💡Okay, so I have this problem here about binomial expansions. The expression is (2x + 1/∛x)^n, and it says that only the 5th term has the largest binomial coefficient. I need to find two things: the constant term in the expansion and the term with the largest coefficient. Hmm, let's break this down step by step.First, I remember that in a binomial expansion of (a + b)^n, the terms are given by the binomial coefficients C(n, k) multiplied by a^(n-k) and b^k. The binomial coefficients are the numbers from Pascal's triangle, right? So, the k-th term is C(n, k-1) * a^(n - (k-1)) * b^(k-1). The problem mentions that only the 5th term has the largest binomial coefficient. That means that C(n, 4) is the largest coefficient, and it's the only one. So, I need to figure out what n is. I recall that in the binomial coefficients, the largest coefficient(s) occur around the middle of the expansion. If only the 5th term is the largest, that suggests that n is such that the middle term is the 5th term. Wait, so if the 5th term is the middle term, then the number of terms is 2*5 -1 = 9? No, that doesn't sound right. Let me think again. The position of the largest coefficient in the binomial expansion is around n/2. So, if the 5th term is the largest, then n/2 should be approximately 4 (since the 5th term corresponds to k=4 in the coefficients). So, n should be around 8? Hmm, but let me check.Wait, actually, the term number is k+1, so the 5th term corresponds to k=4. So, the binomial coefficient C(n, 4) is the largest. For the binomial coefficients, the largest occurs at k = floor(n/2). So, if C(n, 4) is the largest, then floor(n/2) = 4. So, n/2 is approximately 4, so n is 8 or 9. But the problem says only the 5th term is the largest, meaning that n must be such that the coefficients increase up to the 5th term and then decrease. So, if n is 8, then the coefficients are symmetric around the 5th term, which would mean that the 5th term is the middle term, and it's the largest. But wait, for n=8, the expansion has 9 terms, so the 5th term is indeed the middle one. So, that makes sense. So, n=8.Wait, but let me confirm. If n=8, then the coefficients are C(8,0), C(8,1), ..., C(8,8). The largest coefficient is C(8,4)=70, which is indeed the 5th term. So, that seems correct. So, n=8.Okay, so now I know that n=8. Now, moving on to part (1): Find the constant term in the expansion.The general term in the expansion of (2x + 1/∛x)^8 is given by:T_{r+1} = C(8, r) * (2x)^{8 - r} * (1/∛x)^rSimplify this:First, let's write (1/∛x) as x^{-1/3}. So, (1/∛x)^r = x^{-r/3}.Similarly, (2x)^{8 - r} = 2^{8 - r} * x^{8 - r}.So, combining these, the term becomes:C(8, r) * 2^{8 - r} * x^{8 - r} * x^{-r/3} = C(8, r) * 2^{8 - r} * x^{8 - r - r/3}Simplify the exponent:8 - r - r/3 = 8 - (4r)/3So, the exponent of x is 8 - (4r)/3.We need the constant term, which means the exponent of x should be zero. So, set 8 - (4r)/3 = 0.Solving for r:8 = (4r)/3Multiply both sides by 3:24 = 4rDivide both sides by 4:r = 6So, the constant term is the term when r=6, which is T_{7}.Compute T_{7}:C(8,6) * 2^{8 - 6} = C(8,6) * 2^2C(8,6) is equal to C(8,2) because C(n, k) = C(n, n - k). C(8,2) is (8*7)/2 = 28.So, 28 * 4 = 112.Therefore, the constant term is 112.Okay, that seems straightforward. Now, moving on to part (2): Find the term with the largest coefficient in the expansion.Wait, the problem already mentioned that the binomial coefficient of only the 5th term is the largest. But here, it's asking for the term with the largest coefficient, which might not necessarily be the binomial coefficient alone, because each term also has a factor of 2^{8 - r}.So, the coefficient of each term is C(8, r) * 2^{8 - r}. We need to find the value of r that maximizes this expression.So, let's denote the coefficient as:C(8, r) * 2^{8 - r}We need to find r such that this is maximized.One way to approach this is to compute the ratio of consecutive coefficients and see where the ratio changes from greater than 1 to less than 1. That will indicate the maximum.So, let's compute the ratio of C(8, r+1) * 2^{8 - (r+1)} / [C(8, r) * 2^{8 - r}]Simplify this ratio:[C(8, r+1)/C(8, r)] * [2^{7 - r}/2^{8 - r}] = [ (8 - r)/(r + 1) ) ] * (1/2)So, the ratio is (8 - r)/(2(r + 1))We can set this ratio equal to 1 to find the critical point.