💡Okay, so I have this logic problem here, and I need to find the inverse negation of the statement "If ( x geqslant 3 ), then ( x < 0 )." The options are A, B, C, and D, each with different conditional statements. Hmm, I remember that in logic, there are different types of related statements like converse, inverse, and contrapositive. Let me try to recall what each of these means.First, the original statement is "If ( x geqslant 3 ), then ( x < 0 )." In logical terms, this can be written as ( P rightarrow Q ), where ( P ) is ( x geqslant 3 ) and ( Q ) is ( x < 0 ).Now, the inverse of a statement is formed by negating both the hypothesis and the conclusion of the original statement. So, the inverse would be ( neg P rightarrow neg Q ). That means, for our problem, the inverse would be "If ( x < 3 ), then ( x geqslant 0 )." Wait, but that's not one of the options. Let me check the options again.Option A: If ( x geqslant 0 ), then ( x < 3 ).Option B: If ( x < 3 ), then ( x leqslant 0 ).Option C: If ( x < 0 ), then ( x leqslant 3 ).Option D: If ( x > 3 ), then ( x geqslant 0 ).Hmm, none of these seem to match the inverse I just wrote. Maybe I misunderstood what inverse negation means. Let me think again.Wait, maybe inverse negation refers to the contrapositive? The contrapositive of ( P rightarrow Q ) is ( neg Q rightarrow neg P ). So, applying that to our statement, the contrapositive would be "If ( x geqslant 0 ), then ( x < 3 )." Oh, that's option A. So, is the inverse negation the same as the contrapositive? I think that might be the case.But just to be sure, let me verify. The original statement is ( P rightarrow Q ). The contrapositive is ( neg Q rightarrow neg P ), which is logically equivalent to the original statement. The inverse is ( neg P rightarrow neg Q ), which is not necessarily equivalent. So, if the question is asking for the inverse negation, maybe it's referring to the contrapositive.Alternatively, maybe "inverse negation" is a term I'm not familiar with. Let me break it down. Inverse would mean negating both parts, and negation would be the opposite. So, perhaps it's the same as the contrapositive.Looking back at the options, option A is "If ( x geqslant 0 ), then ( x < 3 )," which matches the contrapositive. Option B is "If ( x < 3 ), then ( x leqslant 0 )," which is the inverse but not the contrapositive. Option C is "If ( x < 0 ), then ( x leqslant 3 )," which seems like a different statement altogether. Option D is "If ( x > 3 ), then ( x geqslant 0 )," which also doesn't fit.So, based on this, I think option A is the correct answer because it corresponds to the contrapositive of the original statement, which is logically equivalent and might be what the question means by inverse negation.Just to double-check, let's analyze the original statement and its contrapositive. The original statement says that if ( x ) is at least 3, then it's less than 0. But wait, that doesn't make sense because if ( x ) is 3 or more, it can't be less than 0. So, the original statement is actually false because there are no real numbers that are both ( geq 3 ) and ( < 0 ). However, the contrapositive should have the same truth value as the original statement. So, if the original is false, the contrapositive should also be false.Looking at option A: "If ( x geqslant 0 ), then ( x < 3 )." This is also not necessarily true because ( x ) could be 3 or more, which would make the hypothesis true but the conclusion false. So, both the original statement and its contrapositive are false, which makes sense because they are logically equivalent.Therefore, even though both statements are false, the form of the contrapositive matches option A, which is what the question is asking for.

💡Alright, so I'm trying to figure out this syllogistic reasoning problem. Let me start by understanding what's being asked here. The problem states:"The square of any real number is greater than 0, a ∈ R, then a² > 0. Among the following statements, the correct one is ( )."And then it gives four options:A: Major premise error, causing conclusion errorB: Minor premise error, causing conclusion errorC: Error in reasoning form, causing conclusion errorD: The reasoning is fine, and the conclusion is correctOkay, so I need to analyze this syllogism and determine which of these statements is correct. First, let's break down what a syllogism is. A syllogism is a form of deductive reasoning consisting of two premises and a conclusion. The major premise is a general statement, the minor premise is a specific statement, and the conclusion is what logically follows from the two premises.In this case, the major premise is: "The square of any real number is greater than 0." The minor premise is: "a ∈ R," meaning "a is a real number." The conclusion is: "a² > 0."So, the structure is:1. Major premise: All real numbers squared are greater than 0.2. Minor premise: a is a real number.3. Conclusion: Therefore, a squared is greater than 0.Now, I need to evaluate whether this syllogism is correct or if there's an error in the premises or the reasoning form.First, let's look at the major premise: "The square of any real number is greater than 0." Hmm, is this always true? Well, if I take any real number and square it, will it always be greater than 0? Let's think about some examples.If a = 2, then a² = 4, which is greater than 0. If a = -3, then a² = 9, which is also greater than 0. But wait, what if a = 0? Then a² = 0, which is not greater than 0. So, the square of 0 is equal to 0, not greater than 0. Therefore, the major premise is not entirely correct because it doesn't hold true when a = 0.So, the major premise has an error because it claims that the square of any real number is greater than 0, but this isn't the case when the real number is 0. This means that the major premise is flawed.Now, moving on to the minor premise: "a ∈ R," which means "a is a real number." This seems straightforward and doesn't have any obvious errors. It's just stating that a is a real number, which could be any number on the real number line, including positive numbers, negative numbers, and zero.Given that the minor premise is correct, but the major premise is incorrect, what does that mean for the conclusion? The conclusion is: "a² > 0." If the major premise is incorrect because it doesn't account for a = 0, then the conclusion might not always be true. Specifically, when a = 0, a² = 0, which contradicts the conclusion that a² > 0.Therefore, the error in the major premise leads to an incorrect conclusion. The syllogism assumes that squaring any real number will always result in a number greater than 0, but this isn't true when the number is 0.Now, let's look at the options given:A: Major premise error, causing conclusion errorB: Minor premise error, causing conclusion errorC: Error in reasoning form, causing conclusion errorD: The reasoning is fine, and the conclusion is correctFrom my analysis, the major premise is incorrect because it doesn't hold true for all real numbers (specifically, when a = 0). The minor premise is correct, and there doesn't seem to be an error in the reasoning form itself; the structure of the syllogism is logical. The issue lies with the major premise being inaccurate.Therefore, the correct answer should be option A: Major premise error, causing conclusion error.But just to double-check, let's consider the other options to make sure I'm not missing something.Option B suggests that the minor premise is incorrect. However, the minor premise is simply stating that a is a real number, which is a valid statement. There's no error here.Option C talks about an error in the reasoning form. The syllogism follows a standard structure: major premise, minor premise, conclusion. There's no logical fallacy or incorrect form here; the problem is with the content of the major premise, not the structure.Option D claims that the reasoning is fine and the conclusion is correct. But as we saw, the conclusion isn't always correct because when a = 0, a² = 0, which contradicts the conclusion that a² > 0. So, this option is incorrect.Therefore, after carefully analyzing each part of the syllogism and considering the given options, I'm confident that option A is the correct answer.

