intersecting; they must be skew.But hold on, the question says line a is parallel to plane α. Does that mean line a is parallel to every line in plane α? I don't think so. It just means that line a doesn't intersect plane α. So, line a could be parallel to some lines in plane α and skew to others.So, if line a is parallel to plane α, it can be parallel to line b if they have the same direction, or it can be skew if they don't. Therefore, the relationship between a and b is either parallel or skew.Looking back at the options, D says "Parallel or skew," which matches what I just thought. So, I think the answer is D.But wait, let me double-check. If line a is parallel to plane α, and line b is in plane α, can they ever intersect? Since line a is parallel to the entire plane, it can't intersect any point in the plane, including line b. So, they can't intersect. That means the only possibilities are parallel or skew.Yes, that makes sense. So, the answer should be D: Parallel or skew.

💡Okay, so I have this sequence problem here. The sequence is denoted by {a_n}, and it satisfies the condition that a_n + a_{n+1} = 1/2 for all natural numbers n where n is greater than or equal to 1. Additionally, I'm given that a_2 = 1. I need to find the value of S_{21}, which I assume is the sum of the first 21 terms of the sequence. The options given are A: 9/2, B: 11/2, C: 6, and D: 5.Alright, let me start by understanding the problem step by step. First, the sequence has a recurrence relation: each term plus the next term equals 1/2. So, for any n, a_n + a_{n+1} = 1/2. That seems like a recursive formula that relates consecutive terms. Also, I know that a_2 is 1, which is a specific value given.Since the problem is about finding the sum of the first 21 terms, S_{21}, I need to figure out a way to compute that. To do that, I might need to find a general formula for a_n or at least find a pattern in the sequence so that I can compute the sum without having to calculate each term individually, which would be time-consuming.Let me write down what I know:1. a_n + a_{n+1} = 1/2 for all n ≥ 1.2. a_2 = 1.I need to find S_{21} = a_1 + a_2 + a_3 + ... + a_{21}.Given that a_n + a_{n+1} = 1/2, this suggests that the sum of consecutive terms is constant. So, each pair of consecutive terms adds up to 1/2. That might be useful for grouping terms when calculating the sum.Let me try to find a few terms of the sequence to see if I can spot a pattern.Starting with n=1:a_1 + a_2 = 1/2.But I know that a_2 = 1, so plugging that in:a_1 + 1 = 1/2.Solving for a_1:a_1 = 1/2 - 1 = -1/2.Okay, so a_1 is -1/2.Now, let's find a_3. Using n=2:a_2 + a_3 = 1/2.We know a_2 is 1, so:1 + a_3 = 1/2.Therefore, a_3 = 1/2 - 1 = -1/2.Hmm, interesting. So a_3 is also -1/2.Let's find a_4. Using n=3:a_3 + a_4 = 1/2.We have a_3 = -1/2, so:-1/2 + a_4 = 1/2.Solving for a_4:a_4 = 1/2 + 1/2 = 1.So, a_4 is 1.Wait a second, this seems like a pattern. Let me write down the terms I have so far:a_1 = -1/2a_2 = 1a_3 = -1/2a_4 = 1If I continue this pattern, it seems like the sequence alternates between -1/2 and 1. Let's test this with n=4:a_4 + a_5 = 1/2.We know a_4 is 1, so:1 + a_5 = 1/2.Therefore, a_5 = 1/2 - 1 = -1/2.Yes, so a_5 is -1/2, which continues the pattern. So, the sequence alternates between -1/2 and 1 starting from a_1.So, the sequence is: -1/2, 1, -1/2, 1, -1/2, 1, ..., and so on.Therefore, the terms alternate between -1/2 and 1. So, every odd term is -1/2, and every even term is 1.Let me confirm this with another term. Let's compute a_6:Using n=5:a_5 + a_6 = 1/2.a_5 is -1/2, so:-1/2 + a_6 = 1/2.Therefore, a_6 = 1/2 + 1/2 = 1.Yes, a_6 is 1, which fits the pattern.So, the sequence is periodic with period 2: -1/2, 1, -1/2, 1, etc.Given that, I can model the sequence as:a_n = -1/2 if n is odd,a_n = 1 if n is even.So, for n=1,3,5,...,21, a_n = -1/2,and for n=2,4,6,...,20, a_n = 1.Wait, but n=21 is odd, so a_{21} = -1/2.So, in the sum S_{21}, which is the sum from n=1 to n=21, we have 21 terms.Since the sequence alternates every term, in 21 terms, how many odd terms and how many even terms do we have?Well, from n=1 to n=21, there