(8 - r)/(2(r + 1)) = 1Multiply both sides by 2(r + 1):8 - r = 2(r + 1)8 - r = 2r + 2Bring all terms to one side:8 - 2 = 2r + r6 = 3rr = 2So, when r=2, the ratio is equal to 1. So, for r < 2, the ratio is greater than 1, meaning the coefficients are increasing. For r > 2, the ratio is less than 1, meaning the coefficients are decreasing.Therefore, the maximum coefficient occurs at r=2 and r=3? Wait, because when r=2, the ratio is 1, so the coefficients at r=2 and r=3 are equal?Wait, let's compute the coefficients at r=2 and r=3.Compute coefficient at r=2:C(8,2)*2^{6} = 28 * 64 = 1792Coefficient at r=3:C(8,3)*2^{5} = 56 * 32 = 1792So, both r=2 and r=3 give the same coefficient of 1792.Therefore, the terms with the largest coefficients are the 3rd and 4th terms, corresponding to r=2 and r=3.But the problem says "the term with the largest coefficient", so maybe both are acceptable? Or perhaps the question expects both.Wait, let me check the ratio again.We found that the ratio is (8 - r)/(2(r + 1)). So, when r=2, the ratio is (8 - 2)/(2*(2 + 1)) = 6/6 = 1.So, the coefficient at r=3 is equal to the coefficient at r=2. So, both are maximum.Therefore, the terms with the largest coefficients are the 3rd and 4th terms.But let me confirm by computing the coefficients for r=1,2,3,4.Compute coefficient at r=1:C(8,1)*2^{7} = 8 * 128 = 1024r=2: 28*64=1792r=3: 56*32=1792r=4: 70*16=1120So, yes, r=2 and r=3 give the maximum coefficient of 1792.Therefore, the terms with the largest coefficients are the 3rd and 4th terms.But the question says "the term with the largest coefficient", so maybe we need to write both.Alternatively, perhaps the problem expects only one term, but in this case, there are two terms with the same maximum coefficient.So, in conclusion, the constant term is 112, and the terms with the largest coefficients are the 3rd and 4th terms, both equal to 1792.Wait, but let me make sure about the general term.Earlier, I had T_{r+1} = C(8, r) * 2^{8 - r} * x^{8 - (4r)/3}So, for r=2, the exponent is 8 - 8/3 = 16/3, so x^{16/3}For r=3, the exponent is 8 - 12/3 = 8 - 4 = 4, so x^4.Therefore, the terms are 1792 x^{16/3} and 1792 x^4.So, both are terms with the largest coefficients.Therefore, the answers are:(1) The constant term is 112.(2) The terms with the largest coefficients are 1792 x^{16/3} and 1792 x^4.Wait, but the problem says "the term with the largest coefficient", singular. So, maybe it's expecting both? Or perhaps it's considering them as separate terms.Alternatively, maybe I made a mistake in interpreting n. Earlier, I thought n=8 because the 5th term is the largest binomial coefficient. But let me double-check.Wait, the problem says "the binomial coefficient of only the 5th term in the expansion is the largest". So, in the expansion, the 5th term has the largest binomial coefficient, and no other term has the same.But in our case, when n=8, the binomial coefficients are symmetric, so C(8,4)=70 is the largest, and it's unique. So, that's correct.But when we consider the coefficients multiplied by 2^{8 - r}, we get that both r=2 and r=3 give the same maximum coefficient. So, that's correct.Therefore, the answers are as above.But let me just recap:Given (2x + 1/∛x)^8.(1) Constant term: r=6, T7=112.(2) Largest coefficients: r=2 and r=3, giving terms 1792 x^{16/3} and 1792 x^4.Yes, that seems correct.I think I've covered all the steps. I started by determining n based on the position of the largest binomial coefficient, then found the constant term by setting the exponent to zero, and finally found the term(s) with the largest coefficient by analyzing the ratio of consecutive coefficients. Everything checks out.