💡Alright, so I've got this geometry problem here, and I need to figure out the ratio in which the line BH divides the segment CD in this convex quadrilateral ABCD. Let me try to unpack all the given information step by step.First, let's list out what's given:1. **Convex Quadrilateral ABCD**: So, all interior angles are less than 180 degrees, and the sides don't cross each other.2. **Angle ABC is 90 degrees**: That means at vertex B, the angle between sides AB and BC is a right angle.3. **Angle BAC equals Angle CAD**: So, the angle at vertex A between BA and AC is equal to the angle at vertex A between CA and AD. This suggests that AC is the angle bisector of angle BAD.4. **AC equals AD**: So, the length of diagonal AC is equal to the length of side AD.5. **DH is the altitude of triangle ACD**: That means DH is perpendicular to AC, and H is the foot of this perpendicular from D onto AC.And the question is: In what ratio does the line BH divide the segment CD?Okay, let's try to visualize this. Maybe drawing a sketch would help. Since it's a convex quadrilateral, I can imagine points A, B, C, D arranged in order without crossing.Starting with point A. From A, we have two lines: one going to B and another going to D. Since angle BAC equals angle CAD, AC is the angle bisector of angle BAD. Also, AC equals AD, which is interesting.Given that angle ABC is 90 degrees, triangle ABC is a right-angled triangle at B. So, AB is perpendicular to BC.Now, since AC is equal to AD, triangle ACD is an isosceles triangle with AC = AD. That means angles at C and D in triangle ACD are equal. Wait, but we also know that DH is the altitude from D to AC. So, in triangle ACD, DH is perpendicular to AC, making triangle AHD and triangle CHD both right-angled triangles.Given that AC = AD, triangle AHD is congruent to triangle ACD? Hmm, not sure about that. Wait, no, because triangle AHD is part of triangle ACD. Maybe I need to look for some congruency or similarity here.Since angle BAC equals angle CAD, and AC is common to both triangles ABC and ACD, maybe there's some similarity or congruency between these triangles.Wait, triangle ABC is right-angled at B, and triangle AHD is also right-angled at H. If I can show that these two triangles are congruent, that might help.Let's see: In triangle ABC, we have AB, BC, and AC. In triangle AHD, we have AH, HD, and AD. Given that AC = AD, maybe if AB = AH and BC = HD, then triangles ABC and AHD would be congruent by SAS or something.But how do I know AB = AH? Hmm, maybe not directly. Alternatively, since AC is the angle bisector, maybe I can use the Angle Bisector Theorem.The Angle Bisector Theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. In triangle ABD, AC is the angle bisector of angle BAD, so it should divide BD into segments proportional to AB and AD.Wait, but BD isn't necessarily a side here. Maybe I need to think differently.Alternatively, since AC = AD, triangle ACD is isosceles, so angles at C and D are equal. Let's denote angle BAC = angle CAD = θ. Then, angle BAD = 2θ.In triangle ABC, angle BAC = θ, angle ABC = 90 degrees, so angle BCA = 90 - θ.Similarly, in triangle ACD, angle CAD = θ, and since AC = AD, angles at C and D are equal. Let's denote angle ACD = angle ADC = φ. Then, in triangle ACD, the sum of angles is θ + φ + φ = 180, so 2φ = 180 - θ, which means φ = (180 - θ)/2 = 90 - θ/2.Wait, so angle ACD = 90 - θ/2.But in triangle ABC, angle BCA = 90 - θ. So, angle BCA is 90 - θ, and angle ACD is 90 - θ/2. That suggests that angle BCA is larger than angle ACD by θ/2.Hmm, not sure if that helps directly.Let me think about the coordinates. Maybe assigning coordinates to points would make this easier.Let's place point A at the origin (0,0). Since angle ABC is 90 degrees, let's place point B along the x-axis and point C along the y-axis. So, let’s say point B is at (b, 0) and point C is at (0, c), where b and c are positive real numbers.Given that AC = AD, point D must be somewhere such that the distance from A to D is equal to the distance from A to C. Since AC is the distance from (0,0) to (0,c), which is c. So, AD must also be c. Therefore, point D lies somewhere on the circle centered at A with radius c.Also, angle BAC = angle CAD. Since angle BAC is the angle between BA and AC, which in our coordinate system is the angle between the negative x-axis (from A to B) and the positive y-axis (from A to C). So, angle BAC is 90 degrees. Wait, but angle BAC is given to be equal to angle CAD, which would mean angle CAD is also 90 degrees. But that can't be because in triangle ACD, angle at A is 90 degrees, and AC = AD, so triangle ACD would be an isosceles right-angled triangle.Wait, but in that case, point D would be at (c,0), but that would make AD = c, which is equal to AC = c. But then, point D would be at (c,0), which is on the x-axis. But point B is also on the x-axis at (b,0). If b ≠ c, then points B and D would be different points on the x-axis.But in the problem, ABCD is a convex quadrilateral, so points must be arranged in order without crossing. If D is on the x-axis, then the quadrilateral would be A(0,0), B(b,0), C(0,c), D(c,0). But then, connecting D to C would cross over AB if c ≠ b. Hmm, maybe not. Wait, actually, in this coordinate system, if D is at (c,0), then CD would be from (0,c) to (c,0), which is a diagonal line, and BD would be from (b,0) to (c,0), which is along the x-axis. So, the quadrilateral would be A(0,0), B(b,0), C(0,c), D(c,0). Is this convex? Let me see: plotting these points, A is at the origin, B is to the right on the x-axis, C is up on the y-axis, and D is to the right on the x-axis. Connecting them in order, A to B to C to D to A. Hmm, actually, this might not be convex because the angle at D could be greater than 180 degrees if c > b or something. Maybe I need to adjust.Alternatively, maybe I should place point D somewhere else. Since AC = AD, and AC is along the y-axis from (0,0) to (0,c), then AD must be a line of length c from A(0,0) in some direction. Since angle BAC = angle CAD, which is θ, and angle BAC is between BA and AC, which is 90 degrees in my coordinate system. Wait, no, in my initial placement, angle BAC is 90 degrees, but the problem says angle BAC = angle CAD, which would mean angle CAD is also 90 degrees, making angle BAD = 180 degrees, which can't be because ABCD is convex.Wait, maybe my initial coordinate system is flawed. Let me try a different approach.Let me place point A at (0,0). Let me let AC lie along the x-axis for simplicity. So, point C is at (c,0). Then, since angle BAC = angle CAD, and AC = AD, point D must be such that AD = AC = c, and angle CAD = angle BAC.Let me denote angle BAC = θ, so angle CAD = θ as well. Therefore, angle BAD = 2θ.Since AC is along the x-axis from (0,0) to (c,0), then AD must make an angle θ above the x-axis. So, point D would be at (c*cosθ, c*sinθ).Similarly, point B is such that angle ABC = 90 degrees. Since AC is along the x-axis, and angle BAC = θ, point B must be somewhere in the plane such that angle at B is 90 degrees.Wait, maybe I need to define point B in terms of θ as well. Let me think.From point A(0,0), AC is along the x-axis to C(c,0). Angle BAC = θ, so AB makes an angle θ with AC. Therefore, AB is at an angle θ above the x-axis.But angle ABC is 90 degrees, so triangle ABC is right-angled at B.Let me denote point B as (b*cosθ, b*sinθ), where b is the length of AB.Then, point C is at (c,0). So, vector BC would be from B to C: (c - b*cosθ, -b*sinθ).Since angle at B is 90 degrees, vectors BA and BC are perpendicular. Vector BA is from B to A: (-b*cosθ, -b*sinθ). Vector BC is (c - b*cosθ, -b*sinθ).Their dot product should be zero:(-b*cosθ)(c - b*cosθ) + (-b*sinθ)(-b*sinθ) = 0Expanding:- b c cosθ + b² cos²θ + b² sin²θ = 0Simplify:- b c cosθ + b² (cos²θ + sin²θ) = 0Since cos²θ + sin²θ = 1:- b c cosθ + b² = 0So:b² - b c cosθ = 0Factor:b(b - c cosθ) = 0So, b = 0 or b = c cosθBut b = 0 would mean point B coincides with A, which isn't possible. So, b = c cosθTherefore, point B is at (b cosθ, b sinθ) = (c cos²θ, c cosθ sinθ)Okay, so now we have coordinates for points A, B, C, D:- A: (0,0)- B: (c cos²θ, c cosθ sinθ)- C: (c, 0)- D: (c cosθ, c sinθ)Now, we need to find point H, which is the foot of the altitude from D to AC.Since AC is along the x-axis from (0,0) to (c,0), the altitude from D to AC is just the vertical line from D down to AC. So, point H is the projection of D onto AC, which is (c cosθ, 0).Wait, because D is at (c cosθ, c sinθ), so dropping a perpendicular to AC (the x-axis) would land at (c cosθ, 0). So, H is at (c cosθ, 0).Now, we need to find the line BH and see where it intersects CD.Point B is at (c cos²θ, c cosθ sinθ), and point H is at (c cosθ, 0). So, the line BH goes from (c cos²θ, c cosθ sinθ) to (c cosθ, 0).Let me find the parametric equations for line BH.Let parameter t go from 0 to 1, with t=0 at B and t=1 at H.So, x(t) = c cos²θ + t(c cosθ - c cos²θ) = c cos²θ + t c cosθ (1 - cosθ)Similarly, y(t) = c cosθ sinθ + t(0 - c cosθ sinθ) = c cosθ sinθ (1 - t)Now, we need to find where this line intersects CD.Point C is at (c, 0), and point D is at (c cosθ, c sinθ). So, segment CD goes from (c, 0) to (c cosθ, c sinθ).Let me parameterize CD as well. Let parameter s go from 0 to 1, with s=0 at C and s=1 at D.So, x(s) = c + s(c cosθ - c) = c(1 + s(cosθ - 1))y(s) = 0 + s(c sinθ - 0) = c s sinθNow, to find the intersection point N between BH and CD, we need to solve for t and s such that:c cos²θ + t c cosθ (1 - cosθ) = c(1 + s(cosθ - 1))andc cosθ sinθ (1 - t) = c s sinθLet me simplify these equations.First, divide both sides by c:cos²θ + t cosθ (1 - cosθ) = 1 + s(cosθ - 1)andcosθ sinθ (1 - t) = s sinθAssuming sinθ ≠ 0 (since θ is an angle in a convex quadrilateral, so θ ≠ 0 or 180 degrees), we can divide both sides by sinθ:cosθ (1 - t) = sSo, s = cosθ (1 - t)Now, substitute s into the first equation:cos²θ + t cosθ (1 - cosθ) = 1 + cosθ (1 - t)(cosθ - 1)Let me expand the right-hand side:1 + cosθ (1 - t)(cosθ - 1) = 1 + cosθ [ (cosθ - 1) - t(cosθ - 1) ] = 1 + cosθ (cosθ - 1) - t cosθ (cosθ - 1)So, the equation becomes:cos²θ + t cosθ (1 - cosθ) = 1 + cos²θ - cosθ - t cosθ (cosθ - 1)Let me bring all terms to the left-hand side:cos²θ + t cosθ (1 - cosθ) - 1 - cos²θ + cosθ + t cosθ (cosθ - 1) = 0Simplify:(cos²θ - cos²θ) + (t cosθ (1 - cosθ) + t cosθ (cosθ - 1)) + (-1 + cosθ) = 0Notice that t cosθ (1 - cosθ) + t cosθ (cosθ - 1) = t cosθ [ (1 - cosθ) + (cosθ - 1) ] = t cosθ (0) = 0So, we're left with:-1 + cosθ = 0Which implies:cosθ = 1But cosθ = 1 implies θ = 0 degrees, which contradicts our earlier assumption that θ is an angle in a convex quadrilateral (θ cannot be 0). Therefore, this suggests that our assumption might be wrong, or perhaps there's a mistake in the calculations.Wait, maybe I made a mistake in the algebra. Let me go back and check.Starting from:cos²θ + t cosθ (1 - cosθ) = 1 + s(cosθ - 1)ands = cosθ (1 - t)Substituting s:cos²θ + t cosθ (1 - cosθ) = 1 + cosθ (1 - t)(cosθ - 1)Let me expand the right-hand side again:1 + cosθ (1 - t)(cosθ - 1) = 1 + cosθ [ (cosθ - 1) - t(cosθ - 1) ] = 1 + cosθ (cosθ - 1) - t cosθ (cosθ - 1)So, the equation is:cos²θ + t cosθ (1 - cosθ) = 1 + cos²θ - cosθ - t cosθ (cosθ - 1)Now, subtract cos²θ from both sides:t cosθ (1 - cosθ) = 1 - cosθ - t cosθ (cosθ - 1)Notice that (cosθ - 1) = -(1 - cosθ), so:t cosθ (1 - cosθ) = 1 - cosθ + t cosθ (1 - cosθ)Bring all terms to the left:t cosθ (1 - cosθ) - t cosθ (1 - cosθ) - (1 - cosθ) = 0Simplify:0 - (1 - cosθ) = 0Which gives:- (1 - cosθ) = 0 => 1 - cosθ = 0 => cosθ = 1Again, same result. This suggests that the only solution is when cosθ = 1, which is not possible in this context. Therefore, perhaps our initial assumption about the coordinates is flawed, or maybe the lines BH and CD are parallel, meaning they don't intersect, which contradicts the problem statement since BH must intersect CD at some point N.Wait, but in a convex quadrilateral, BH should intersect CD somewhere between C and D. So, perhaps there's a mistake in how I've set up the coordinates.Let me reconsider the coordinate system. Maybe placing AC along the x-axis is causing some issues because of the way the angles are set up. Alternatively, perhaps I should place point A at (0,0), point B at (b,0), point C at (0,c), and then find point D accordingly.Let me try that.So, point A: (0,0)Point B: (b,0)Point C: (0,c)Now, since AC = AD, point D must be such that the distance from A to D is equal to AC, which is sqrt(0² + c²) = c. So, AD = c, meaning point D lies on the circle centered at A with radius c.Also, angle BAC = angle CAD. Let's denote angle BAC = θ, so angle CAD = θ as well.Since angle BAC is the angle between BA and AC. BA is from A(0,0) to B(b,0), which is along the positive x-axis. AC is from A(0,0) to C(0,c), which is along the positive y-axis. Therefore, angle BAC is 90 degrees. Wait, but the problem states angle BAC = angle CAD, so angle CAD is also 90 degrees. That would mean that angle BAD = angle BAC + angle CAD = 90 + 90 = 180 degrees, which would make points B, A, D colinear, which contradicts the convexity of the quadrilateral.Hmm, this suggests that my initial assumption of placing B along the x-axis and C along the y-axis might not be suitable because it forces angle BAC to be 90 degrees, which then makes angle CAD also 90 degrees, leading to a straight line at A, which isn't allowed in a convex quadrilateral.Therefore, perhaps I need to adjust the coordinate system so that angle BAC is not 90 degrees, but some other angle θ, and then AC is not along the y-axis.Let me try a different approach. Let me place point A at (0,0), point B at (b,0), and point C somewhere in the plane such that angle ABC is 90 degrees. Then, AC is the angle bisector of angle BAD, and AC = AD.Let me denote angle BAC = angle CAD = θ.Since angle ABC = 90 degrees, triangle ABC is right-angled at B.Let me denote AB = x, BC = y, so AC = sqrt(x² + y²).Given that AC = AD, so AD = sqrt(x² + y²).Also, since angle BAC = angle CAD = θ, point D must be such that AD = AC and angle CAD = θ.Let me try to find coordinates for point D.From point A(0,0), AC is at an angle θ from AB. Since AB is along the x-axis from A(0,0) to B(b,0), then AC makes an angle θ above the x-axis.Therefore, point C can be represented in polar coordinates as (AC, θ) from A. So, coordinates of C would be (AC cosθ, AC sinθ).But AC = sqrt(x² + y²), where x = AB = b, and y = BC.Wait, in triangle ABC, AB = b, BC = y, and AC = sqrt(b² + y²).So, point C is at (sqrt(b² + y²) cosθ, sqrt(b² + y²) sinθ).But also, in triangle ABC, angle at B is 90 degrees, so by definition, AB and BC are perpendicular.Wait, but if I place point B at (b,0), then point C must be at (b, y) to make BC vertical, but then AC would be the hypotenuse from (0,0) to (b, y), which has length sqrt(b² + y²).But in that case, angle BAC is the angle between AB (along x-axis) and AC (from (0,0) to (b,y)). So, angle BAC = arctan(y/b).Similarly, angle CAD = angle BAC = arctan(y/b), so point D must be such that from A(0,0), AD makes an angle 2θ with AB, where θ = arctan(y/b).Wait, no. Since angle BAC = angle CAD = θ, then angle BAD = 2θ.So, point D is located such that from A(0,0), it's at an angle 2θ from AB, and at a distance AC = sqrt(b² + y²).Therefore, coordinates of D would be (sqrt(b² + y²) cos(2θ), sqrt(b² + y²) sin(2θ)).But θ = arctan(y/b), so 2θ = 2 arctan(y/b).Using the double angle formula:cos(2θ) = (1 - tan²θ)/(1 + tan²θ) = (1 - (y²/b²))/(1 + (y²/b²)) = (b² - y²)/(b² + y²)sin(2θ) = 2 tanθ/(1 + tan²θ) = 2(y/b)/(1 + y²/b²) = 2by/(b² + y²)Therefore, coordinates of D are:x = sqrt(b² + y²) * (b² - y²)/(b² + y²) = (b² - y²)/sqrt(b² + y²)y = sqrt(b² + y²) * 2by/(b² + y²) = 2by / sqrt(b² + y²)So, D is at ((b² - y²)/sqrt(b² + y²), 2by / sqrt(b² + y²))Now, we need to find point H, which is the foot of the altitude from D to AC.Since AC is from A(0,0) to C(b, y), the line AC can be parameterized as t*(b, y), where t ranges from 0 to 1.The altitude from D to AC is perpendicular to AC. So, the vector AC is (b, y), and the direction vector is (b, y). The slope of AC is y/b, so the slope of the altitude DH is -b/y.Point D is at ((b² - y²)/sqrt(b² + y²), 2by / sqrt(b² + y²)). Let me denote this as (d_x, d_y).The equation of line AC is y = (y/b)x.The equation of altitude DH is y - d_y = (-b/y)(x - d_x)We need to find the intersection point H between AC and DH.So, set y = (y/b)x into the equation of DH:(y/b)x - d_y = (-b/y)(x - d_x)Multiply both sides by y to eliminate denominators:y*(y/b)x - y*d_y = -b*(x - d_x)Simplify:(y²/b)x - y d_y = -b x + b d_xBring all terms to one side:(y²/b)x + b x - y d_y - b d_x = 0Factor x:x(y²/b + b) = y d_y + b d_xSo,x = (y d_y + b d_x) / (y²/b + b)Simplify denominator:y²/b + b = (y² + b²)/bSo,x = (y d_y + b d_x) * (b / (y² + b²)) = b(y d_y + b d_x)/(y² + b²)Similarly, y = (y/b)x = (y/b) * [b(y d_y + b d_x)/(y² + b²)] = y(y d_y + b d_x)/(y² + b²)Now, let's compute d_x and d_y:d_x = (b² - y²)/sqrt(b² + y²)d_y = 2by / sqrt(b² + y²)So,y d_y = y * 2by / sqrt(b² + y²) = 2b y² / sqrt(b² + y²)b d_x = b * (b² - y²)/sqrt(b² + y²) = b(b² - y²)/sqrt(b² + y²)Therefore,y d_y + b d_x = [2b y² + b(b² - y²)] / sqrt(b² + y²) = [2b y² + b³ - b y²] / sqrt(b² + y²) = [b³ + b y²] / sqrt(b² + y²) = b(b² + y²)/sqrt(b² + y²) = b sqrt(b² + y²)Therefore,x = b * [b sqrt(b² + y²)] / (y² + b²) = b² sqrt(b² + y²) / (b² + y²) = b² / sqrt(b² + y²)Similarly,y = y * [b sqrt(b² + y²)] / (b² + y²) = b y sqrt(b² + y²) / (b² + y²) = b y / sqrt(b² + y²)So, point H is at (b² / sqrt(b² + y²), b y / sqrt(b² + y²))Now, we need to find the equation of line BH.Point B is at (b,0), and point H is at (b² / sqrt(b² + y²), b y / sqrt(b² + y²))Let me denote sqrt(b² + y²) as k for simplicity.So, H is at (b²/k, b y/k)The vector from B to H is (b²/k - b, b y/k - 0) = (b(b/k - 1), b y/k)So, the parametric equation of BH can be written as:x = b + t*(b(b/k - 1))y = 0 + t*(b y/k)Where t ranges from 0 to 1.We need to find where this line intersects CD.Point C is at (b, y), and point D is at ((b² - y²)/k, 2b y /k)So, segment CD goes from (b, y) to ((b² - y²)/k, 2b y /k)Let me parameterize CD with parameter s from 0 to 1:x = b + s*((b² - y²)/k - b) = b + s*( (b² - y² - b k)/k )y = y + s*(2b y /k - y) = y + s*( (2b y - y k)/k )Simplify:x = b + s*( (b² - y² - b k)/k )y = y + s*( y(2b - k)/k )Now, we need to find t and s such that:b + t*(b(b/k - 1)) = b + s*( (b² - y² - b k)/k )andt*(b y/k) = y + s*( y(2b - k)/k )Let me simplify the first equation:b + t*(b(b/k - 1)) = b + s*( (b² - y² - b k)/k )Subtract b from both sides:t*(b(b/k - 1)) = s*( (b² - y² - b k)/k )Similarly, the second equation:t*(b y/k) = y + s*( y(2b - k)/k )Let me solve the second equation for t:t = [ y + s*( y(2b - k)/k ) ] / (b y/k ) = [ y + s y(2b - k)/k ] * (k / b y ) = [1 + s(2b - k)/k ] * (k / b )= (k / b ) + s(2b - k)/bNow, substitute t into the first equation:t*(b(b/k - 1)) = s*( (b² - y² - b k)/k )Substitute t:[ (k / b ) + s(2b - k)/b ] * b(b/k - 1) = s*( (b² - y² - b k)/k )Simplify:[ (k / b ) + s(2b - k)/b ] * (b/k - 1) = s*( (b² - y² - b k)/k )Let me compute (b/k - 1):b/k - 1 = (b - k)/kSo,[ (k / b ) + s(2b - k)/b ] * (b - k)/k = s*( (b² - y² - b k)/k )Multiply both sides by k to eliminate denominators:[ (k / b ) + s(2b - k)/b ] * (b - k) = s*(b² - y² - b k )Let me expand the left-hand side:(k / b )(b - k) + s(2b - k)/b (b - k) = s*(b² - y² - b k )Simplify each term:First term: (k / b )(b - k) = k - k²/bSecond term: s(2b - k)(b - k)/bSo,k - k²/b + s(2b - k)(b - k)/b = s*(b² - y² - b k )Let me compute (2b - k)(b - k):= 2b(b - k) - k(b - k) = 2b² - 2b k - b k + k² = 2b² - 3b k + k²Therefore, the equation becomes:k - k²/b + s(2b² - 3b k + k²)/b = s*(b² - y² - b k )Let me bring all terms to one side:k - k²/b + s(2b² - 3b k + k²)/b - s*(b² - y² - b k ) = 0Factor s:k - k²/b + s[ (2b² - 3b k + k²)/b - (b² - y² - b k ) ] = 0Simplify the expression inside the brackets:(2b² - 3b k + k²)/b - (b² - y² - b k ) = (2b² - 3b k + k²)/b - b² + y² + b k= 2b - 3k + k²/b - b² + y² + b k= (2b + b k) - 3k + k²/b - b² + y²= b(2 + k) - 3k + k²/b - b² + y²This seems complicated. Maybe I need to express y² in terms of k.Recall that k = sqrt(b² + y²), so y² = k² - b²Substitute y²:= b(2 + k) - 3k + k²/b - b² + (k² - b²)= 2b + b k - 3k + k²/b - b² + k² - b²= 2b + b k - 3k + k²/b + k² - 2b²This is getting too messy. Maybe there's a better approach.Alternatively, since we've expressed everything in terms of b and y, perhaps we can choose specific values for b and y to simplify the calculations. Let's assume b = 1 and y = 1 for simplicity.So, let b = 1, y = 1.Then, k = sqrt(1 + 1) = sqrt(2)Point C is at (1,1)Point D is at ((1 - 1)/sqrt(2), 2*1*1 / sqrt(2)) = (0, 2/sqrt(2)) = (0, sqrt(2))Wait, that can't be right because if D is at (0, sqrt(2)), then AD is from (0,0) to (0, sqrt(2)), which is along the y-axis, and AC is from (0,0) to (1,1), which is along the line y = x. So, angle BAC is 45 degrees, and angle CAD is also 45 degrees, making angle BAD = 90 degrees.But in this case, point D is at (0, sqrt(2)), and point C is at (1,1). So, segment CD goes from (1,1) to (0, sqrt(2)).Point H is the foot of the altitude from D to AC. Since AC is the line y = x, the altitude from D(0, sqrt(2)) to AC is the line perpendicular to y = x, which has slope -1. The equation of DH is y - sqrt(2) = -1(x - 0), so y = -x + sqrt(2).Intersection with AC (y = x):x = -x + sqrt(2) => 2x = sqrt(2) => x = sqrt(2)/2So, H is at (sqrt(2)/2, sqrt(2)/2)Now, line BH goes from B(1,0) to H(sqrt(2)/2, sqrt(2)/2)Let me find the parametric equations for BH.Let parameter t go from 0 to 1:x(t) = 1 + t(sqrt(2)/2 - 1) = 1 - t(1 - sqrt(2)/2)y(t) = 0 + t(sqrt(2)/2 - 0) = t sqrt(2)/2We need to find where this line intersects CD.Segment CD goes from C(1,1) to D(0, sqrt(2))Parametrize CD with parameter s from 0 to 1:x(s) = 1 - s(1 - 0) = 1 - sy(s) = 1 + s(sqrt(2) - 1)Now, set x(t) = x(s) and y(t) = y(s):1 - t(1 - sqrt(2)/2) = 1 - sandt sqrt(2)/2 = 1 + s(sqrt(2) - 1)From the first equation:1 - t(1 - sqrt(2)/2) = 1 - s => t(1 - sqrt(2)/2) = sFrom the second equation:t sqrt(2)/2 = 1 + s(sqrt(2) - 1)Substitute s = t(1 - sqrt(2)/2):t sqrt(2)/2 = 1 + t(1 - sqrt(2)/2)(sqrt(2) - 1)Let me compute (1 - sqrt(2)/2)(sqrt(2) - 1):= 1*sqrt(2) - 1*1 - (sqrt(2)/2)*sqrt(2) + (sqrt(2)/2)*1= sqrt(2) - 1 - (2/2) + sqrt(2)/2= sqrt(2) - 1 - 1 + sqrt(2)/2= (sqrt(2) + sqrt(2)/2) - 2= (3 sqrt(2)/2) - 2So, the equation becomes:t sqrt(2)/2 = 1 + t(3 sqrt(2)/2 - 2)Bring all terms to one side:t sqrt(2)/2 - t(3 sqrt(2)/2 - 2) = 1Factor t:t [ sqrt(2)/2 - 3 sqrt(2)/2 + 2 ] = 1Simplify inside the brackets:sqrt(2)/2 - 3 sqrt(2)/2 = (-2 sqrt(2))/2 = -sqrt(2)So,t (-sqrt(2) + 2) = 1Therefore,t = 1 / (2 - sqrt(2)) = (2 + sqrt(2)) / ( (2 - sqrt(2))(2 + sqrt(2)) ) = (2 + sqrt(2))/ (4 - 2) ) = (2 + sqrt(2))/2 = 1 + (sqrt(2)/2)But t = 1 + (sqrt(2)/2) > 1, which is outside the range [0,1] for the parameter t on segment BH. This suggests that the intersection point N is beyond point H on the extension of BH, which contradicts the convexity of the quadrilateral since N should lie on segment CD.This indicates a mistake in our assumption or calculations. Let me check the coordinates again.Wait, when b = 1 and y = 1, point D is at (0, sqrt(2)), which is correct. Point C is at (1,1). So, segment CD is from (1,1) to (0, sqrt(2)). The line BH goes from (1,0) to (sqrt(2)/2, sqrt(2)/2). When we parameterize BH, t=0 is at B(1,0), and t=1 is at H(sqrt(2)/2, sqrt(2)/2). The intersection with CD occurs at t > 1, which is outside the segment BH, meaning that within the segment BH, it doesn't intersect CD. But in reality, in the convex quadrilateral, BH should intersect CD somewhere between C and D. Therefore, perhaps my coordinate system is still flawed.Alternatively, maybe choosing b = 1 and y = 1 is not suitable because it leads to a degenerate case where the intersection is outside the segment. Let me try different values.Let me choose b = 2 and y = 1.Then, k = sqrt(4 + 1) = sqrt(5)Point C is at (2,1)Point D is at ((4 - 1)/sqrt(5), 2*2*1 / sqrt(5)) = (3/sqrt(5), 4/sqrt(5))Point H is the foot of the altitude from D to AC.Line AC is from (0,0) to (2,1), so its slope is 1/2. The altitude from D has slope -2.Equation of AC: y = (1/2)xEquation of DH: y - 4/sqrt(5) = -2(x - 3/sqrt(5))Find intersection H:(1/2)x = -2x + 6/sqrt(5) + 4/sqrt(5)(1/2)x + 2x = 10/sqrt(5)(5/2)x = 10/sqrt(5)x = (10/sqrt(5)) * (2/5) = (20)/(5 sqrt(5)) = 4/sqrt(5)Then, y = (1/2)(4/sqrt(5)) = 2/sqrt(5)So, H is at (4/sqrt(5), 2/sqrt(5))Now, line BH goes from B(2,0) to H(4/sqrt(5), 2/sqrt(5))Parametric equations for BH:x(t) = 2 + t(4/sqrt(5) - 2) = 2 + t(4/sqrt(5) - 2)y(t) = 0 + t(2/sqrt(5) - 0) = t(2/sqrt(5))Segment CD goes from C(2,1) to D(3/sqrt(5), 4/sqrt(5))Parametrize CD with parameter s:x(s) = 2 + s(3/sqrt(5) - 2) = 2 + s(3/sqrt(5) - 2)y(s) = 1 + s(4/sqrt(5) - 1)Set x(t) = x(s) and y(t) = y(s):2 + t(4/sqrt(5) - 2) = 2 + s(3/sqrt(5) - 2)andt(2/sqrt(5)) = 1 + s(4/sqrt(5) - 1)From the first equation:t(4/sqrt(5) - 2) = s(3/sqrt(5) - 2)From the second equation:t(2/sqrt(5)) = 1 + s(4/sqrt(5) - 1)Let me solve the first equation for t:t = s(3/sqrt(5) - 2)/(4/sqrt(5) - 2)Let me compute the denominator and numerator:Numerator: 3/sqrt(5) - 2 ≈ 3/2.236 - 2 ≈ 1.342 - 2 ≈ -0.658Denominator: 4/sqrt(5) - 2 ≈ 4/2.236 - 2 ≈ 1.789 - 2 ≈ -0.211So, t ≈ s*(-0.658)/(-0.211) ≈ s*3.118Now, substitute t into the second equation:(3.118 s)(2/sqrt(5)) = 1 + s(4/sqrt(5) - 1)Compute 2/sqrt(5) ≈ 0.894So,3.118 s * 0.894 ≈ 1 + s(1.789 - 1)≈ 1 + s(0.789)Compute 3.118 * 0.894 ≈ 2.789So,2.789 s ≈ 1 + 0.789 sSubtract 0.789 s:2.789 s - 0.789 s ≈ 12 s ≈ 1s ≈ 0.5Then, t ≈ 3.118 * 0.5 ≈ 1.559But t ≈ 1.559 > 1, which again is outside the segment BH. This suggests that in this coordinate system, the intersection point N is beyond H on the extension of BH, which contradicts the convexity.This indicates that perhaps my approach is flawed, or maybe the ratio is always 1:1 regardless of the specific values, meaning that BH bisects CD.Alternatively, perhaps there's a symmetry or congruence that I'm missing.Given that AC = AD and angle BAC = angle CAD, triangle ACD is isosceles with AC = AD, and AC is the angle bisector. Also, DH is the altitude, so H is the midpoint of AC if triangle ACD is isosceles, but wait, in an isosceles triangle, the altitude from the apex (D) to the base (AC) is also the median and the angle bisector, so H would be the midpoint of AC.Wait, if H is the midpoint of AC, then in triangle ACD, DH is the median and altitude, so AC is bisected by DH at H.Given that, then AC is divided equally by H.Now, considering triangle ABC, which is right-angled at B, and AC is the hypotenuse.If we can show that triangles ABC and AHD are congruent, then BH would be equal to AH, making triangle ABH isosceles, and thus BH would bisect CD.Wait, let's see:In triangle ABC, AB is perpendicular to BC, and AC is the hypotenuse.In triangle AHD, AH is part of AC, and HD is the altitude, so triangle AHD is also right-angled at H.If AC = AD, and if AB = AH, then triangles ABC and AHD would be congruent by hypotenuse-leg theorem.But how do we know AB = AH?Wait, since H is the midpoint of AC (because in isosceles triangle ACD, the altitude from D is also the median), then AH = HC = AC/2.But AC = AD, so AH = AD/2.But unless AB = AD/2, which isn't necessarily given, this might not hold.Alternatively, maybe there's a reflection or rotation symmetry.Given that AC = AD and angle BAC = angle CAD, perhaps reflecting point B over the angle bisector AC would map it to point D.If that's the case, then BH would map to DH, and their intersection point N would be the midpoint of CD.Therefore, the ratio CN:ND would be 1:1.Alternatively, considering the properties of angle bisectors and congruent triangles, it's plausible that BH divides CD into two equal parts.Given the complexity of the coordinate approach leading to contradictions, and considering the symmetry and properties of the given triangles, it's reasonable to conclude that BH divides CD in the ratio 1:1.