are 21 terms. Since 21 is odd, the number of odd terms is (21 + 1)/2 = 11 terms, and the number of even terms is 10 terms.Wait, let me verify that. From 1 to 21, inclusive, how many odd numbers are there? The number of odd numbers from 1 to m is given by ceil(m/2). So, for m=21, ceil(21/2) = 11. So, 11 odd terms and 10 even terms.Therefore, in S_{21}, we have 11 terms of -1/2 and 10 terms of 1.Therefore, S_{21} = 11*(-1/2) + 10*(1).Calculating that:11*(-1/2) = -11/2,10*(1) = 10.So, S_{21} = -11/2 + 10.Convert 10 to halves: 10 = 20/2.So, S_{21} = (-11/2) + (20/2) = (20 - 11)/2 = 9/2.So, S_{21} is 9/2, which is option A.Wait, but let me make sure I didn't make a mistake in counting the number of odd and even terms.From n=1 to n=21:Number of odd terms: For n=1,3,5,...,21. The number of terms is (21 - 1)/2 + 1 = (20)/2 +1=10 +1=11.Yes, 11 odd terms.Number of even terms: n=2,4,...,20. The number of terms is (20 - 2)/2 +1= (18)/2 +1=9 +1=10.Yes, 10 even terms.So, 11*(-1/2) + 10*(1) = -11/2 +10= (-11 +20)/2=9/2.So, 9/2 is the sum.Therefore, the correct answer is A: 9/2.Alternatively, another way to think about this is that each pair of terms (a_1 + a_2), (a_3 + a_4), ..., (a_{21} + a_{22}) would each sum to 1/2. But wait, since we're only summing up to a_{21}, which is odd, we have 10 full pairs and one extra term.Wait, let's see:If we group the terms as (a_1 + a_2) + (a_3 + a_4) + ... + (a_{21} + a_{22}), but since we only go up to a_{21}, the last term a_{21} is alone.But wait, actually, in S_{21}, we have 21 terms, which is an odd number, so when grouping into pairs, we have 10 pairs and one single term.Each pair (a_n + a_{n+1}) is 1/2, so 10 pairs would give 10*(1/2)=5.Then, the single term is a_{21}, which is -1/2.Therefore, S_{21}=5 + (-1/2)=5 - 1/2=9/2.Yes, that also gives 9/2, which confirms the previous result.So, regardless of the method, the sum is 9/2.Therefore, the correct answer is A: 9/2.**Final Answer**boxed{A}

💡Okay, so I have this problem here: "Determine the maximum number of unique planes that can be defined by 15 points in general position in 3-dimensional space (no four points are coplanar)." Hmm, let me try to figure this out step by step.First, I remember that in three-dimensional space, three non-collinear points define a unique plane. So, if I have 15 points, and no four of them are coplanar, that means every set of three points will define a different plane, right? Because if four points were coplanar, then multiple sets of three points would lie on the same plane, which would reduce the total number of unique planes.So, the key here is that since no four points are coplanar, every combination of three points gives a unique plane. Therefore, to find the maximum number of unique planes, I just need to calculate how many ways I can choose three points out of 15.I think this is a combination problem because the order of the points doesn't matter when defining a plane. So, the formula for combinations is:[binom{n}{k} = frac{n!}{k!(n - k)!}]Where ( n ) is the total number of items, and ( k ) is the number of items to choose. In this case, ( n = 15 ) and ( k = 3 ).Plugging in the numbers:[binom{15}{3} = frac{15!}{3!(15 - 3)!} = frac{15!}{3! times 12!}]I can simplify this by canceling out the ( 12! ) in the numerator and denominator:[frac{15 times 14 times 13 times 12!}{3! times 12!} = frac{15 times 14 times 13}{3!}]Since ( 3! = 6 ), this becomes:[frac{15 times 14 times 13}{6}]Now, let me compute this step by step. First, multiply 15, 14, and 13:15 multiplied by 14 is 210.210 multiplied by 13 is... let's see, 210 times 10 is 2100, and 210 times 3 is 630, so adding those together gives 2730.Now, divide 2730 by 6:2730 divided by 6 is 455.So, the maximum number of unique planes that can be defined by 15 points in general position is 455.Wait, let me double-check my calculations to make sure I didn't make a mistake. 15 choose 3 is indeed 455. Yeah, that seems right. I don't think I made any errors in the multiplication or division steps. So, I'm confident that 455 is the correct answer.**Final Answer**The maximum number of unique planes is boxed{455}.