💡Okay, so I have this trapezoid EFGH. Let me visualize it first. In a trapezoid, two sides are parallel. The problem says GH is parallel to EF, and both EF and GH are equal in length. Wait, that's interesting because usually in a trapezoid, the two parallel sides are called the bases, and they are of different lengths. But here, EF and GH are equal. Hmm, so EF is 10 units, and GH is 22 units. Wait, hold on, that can't be right. If EF and GH are equal, but EF is 10 and GH is 22, that contradicts each other. Maybe I misread the problem.Let me check again: "In trapezoid EFGH, sides EF and GH are equal. GH is parallel to EF, EF = 10 units, and GH = 22 units." Wait, that doesn't make sense because if EF and GH are equal, they must be the same length. But EF is 10 and GH is 22. There must be a mistake here. Maybe the problem meant that the non-parallel sides are equal? Because in a trapezoid, usually, the non-parallel sides are called the legs, and sometimes they can be equal, making it an isosceles trapezoid.Let me read the problem again: "In trapezoid EFGH, sides EF and GH are equal. GH is parallel to EF, EF = 10 units, and GH = 22 units. The perpendicular distance between EF and GH is 5 units. Calculate the perimeter of trapezoid EFGH."Wait, so EF and GH are equal, but EF is 10 and GH is 22? That's conflicting. Maybe it's a typo, and they meant that the non-parallel sides are equal? Because EF and GH are the bases, and if they are equal, they should have the same length. But here, EF is 10 and GH is 22, which are different. So perhaps the problem meant that the legs, which are the non-parallel sides, are equal. That would make more sense because in an isosceles trapezoid, the legs are equal.So, assuming that, let's proceed. So, trapezoid EFGH has bases EF = 10 units and GH = 22 units, which are parallel. The legs, which are the non-parallel sides, are equal. The perpendicular distance (height) between the bases is 5 units. We need to find the perimeter.Okay, so to find the perimeter, we need the lengths of all four sides. We know the lengths of the two bases: EF = 10 and GH = 22. We need to find the lengths of the legs, which are equal. Let's denote the legs as FG and HE. Since it's an isosceles trapezoid, FG = HE.To find the length of the legs, we can use the Pythagorean theorem. If we drop perpendiculars from the endpoints of the shorter base EF to the longer base GH, we'll form two right triangles on either side of the trapezoid. The height of these triangles is the perpendicular distance between the bases, which is 5 units.The difference in the lengths of the two bases is 22 - 10 = 12 units. Since the trapezoid is isosceles, this extra length is split equally on both sides. So, each of these right triangles will have a base of 12/2 = 6 units.Now, each leg of the trapezoid is the hypotenuse of a right triangle with one leg being the height (5 units) and the other leg being 6 units. So, using the Pythagorean theorem:Leg length = √(6² + 5²) = √(36 + 25) = √61So, each leg is √61 units long. Since there are two legs, their combined length is 2√61.Now, adding up all the sides for the perimeter:Perimeter = EF + FG + GH + HE = 10 + √61 + 22 + √61 = 32 + 2√61So, the perimeter of trapezoid EFGH is 32 + 2√61 units.

💡To solve the problem, we start by factorizing 162 into its prime factors:[162 = 2 cdot 3^4]Given that (162p = q^3), (162p) must be a perfect cube. This means each prime factor's exponent in (162p) must be a multiple of 3. - The exponent of 2 in 162 is 1. To make it a multiple of 3, we need two more 2's.- The exponent of 3 in 162 is 4. To make it a multiple of 3, we need two more 3's.Thus, (p) must be:[p = 2^2 cdot 3^2 = 4 cdot 9 = 36]Substituting (p = 36) back into the equation (162p = q^3):[162 cdot 36 = 5832 = q^3]Factoring 5832, we find:[5832 = 2^3 cdot 3^6 = 18^3]Thus, (q = 18). Calculating (p + q):[p + q = 36 + 18 = 54]The minimum possible value of (p + q) is (boxed{54}).