💡Okay, so I have this problem here about vectors. It says that vectors a and b are both unit vectors, which means their magnitudes are 1. The angle between them is 60 degrees. I need to find the magnitude of the vector a minus 2 times vector b. The options are A: 2, B: 4 minus square root of 3, C: square root of 13, and D: square root of 3.Hmm, let me think. I remember that to find the magnitude of a vector difference, I can use the formula that involves the dot product. The formula is |a - 2b| squared equals (a - 2b) dot (a - 2b). That should give me the square of the magnitude, and then I can take the square root at the end.So, expanding that dot product, it should be a dot a minus 2 times a dot b minus 2 times b dot a plus 4 times b dot b. Wait, actually, no, that's not quite right. Let me write it out step by step.First, (a - 2b) dot (a - 2b) equals a dot a minus 2a dot b minus 2b dot a plus 4b dot b. But since dot product is commutative, a dot b is the same as b dot a, so that simplifies to a dot a minus 4a dot b plus 4b dot b.Now, since a and b are unit vectors, a dot a is 1 and b dot b is also 1. So that part is straightforward. The tricky part is finding a dot b. I remember that the dot product of two vectors is equal to the product of their magnitudes times the cosine of the angle between them. Since both are unit vectors, their magnitudes are 1, so a dot b is just cos(theta), where theta is 60 degrees.Cosine of 60 degrees is 0.5, so a dot b is 0.5. Plugging that back into the equation, we have:|a - 2b| squared equals 1 minus 4 times 0.5 plus 4 times 1. Let me calculate that:1 minus 4 times 0.5 is 1 minus 2, which is -1. Then, 4 times 1 is 4. So, -1 plus 4 is 3. Therefore, |a - 2b| squared is 3, which means |a - 2b| is the square root of 3.Looking back at the options, D is square root of 3, so that should be the answer. But wait, let me double-check my calculations to make sure I didn't make a mistake.Starting again, (a - 2b) dot (a - 2b) is a dot a minus 2a dot b minus 2b dot a plus 4b dot b. Since a dot a is 1, b dot b is 1, and a dot b is 0.5. So, substituting, it's 1 - 2*(0.5) - 2*(0.5) + 4*1. That simplifies to 1 - 1 - 1 + 4, which is 1 - 1 is 0, 0 -1 is -1, -1 +4 is 3. Yep, same result.So, the magnitude is square root of 3, which is option D. I think that's correct.**Final Answer**boxed{D}

💡Okay, so I have this problem here where a sequence is defined recursively. It starts with x₀ = 5, and then each subsequent term is given by the formula x_{n+1} = (x_n² + 5x_n + 4)/(x_n + 6). I need to find the smallest positive integer m such that x_m ≤ 4 + 1/(2²⁰). Then, I have to figure out which interval m falls into from the given options.Hmm, let me try to understand what's going on here. The sequence starts at 5, and each term is calculated using that fraction. I guess I need to see how this sequence behaves. Does it converge? If so, to what? And how quickly does it approach that limit?First, maybe I should try computing the first few terms to get a sense of the pattern. Let's see:x₀ = 5.x₁ = (5² + 5*5 + 4)/(5 + 6) = (25 + 25 + 4)/11 = 54/11 ≈ 4.9091.x₂ = (x₁² + 5x₁ + 4)/(x₁ + 6). Let me compute that:x₁ ≈ 4.9091, so x₁² ≈ 24.1, 5x₁ ≈ 24.5455, so numerator ≈ 24.1 + 24.5455 + 4 ≈ 52.6455. Denominator is x₁ + 6 ≈ 4.9091 + 6 ≈ 10.9091. So x₂ ≈ 52.6455 / 10.9091 ≈ 4.825.Hmm, it's decreasing. Let me do one more:x₂ ≈ 4.825, so x₂² ≈ 23.29, 5x₂ ≈ 24.125, so numerator ≈ 23.29 + 24.125 + 4 ≈ 51.415. Denominator is x₂ + 6 ≈ 4.825 + 6 ≈ 10.825. So x₃ ≈ 51.415 / 10.825 ≈ 4.75.It's still decreasing. So it seems like the sequence is decreasing from 5 towards some limit. Maybe it converges to 4? Let me check.If the sequence converges to some limit L, then taking limits on both sides of the recursive formula:L = (L² + 5L + 4)/(L + 6).Multiply both sides by (L + 6):L(L + 6) = L² + 5L + 4.Expanding left side: L² + 6L = L² + 5L + 4.Subtract L² from both sides: 6L = 5L + 4.Subtract 5L: L = 4.So yes, the sequence converges to 4. That makes sense. So as n increases, x_n approaches 4. But we need to find when it gets within 1/(2²⁰) of 4, which is a very small number, about 9.5367e-7.So we need to find the smallest m such that x_m ≤ 4 + 1/(2²⁰). Since the sequence is decreasing, once it passes below 4 + 1/(2²⁰), it will stay there. So we need to find when it first gets that close to 4.But computing each term one by one up to that point would be impractical because m could be very large. So I need a smarter way to estimate m.Maybe I can analyze the recursive formula more carefully. Let me define y_n = x_n - 4. Then, since x_n = y_n + 4, substituting into the recursive formula:x_{n+1} = ( (y_n + 4)² + 5(y_n + 4) + 4 ) / (y_n + 4 + 6 )Let me expand the numerator:(y_n + 4)² = y_n² + 8y_n + 165(y_n + 4) = 5y_n + 20Adding the 4: total numerator is y_n² + 8y_n + 16 + 5y_n + 20 + 4 = y_n² + 13y_n + 40Denominator is y_n + 10.So, x_{n+1} = (y_n² + 13y_n + 40)/(y_n + 10)But x_{n+1} = y_{n+1} + 4, so:y_{n+1} + 4 = (y_n² + 13y_n + 40)/(y_n + 10)Subtract 4 from both sides:y_{n+1} = (y_n² + 13y_n + 40)/(y_n + 10) - 4Let me compute that:= (y_n² + 13y_n + 40 - 4(y_n + 10)) / (y_n + 10)= (y_n² + 13y_n + 40 - 4y_n - 40) / (y_n + 10)Simplify numerator:y_n² + 9y_nSo y_{n+1} = (y_n² + 9y_n)/(y_n + 10)Factor numerator:y_n(y_n + 9)/(y_n + 10)So y_{n+1} = y_n(y_n + 9)/(y_n + 10)Hmm, that's interesting. So y_{n+1} = y_n * (y_n + 9)/(y_n + 10)Since y_n is positive (because x_n > 4 and decreasing towards 4), we can analyze this recursion.Let me see if I can write this as y_{n+1} = y_n * (1 - 1/(y_n + 10))Wait, let's see:(y_n + 9)/(y_n + 10) = 1 - 1/(y_n + 10)Yes, because (y_n + 9) = (y_n + 10) - 1, so (y_n + 9)/(y_n + 10) = 1 - 1/(y_n + 10)Therefore, y_{n+1} = y_n * (1 - 1/(y_n + 10))So, y_{n+1} = y_n - y_n/(y_n + 10)This shows that y_{n+1} < y_n, which we already knew because the sequence is decreasing.But perhaps we can approximate this recursion for small y_n. Since we're interested in when y_n becomes very small (on the order of 1/(2²⁰)), maybe we can approximate the recursion for small y_n.When y_n is small, y_n + 10 ≈ 10, so y_{n+1} ≈ y_n - y_n/10 = (9/10)y_nSo, for small y_n, the sequence behaves roughly like y_{n+1} ≈ (9/10)y_n, which is a geometric sequence with ratio 9/10.Therefore, the decay is roughly exponential with base 9/10.But wait, when y_n is small, y_n + 10 ≈ 10, so 1/(y_n + 10) ≈ 1/10, so y_{n+1} ≈ y_n*(1 - 1/10) = (9/10)y_n.But actually, when y_n is not that small, the approximation isn't as good. So maybe we can model this as a differential equation for better approximation.Let me think of y_n as a function y(n), and approximate the difference equation with a differential equation.We have y_{n+1} - y_n = - y_n/(y_n + 10)So, dy/dn ≈ - y/(y + 10)This is a separable differential equation:dy / (y/(y + 10)) = - dnWhich is:(y + 10)/y dy = - dnIntegrate both sides:∫(1 + 10/y) dy = - ∫ dnWhich gives:y + 10 ln y = -n + CSo, y + 10 ln y = -n + CWe can use this to approximate y(n). But we need to find the constant C using initial conditions.At n = 0, y_0 = x_0 - 4 = 5 - 4 = 1.So, plug in y = 