💡Okay, so I have this problem about geometric sequences. Let me try to understand it step by step. First, the problem says: Given a geometric sequence {a_n} with the first term a_1 and common ratio q, then "if a_1 < 0 and 0 < q < 1" is a condition for "for any n ∈ ℕ*, a_{n+1} > a_n" to be (     )      (     ). The options are A, B, C, D.Alright, so I need to figure out what kind of condition "a_1 < 0 and 0 < q < 1" is for the statement that the sequence is increasing, meaning each term is greater than the previous one.Let me recall what a geometric sequence is. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, q. So, the nth term is a_n = a_1 * q^{n-1}.Now, the problem is about the condition for the sequence to be increasing. That is, for every natural number n, a_{n+1} > a_n. So, I need to find under what conditions on a_1 and q does this inequality hold.The given condition is a_1 < 0 and 0 < q < 1. Let me analyze this.First, let's think about what it means for a geometric sequence to be increasing. If the sequence is increasing, then each term is larger than the previous one. So, a_{n+1} = a_n * q must be greater than a_n.So, a_{n+1} > a_n implies that a_n * q > a_n. Let's rearrange this inequality:a_n * q > a_n => a_n * q - a_n > 0 => a_n (q - 1) > 0So, the product of a_n and (q - 1) must be positive. For this product to be positive, either both factors are positive or both are negative.Case 1: Both a_n and (q - 1) are positive.If a_n is positive, then (q - 1) must also be positive, which means q > 1.Case 2: Both a_n and (q - 1) are negative.If a_n is negative, then (q - 1) must also be negative, which means q < 1.So, for the sequence to be increasing, either:1. a_n is positive and q > 1, or2. a_n is negative and q < 1.But wait, in a geometric sequence, the sign of a_n depends on the sign of a_1 and the value of q. If a_1 is positive and q is positive, all terms will be positive. If a_1 is negative and q is positive, all terms will be negative. If q is negative, the terms will alternate in sign.So, if we have a_1 < 0 and 0 < q < 1, then all terms a_n will be negative because a_1 is negative and q is positive. Also, since 0 < q < 1, each subsequent term is a multiple of the previous term by a number less than 1 but greater than 0, which makes the terms closer to zero. Since the terms are negative, getting closer to zero means they are increasing.For example, let's take a_1 = -4 and q = 1/2. Then the sequence would be:a_1 = -4 a_2 = -4 * 1/2 = -2 a_3 = -2 * 1/2 = -1 a_4 = -1 * 1/2 = -0.5 and so on.Each term is greater than the previous one because they are moving towards zero from the negative side. So, in this case, the sequence is indeed increasing.Now, is this condition "a_1 < 0 and 0 < q < 1" a sufficient condition for the sequence to be increasing? Yes, because whenever these two conditions are met, the sequence will be increasing.But is it a necessary condition? That is, are there other conditions where the sequence is increasing without a_1 being negative and q between 0 and 1?Yes, for example, if a_1 is positive and q > 1, the sequence is also increasing. For instance, a_1 = 2 and q = 3:a_1 = 2 a_2 = 6 a_3 = 18 a_4 = 54 and so on.Each term is greater than the previous one, so the sequence is increasing. However, in this case, a_1 is positive and q > 1, which is a different set of conditions.Therefore, the condition "a_1 < 0 and 0 < q < 1" is sufficient but not necessary for the sequence to be increasing. There are other conditions that can also make the sequence increasing.So, looking back at the options:A: A sufficient but not necessary condition B: A necessary but not sufficient condition C: A more sufficient than necessary condition D: Neither sufficient nor necessary conditionThe correct answer should be A: A sufficient but not necessary condition.Wait, let me double-check. The problem states "if a_1 < 0 and 0 < q < 1" is a condition for "for any n ∈ ℕ*, a_{n+1} > a_n" to be (     ). So, is the condition sufficient, necessary, both, or neither?We established that if a_1 < 0 and 0 < q < 1, then the sequence is increasing. So, it's sufficient. But it's not necessary because, as shown, the sequence can be increasing under different conditions. Therefore, it's a sufficient but not necessary condition. So, A is correct.I think that's solid. I don't see any flaws in this reasoning.**Final Answer**boxed{A}

💡Alright, so I have this problem: In a plane, there's a finite set of polygons, and every pair of these polygons shares at least one common point. I need to prove that there's a line that intersects all these polygons. Hmm, okay. Let me try to wrap my head around this.First, let's break down what's given. We have multiple polygons, and any two of them have at least one point in common. So, they all overlap with each other in some way. The goal is to find a single line that cuts through all of them. That sounds like it should be possible, but I need to figure out why.Maybe I can start by thinking about simpler shapes, like line segments or circles, and see if a similar principle applies. For example, if I have multiple line segments, and every pair intersects, does there exist a line that intersects all of them? Hmm, not necessarily. Imagine several line segments arranged like spokes on a wheel; each pair intersects at the center, but there's no single line that intersects all of them unless it's the center itself. But in this case, the center is a point, not a line.Wait, but polygons are more complex than line segments. They have multiple sides and can cover more area. Maybe the fact that they're polygons with multiple edges gives us something to work with.Another thought: if every pair of polygons shares a common point, maybe all these points are related in some way. Perhaps there's a common intersection point that lies in all polygons? If that's the case, then any line passing through that point would intersect all polygons. But the problem doesn't state that there's a single common point for all polygons, only that every pair shares at least one point. So, it's possible that different pairs share different points.Hmm, so maybe there isn't a single common point, but there's still a line that intersects all polygons. How can that be?Let me think about projections. If I project all polygons onto a certain line, say the x-axis, each polygon would project to an interval on that line. If every pair of polygons intersects, then their projections should overlap, right? Because if two polygons intersect, their projections on the x-axis must overlap. So, if I have multiple intervals on the x-axis, and every pair overlaps, then there must be a common point where all intervals overlap. That sounds familiar—it's similar to the Helly's theorem in one dimension.Helly's theorem says that for a family of convex sets in ℝⁿ, if the intersection of every n+1 of them is non-empty, then the whole family has a common point. In one dimension, for intervals, if every pair overlaps, then there's a common point. So, applying that here, if I project all polygons onto the x-axis, the projections are intervals, and since every pair overlaps, there must be a common point on the x-axis where all projections overlap.Okay, so there's a point on the x-axis that's in the projection of every polygon. That means for each polygon, there's at least one point in the polygon that projects to this common point on the x-axis. So, if I draw a vertical line through this common point on the x-axis, this line must intersect every polygon because each polygon has at least one point on this vertical line.Wait, but polygons are 2D shapes, so a vertical line might not necessarily intersect all of them unless the projection ensures that. But since the projection of each polygon onto the x-axis includes this common point, and the polygon is a 2D shape, there must be some point in the polygon that lies on the vertical line through that common x-coordinate.So, does that mean that the vertical line intersects all polygons? It seems so. But I need to make sure this logic holds. Let me think about it again.If I have a polygon, and I project it onto the x-axis, the projection is an interval. If every pair of polygons shares a point, their projections share an interval. By Helly's theorem in one dimension, all projections share at least one common point. Therefore, there exists an x-coordinate such that every polygon has at least one point with that x-coordinate. Drawing a vertical line at that x-coordinate would intersect every polygon because each polygon has a point on that line.That makes sense. But what if the polygons are arranged in such a way that their projections don't all overlap? Wait, no, because the problem states that every pair shares a common point, so their projections must overlap pairwise, which by Helly's theorem implies a common intersection point for all projections.Okay, so I think this approach works. By projecting onto a line, using Helly's theorem to find a common point in the projections, and then drawing a line perpendicular to the projection line through that common point, we ensure that this line intersects all polygons.But let me consider if there's another way to think about this. Maybe using duality or some other geometric transformation. Or perhaps considering the arrangement of polygons and their intersections.Another idea: since every pair of polygons intersects, maybe there's a point where infinitely many polygons intersect, but that's not necessarily the case. The problem only states that every pair intersects, not that there's a universal intersection point.But going back to the projection idea, it seems solid. Let me formalize it a bit more.Suppose I choose an arbitrary direction and project all polygons onto a line in that direction. Each polygon's projection is a segment on that line. Since every pair of polygons intersects, their projections must overlap. By Helly's theorem for intervals on a line, there must be a common point in all projections. Therefore, there exists a point on the projection line that is covered by all projections. Drawing a line perpendicular to the projection line through this common point will intersect all polygons because each polygon has at least one point on this perpendicular line.This seems like a valid proof. I think I can stop here.**Final Answer**boxed{text{Such a line exists.}}

💡First, I'll denote the principal amount as ( P ) and the original rate of interest as ( R% ).The simple interest for the original rate over 10 years is calculated as:[text{SI} = frac{P times R times 10}{100}]If the interest rate is increased by 5%, the new rate becomes ( (R + 5)% ). The simple interest with the new rate is:[text{SI}_{text{new}} = frac{P times (R + 5) times 10}{100}]According to the problem, the new simple interest is Rs. 100 more than the original simple interest:[frac{P times (R + 5) times 10}{100} = frac{P times R times 10}{100} + 100]Subtracting the original simple interest from both sides:[frac{P times (R + 5) times 10}{100} - frac{P times R times 10}{100} = 100]Simplifying the equation:[frac{P times 5 times 10}{100} = 100][frac{50P}{100} = 100][50P = 100 times 100][P = frac{10000}{50}][P = 200]Therefore, the principal amount is Rs. 200.