1, n = 0:1 + 10 ln 1 = 0 + C => 1 + 0 = C => C = 1Therefore, the approximate equation is:y + 10 ln y = -n + 1We need to solve for n when y = 1/(2²⁰). Let's denote y = ε = 1/(2²⁰).So,ε + 10 ln ε = -n + 1Therefore,n = 1 - ε - 10 ln εSince ε is very small (≈ 9.5367e-7), ε is negligible compared to 10 ln ε.So,n ≈ -10 ln εCompute ln ε:ln(1/(2²⁰)) = -20 ln 2 ≈ -20 * 0.6931 ≈ -13.862Therefore,n ≈ -10*(-13.862) ≈ 138.62So, approximately, n ≈ 139.But wait, this is an approximation. The actual recursion might decay slightly faster or slower.But let's see. The options given are intervals:(A) [9,26](B) [27,80](C) [81,242](D) [243,728](E) [729, ∞)So, 139 is in [81,242], which is option C.But let me check if this approximation is valid. Because when y_n is not extremely small, the differential equation approximation might not be accurate.Alternatively, maybe we can bound the sequence.From the recursion:y_{n+1} = y_n - y_n/(y_n + 10)So, y_{n+1} = y_n * (1 - 1/(y_n + 10))Note that 1 - 1/(y_n + 10) = (y_n + 9)/(y_n + 10)So, y_{n+1} = y_n * (y_n + 9)/(y_n + 10)We can see that for y_n > 0, (y_n + 9)/(y_n + 10) < 1, so y_{n+1} < y_n, which we already knew.Moreover, for y_n ≤ 10, (y_n + 9)/(y_n + 10) ≥ (1 + 9)/(1 + 10) = 10/11 ≈ 0.9091Wait, no. Wait, when y_n is small, say y_n = 1, then (1 + 9)/(1 + 10) = 10/11 ≈ 0.9091When y_n is larger, say y_n = 10, then (10 + 9)/(10 + 10) = 19/20 = 0.95So, actually, the ratio (y_n + 9)/(y_n + 10) is increasing as y_n increases.Wait, so when y_n is small, the ratio is about 10/11 ≈ 0.9091, and when y_n is larger, it approaches 1.So, the decay rate is faster when y_n is smaller.Therefore, the sequence y_n decreases faster when it's smaller, so the decay accelerates as y_n gets smaller.Therefore, the differential equation approximation might underestimate the decay because it assumes a constant decay rate, but in reality, the decay rate increases as y_n decreases.Therefore, the actual n needed might be less than the approximation of ~139.But let's see. Maybe we can bound it.Let me consider that for y_n ≤ 1, (y_n + 9)/(y_n + 10) ≥ 10/11.So, y_{n+1} ≥ y_n * 10/11Therefore, y_n ≥ y_0 * (10/11)^nBut y_0 = 1, so y_n ≥ (10/11)^nWe need y_n ≤ 1/(2²⁰). So,(10/11)^n ≤ 1/(2²⁰)Take natural logs:n ln(10/11) ≤ -20 ln 2Since ln(10/11) is negative, we can divide both sides:n ≥ (-20 ln 2)/ln(10/11)Compute:ln(10/11) ≈ ln(0.9091) ≈ -0.09531So,n ≥ (-20 * 0.6931)/(-0.09531) ≈ (13.862)/(0.09531) ≈ 145.4So, n ≥ 146Wait, that's interesting. So, using the lower bound, we get n ≥ 146.But earlier, the differential equation gave n ≈ 139.Hmm, so the actual n is somewhere between 139 and 146.But wait, let's check the upper bound.We can also note that for y_n ≥ 1, (y_n + 9)/(y_n + 10) ≤ (1 + 9)/(1 + 10) = 10/11 ≈ 0.9091Wait, no, actually, when y_n is larger, the ratio is larger. Wait, when y_n is larger, (y_n + 9)/(y_n + 10) approaches 1.Wait, perhaps I need to think differently.Wait, maybe I can consider that for y_n ≤ 1, (y_n + 9)/(y_n + 10) ≥ 10/11, so y_{n+1} ≥ y_n * 10/11Therefore, y_n ≥ y_0 * (10/11)^nBut we need y_n ≤ 1/(2²⁰), so:(10/11)^n ≤ y_n ≤ 1/(2²⁰)Wait, no, that's not correct. Because y_n is decreasing, so y_n ≤ y_0 * (10/11)^nWait, no, actually, since y_{n+1} ≥ y_n * 10/11, then y_n ≥ y_0 * (10/11)^nBut we need y_n ≤ 1/(2²⁰), so:y_0 * (10/11)^n ≤ 1/(2²⁰)So,(10/11)^n ≤ 1/(2²⁰)Which is the same as before, leading to n ≥ 145.4, so n ≥ 146.But earlier, the differential equation suggested n ≈ 139.So, which one is correct? Or perhaps neither, but the actual n is somewhere in between.But let's think about the behavior. When y_n is large, say y_n = 1, the ratio is 10/11 ≈ 0.9091, but as y_n decreases, the ratio increases towards 1, meaning the decay slows down.Wait, that contradicts my earlier thought. Wait, no, when y_n is small, the ratio (y_n + 9)/(y_n + 10) ≈ 9/10, which is smaller than 10/11, so actually, the decay rate is faster when y_n is small.Wait, no, 9/10 = 0.9, which is less than 10/11 ≈ 0.9091, so actually, when y_n is small, the ratio is smaller, meaning y_{n+1} is smaller, so the decay is faster.Wait, that seems conflicting. Let me clarify.When y_n is small, say y_n approaches 0, then (y_n + 9)/(y_n + 10) approaches 9/10 = 0.9.When y_n is 1, it's 10/11 ≈ 0.9091.When y_n is 10, it's 19/20 = 0.95.So, as y_n increases, the ratio (y_n + 9)/(y_n + 10) increases, approaching 1 as y_n becomes large.Therefore, when y_n is small, the ratio is smaller (0.9), leading to faster decay, and when y_n is larger, the ratio is larger (closer to 1), leading to slower decay.Therefore, the decay is faster when y_n is small, so the sequence decreases more rapidly when it's near 4.Therefore, the differential equation approximation, which assumes a constant decay rate, might overestimate the required n because in reality, once y_n becomes small, the decay accelerates.Wait, but in our earlier calculation, using the differential equation, we got n ≈ 139, and using the lower bound (10/11)^n ≤ y_n, we got n ≥ 146.But since the decay is faster when y_n is small, the actual n needed should be less than 146, perhaps closer to 139.But let's see. Maybe we can use a better approximation.Alternatively, perhaps we can model the recursion as y_{n+1} = y_n - y_n/(y_n + 10)Let me consider that for small y_n, y_n + 10 ≈ 10, so y_{n+1} ≈ y_n - y_n/10 = (9/10)y_nSo, for small y_n, it's approximately a geometric sequence with ratio 9/10.But when y_n is not that small, say y_n = 1, the ratio is 10/11 ≈ 0.9091, which is slightly larger than 9/10.So, perhaps we can model the decay as a combination of these.Alternatively, maybe we can use the approximation that y_n decreases roughly by a factor of 10/11 each time until y_n becomes small, and then by a factor of 9/10.But this might complicate things.Alternatively, perhaps we can use the differential equation approximation but adjust it.Wait, earlier, we had:y + 10 ln y = -n + 1We can use this to approximate y(n). Let's set y = ε = 1/(2²⁰) ≈ 9.5367e-7So,ε + 10 ln ε = -n + 1Compute ln ε:ln(1/(2²⁰)) = -20 ln 2 ≈ -13.86294361So,ε + 10*(-13.86294361) ≈ -n + 1ε is negligible, so:-138.6294361 ≈ -n + 1Therefore,n ≈ 138.6294361 + 1 ≈ 139.6294361So, n ≈ 139.63So, approximately 140.But since the differential equation is an approximation, and the actual recursion might decay slightly faster or slower, we can consider that m is around 140.Given the options, 140 falls into interval [81,242], which is option C.But let me check if this is accurate.Alternatively, perhaps I can compute the number of steps required using the recursion.But computing 140 steps manually is impractical. Alternatively, maybe I can use the fact that the recursion can be transformed into a telescoping product.From the recursion:y_{n+1} = y_n * (y_n + 9)/(y_n + 10)Let me write this as:y_{n+1}/(y_n + 9) = y_n/(y_n + 10)Wait, not sure.Alternatively, perhaps consider the reciprocal:1/y_{n+1} = (y_n + 10)/(y_n(y_n + 9))But not sure.Alternatively, perhaps consider the difference 1/y_{n+1} - 1/y_n.Let me compute:1/y_{n+1} = (y_n + 10)/(y_n(y_n + 9)) = (y_n + 10)/(y_n² + 9y_n)But not sure.Alternatively, perhaps consider the difference 1/y_{n+1} - 1/y_n.Compute:1/y_{n+1} - 1/y_n = (y_n + 10)/(y_n(y_n + 9)) - 1/y_n= [ (y_n + 10) - (y_n + 9) ] / (y_n(y_n + 9))= (1)/(y_n(y_n + 9))So,1/y_{n+1} - 1/y_n = 1/(y_n(y_n + 9))This is a telescoping series.So, summing from n=0 to N-1:Sum_{n=0}^{N-1} [1/y_{n+1} - 1/y_n] = Sum_{n=0}^{N-1} 1/(y_n(y_n + 9))Left side telescopes to 1/y_N - 1/y_0Right side is Sum_{n=0}^{N-1} 1/(y_n(y_n + 9))But this seems complicated because y_n depends on previous terms.Alternatively, perhaps approximate the sum.But since y_n decreases, maybe we can bound the sum.Note that 1/(y_n(y_n + 9)) ≤ 1/(y_n * 9) = 1/(9y_n)Similarly, 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ ...Wait, perhaps not helpful.Alternatively, since y_n is decreasing, y_n ≥ y_{n+1}, so 1/(y_n(y_n + 9)) ≤ 1/(y_{n+1}(y_n + 9))But not sure.Alternatively, perhaps use integral approximation.Let me think of the sum as an approximation of an integral.But this might be getting too complicated.Alternatively, perhaps use the differential equation approximation, which gave n ≈ 140, and given the options, 140 is in [81,242], so the answer is C.But let me see if I can find a better approximation.Wait, earlier, I had:y + 10 ln y = -n + 1So, solving for n:n = 1 - y - 10 ln yWe need y = 1/(2²⁰) ≈ 9.5367e-7Compute:n ≈ 1 - 9.5367e-7 - 10*(-13.86294361)≈ 1 - 0 + 138.6294361≈ 139.6294361So, n ≈ 139.63Therefore, m is approximately 140.But since m must be an integer, and we need x_m ≤ 4 + 1/(2²⁰), which is y_m ≤ 1/(2²⁰), so m is the smallest integer such that y_m ≤ 1/(2²⁰). Since y_n is decreasing, once y_n crosses below 1/(2²⁰), it stays there. So, m is the smallest integer where y_m ≤ 1/(2²⁰). Given the approximation, m ≈ 140.But let's check if this is accurate.Alternatively, perhaps we can use the fact that y_{n+1} = y_n * (y_n + 9)/(y_n + 10)Let me consider that for y_n small, say y_n << 10, then y_{n+1} ≈ y_n * 9/10So, y_n ≈ y_0 * (9/10)^nBut y_0 = 1, so y_n ≈ (9/10)^nWe need y_n ≤ 1/(2²⁰)So,(9/10)^n ≤ 1/(2²⁰)Take natural logs:n ln(9/10) ≤ -20 ln 2Since ln(9/10) ≈ -0.10536So,n ≥ (-20 ln 2)/(-0.10536) ≈ (13.86294)/0.10536 ≈ 131.5So, n ≥ 132But this is a different approximation, giving n ≈ 132.Wait, so now I have two different approximations: one gave n ≈ 139.6, another gave n ≈ 132.Hmm.Alternatively, perhaps the actual decay is somewhere in between.But given that the differential equation gave n ≈ 140, and the geometric approximation with ratio 9/10 gave n ≈ 132, the actual n is likely between 132 and 140.But given that the options are intervals, and 132 to 140 is within [81,242], which is option C, so the answer is C.But let me see if I can get a better estimate.Wait, perhaps I can use the approximation:n ≈ (10/9) * (1/y_n - 1/y_0)Wait, where did I get that?Wait, from the differential equation:y + 10 ln y = -n + 1So, rearranged:n = 1 - y - 10 ln ySo, for y = 1/(2²⁰), n ≈ 1 - 0 - 10*(-13.86294) ≈ 1 + 138.6294 ≈ 139.63So, n ≈ 139.63Therefore, m is approximately 140.But let me check if this is accurate.Alternatively, perhaps I can compute the number of steps required using the recursion.But computing 140 steps manually is impractical. Alternatively, perhaps I can use the fact that the recursion can be transformed into a telescoping product.Wait, earlier, I had:1/y_{n+1} - 1/y_n = 1/(y_n(y_n + 9))So, summing from n=0 to N-1:Sum_{n=0}^{N-1} [1/y_{n+1} - 1/y_n] = Sum_{n=0}^{N-1} 1/(y_n(y_n + 9))Left side telescopes to 1/y_N - 1/y_0Right side is Sum_{n=0}^{N-1} 1/(y_n(y_n + 9))But since y_n is decreasing, maybe we can bound the sum.Note that 1/(y_n(y_n + 9)) ≤ 1/(y_n * 9) = 1/(9y_n)Similarly, 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ 1/(y_n(y_n + 9)) ≥ ...Wait, perhaps not helpful.Alternatively, perhaps approximate the sum as an integral.Let me consider y_n as a function y(n), then the sum Sum_{n=0}^{N-1} 1/(y_n(y_n + 9)) ≈ ∫_{y_0}^{y_N} 1/(y(y + 9)) dyCompute the integral:∫ 1/(y(y + 9)) dy = ∫ [1/9(1/y - 1/(y + 9))] dy = (1/9)(ln y - ln(y + 9)) + CSo,Sum ≈ (1/9)(ln y_N - ln(y_N + 9) - ln y_0 + ln(y_0 + 9))But y_N is very small, so ln(y_N + 9) ≈ ln 9, and ln y_N is large negative.So,Sum ≈ (1/9)(ln y_N - ln 9 - ln y_0 + ln(y_0 + 9))But y_0 = 1, so ln(y_0 + 9) = ln 10Thus,Sum ≈ (1/9)(ln y_N - ln 9 - 0 + ln 10) = (1/9)(ln(y_N * 10 / 9))But y_N = 1/(2²⁰), so:Sum ≈ (1/9)(ln(10/(9*2²⁰))) = (1/9)(ln(10/9) - 20 ln 2)Compute:ln(10/9) ≈ 0.1053620 ln 2 ≈ 13.86294So,Sum ≈ (1/9)(0.10536 - 13.86294) ≈ (1/9)(-13.75758) ≈ -1.5286But the left side of the sum is 1/y_N - 1/y_0 = 2²⁰ - 1 ≈ 1,048,576 - 1 ≈ 1,048,575But the right side is approximately -1.5286, which is clearly not matching.Wait, that can't be right. There must be a mistake in the approximation.Wait, perhaps the integral approximation is not valid here because the function y(n) changes too rapidly.Alternatively, perhaps I should not use the integral approximation for this sum.Alternatively, perhaps I can use the differential equation approximation, which gave n ≈ 140, and given the options, 140 is in [81,242], so the answer is C.But let me see if I can find a better way.Wait, perhaps I can use the fact that y_{n+1} = y_n * (y_n + 9)/(y_n + 10)Let me define z_n = 1/y_nThen,z_{n+1} = (y_n + 10)/(y_n(y_n + 9)) = (y_n + 10)/(y_n² + 9y_n)But z_n = 1/y_n, so y_n = 1/z_nThus,z_{n+1} = (1/z_n + 10)/(1/z_n² + 9/z_n) = ( (1 + 10 z_n)/z_n ) / ( (1 + 9 z_n)/z_n² ) = (1 + 10 z_n)/z_n * z_n²/(1 + 9 z_n) = z_n(1 + 10 z_n)/(1 + 9 z_n)So,z_{n+1} = z_n * (1 + 10 z_n)/(1 + 9 z_n)This seems more complicated, but perhaps we can approximate for small z_n.When z_n is large (since y_n is small, z_n is large), we can approximate:z_{n+1} ≈ z_n * (10 z_n)/(9 z_n) = z_n * 10/9So, for large z_n, z_{n+1} ≈ (10/9) z_nWhich is a geometric sequence with ratio 10/9.So, z_n ≈ z_0 * (10/9)^nBut z_0 = 1/y_0 = 1/1 = 1So, z_n ≈ (10/9)^nWe need z_n ≥ 2²⁰So,(10/9)^n ≥ 2²⁰Take natural logs:n ln(10/9) ≥ 20 ln 2Compute:ln(10/9) ≈ 0.1053620 ln 2 ≈ 13.86294So,n ≥ 13.86294 / 0.10536 ≈ 131.5So, n ≥ 132Therefore, m is approximately 132.But earlier, the differential equation gave m ≈ 140.So, which one is correct?Wait, when z_n is large, z_{n+1} ≈ (10/9) z_n, so z_n grows exponentially with base 10/9.But we need z_n ≥ 2²⁰ ≈ 1,048,576So, solving (10/9)^n ≥ 1,048,576Take ln:n ≥ ln(1,048,576)/ln(10/9) ≈ 13.86294 / 0.10536 ≈ 131.5So, n ≈ 132Therefore, m is approximately 132.But earlier, the differential equation gave m ≈ 140.So, which is more accurate?Wait, when z_n is large, the approximation z_{n+1} ≈ (10/9) z_n is better because the higher-order terms become negligible.Therefore, the approximation z_n ≈ (10/9)^n is more accurate for large n, which is what we need because we're looking for n where z_n is very large.Therefore, m ≈ 132.But wait, let's check:If z_n ≈ (10/9)^n, then when n=132,z_132 ≈ (10/9)^132Compute log base 10:log10(z_132) ≈ 132 * log10(10/9) ≈ 132 * 0.045757 ≈ 6.033So, z_132 ≈ 10^6.033 ≈ 10^6 * 10^0.033 ≈ 1,000,000 * 1.08 ≈ 1,080,000Which is slightly larger than 2²⁰ ≈ 1,048,576Therefore, n=132 gives z_n ≈ 1,080,000, which is just above 1,048,576.Therefore, m=132 is the smallest integer such that z_m ≥ 2²⁰, which corresponds to y_m ≤ 1/(2²⁰)Therefore, m=132.But wait, earlier, the differential equation gave m≈140, and the geometric approximation with ratio 10/9 gave m≈132.But which one is more accurate?Given that for large z_n, the recursion z_{n+1} ≈ (10/9) z_n is a better approximation, I think m≈132 is more accurate.But let's see.Wait, let's compute z_n using the recursion:z_{n+1} = z_n * (1 + 10 z_n)/(1 + 9 z_n)For large z_n, this is approximately z_{n+1} ≈ z_n * (10 z_n)/(9 z_n) = (10/9) z_nSo, the approximation is good for large z_n.Therefore, for n where z_n is large, the recursion behaves like z_{n+1} ≈ (10/9) z_nTherefore, the number of steps needed to reach z_n = 2²⁰ is approximately n ≈ log_{10/9}(2²⁰) ≈ 132Therefore, m≈132.But let's check:If we start with z_0=1, then z_1≈10/9, z_2≈(10/9)^2, ..., z_n≈(10/9)^nSo, to reach z_n=2²⁰, n≈log_{10/9}(2²⁰)=20 log_{10/9}(2)Compute log_{10/9}(2)=ln(2)/ln(10/9)≈0.6931/0.10536≈6.579Therefore, n≈20*6.579≈131.58≈132Therefore, m=132.But wait, earlier, the differential equation gave m≈140.So, which one is correct?Well, the differential equation is an approximation that might not be as accurate for the entire range of y_n.Given that for large z_n, the recursion is well-approximated by z_{n+1}≈(10/9) z_n, which gives m≈132, I think this is a better estimate.Therefore, m≈132, which is in the interval [81,242], so the answer is C.But let me see if I can confirm this.Alternatively, perhaps I can use the fact that the recursion for z_n is z_{n+1}=z_n*(1+10 z_n)/(1+9 z_n)Let me compute the first few terms:z_0=1z_1=1*(1+10*1)/(1+9*1)=11/10=1.1z_2=1.1*(1+10*1.1)/(1+9*1.1)=1.1*(12)/(10.9)=1.1*1.100917≈1.211z_3≈1.211*(1+10*1.211)/(1+9*1.211)=1.211*(13.11)/(11.899)≈1.211*1.101≈1.333z_4≈1.333*(1+10*1.333)/(1+9*1.333)=1.333*(14.33)/(12.997)≈1.333*1.102≈1.470z_5≈1.470*(1+10*1.470)/(1+9*1.470)=1.470*(15.7)/(13.23)≈1.470*1.186≈1.740z_6≈1.740*(1+10*1.740)/(1+9*1.740)=1.740*(18.4)/(16.66)≈1.740*1.105≈1.923z_7≈1.923*(1+10*1.923)/(1+9*1.923)=1.923*(20.23)/(18.307)≈1.923*1.105≈2.125z_8≈2.125*(1+10*2.125)/(1+9*2.125)=2.125*(22.25)/(20.125)≈2.125*1.105≈2.350z_9≈2.350*(1+10*2.350)/(1+9*2.350)=2.350*(24.5)/(21.15)≈2.350*1.158≈2.723z_10≈2.723*(1+10*2.723)/(1+9*2.723)=2.723*(28.23)/(24.507)≈2.723*1.152≈3.136Hmm, so z_n is increasing, but the ratio z_{n+1}/z_n is decreasing.Wait, at n=0, z_1/z_0=1.1n=1, z_2/z_1≈1.1n=2, z_3/z_2≈1.1n=3, z_4/z_3≈1.1n=4, z_5/z_4≈1.1n=5, z_6/z_5≈1.1n=6, z_7/z_6≈1.1n=7, z_8/z_7≈1.1n=8, z_9/z_8≈1.1n=9, z_10/z_9≈1.1Wait, so actually, the ratio z_{n+1}/z_n≈1.1 for all n, which is 10/9≈1.1111So, in reality, z_n is growing by approximately 10/9 each time, which is consistent with our earlier approximation.Therefore, z_n≈(10/9)^nTherefore, to reach z_n=2²⁰≈1,048,576, we need n≈log_{10/9}(2²⁰)=20 log_{10/9}(2)≈20*6.579≈131.58≈132Therefore, m≈132But let's see, if we compute z_n using the recursion, how many steps does it take to reach z_n=1,048,576.But computing 132 steps manually is impractical, but we can see that the ratio is roughly 10/9 each time, so z_n≈(10/9)^nTherefore, m≈132But the options are intervals, and 132 is in [81,242], so the answer is C.Therefore, the least positive integer m such that x_m≤4 +1/(2²⁰) lies in the interval [81,242], so the answer is C.

💡Okay, so I have this sequence problem here. The sequence is defined as ( a_1 = a_2 = 1 ), and for ( n geq 1 ), the recurrence relation is ( a_{n+2} - a_n = 1 ). I need to find the sum of the first 100 terms of this sequence. The options given are A: 2600, B: 2550, C: 2651, D: 2652. Hmm, let me try to figure this out step by step.First, I know that ( a_1 ) and ( a_2 ) are both 1. The recurrence relation ( a_{n+2} - a_n = 1 ) suggests that every two steps, the sequence increases by 1. So, maybe the odd and even terms form separate arithmetic sequences?Let me test this idea. Let's write out the first few terms to see the pattern.- ( a_1 = 1 )- ( a_2 = 1 )- ( a_3 = a_1 + 1 = 1 + 1 = 2 )- ( a_4 = a_2 + 1 = 1 + 1 = 2 )- ( a_5 = a_3 + 1 = 2 + 1 = 3 )- ( a_6 = a_4 + 1 = 2 + 1 = 3 )- ( a_7 = a_5 + 1 = 3 + 1 = 4 )- ( a_8 = a_6 + 1 = 3 + 1 = 4 )- ( a_9 = a_7 + 1 = 4 + 1 = 5 )- ( a_{10} = a_8 + 1 = 4 + 1 = 5 )Okay, so it seems like the odd terms are 1, 2, 3, 4, 5,... and the even terms are also 1, 2, 3, 4, 5,... So both the odd and even subsequences are arithmetic sequences starting at 1 with a common difference of 1.That makes sense because the recurrence relation adds 1 every two terms, so each subsequence (odd and even) increases by 1 each time.Now, since the first 100 terms consist of 50 odd terms and 50 even terms, I can calculate the sum of the odd terms and the sum of the even terms separately, then add them together.Let me recall the formula for the sum of an arithmetic series: ( S_n = frac{n}{2} times (2a + (n - 1)d) ), where ( S_n ) is the sum of the first ( n ) terms, ( a ) is the first term, ( d ) is the common difference, and ( n ) is the number of terms.For the odd terms:- First term (( a )) = 1- Common difference (( d )) = 1- Number of terms (( n )) = 50So, the sum of the odd terms (( S_{odd} )) is:( S_{odd} = frac{50}{2} times (2 times 1 + (50 - 1) times 1) )Simplifying:( S_{odd} = 25 times (2 + 49) = 25 times 51 = 1275 )Similarly, for the even terms:- First term (( a )) = 1- Common difference (( d )) = 1- Number of terms (( n )) = 50So, the sum of the even terms (( S_{even} )) is the same as the sum of the odd terms:( S_{even} = 1275 )Therefore, the total sum of the first 100 terms is:( S_{total} = S_{odd} + S_{even} = 1275 + 1275 = 2550 )Looking back at the options, B is 2550, so that should be the answer.Wait, let me double-check my calculations to make sure I didn't make a mistake.Calculating ( S_{odd} ):( S_{odd} = frac{50}{2} times (2 times 1 + 49 times 1) = 25 times (2 + 49) = 25 times 51 )25 times 50 is 1250, and 25 times 1 is 25, so 1250 + 25 = 1275. That seems correct.Same for ( S_{even} ), it's also 1275. Adding them together gives 2550, which is option B.I think that's solid. The key was recognizing that the sequence splits into two separate arithmetic sequences for odd and even terms, each starting at 1 with a common difference of 1. Then, calculating each sum and adding them together gives the total sum of the first 100 terms.Another way to think about it is that each pair of terms (one odd, one even) increases by 1 each time. So, the first pair is 1,1; the next pair is 2,2; then 3,3; and so on. Each pair sums to an even number, and since there are 50 pairs, the total sum would be the sum of these pairs.Calculating the sum of each pair:- The first pair: 1 + 1 = 2- The second pair: 2 + 2 = 4- The third pair: 3 + 3 = 6- ...- The 50th pair: 50 + 50 = 100So, the sums of the pairs form an arithmetic sequence starting at 2, with a common difference of 2, and 50 terms.Using the sum formula again:( S = frac{n}{2} times (first term + last term) )( S = frac{50}{2} times (2 + 100) = 25 times 102 = 2550 )Yep, same result. That confirms it. So, I feel confident that 2550 is the correct answer.**Final Answer**boxed{B}