💡Okay, so I have this problem here where I need to find the distance between two points, M and F. F is the focus of a given curve, and M is the point (1/2, 0). The curve is defined parametrically with x = 2 cos θ and y = 1 + cos 2θ. Hmm, let's see how I can approach this.First, I remember that parametric equations can sometimes be converted into Cartesian form by eliminating the parameter, which in this case is θ. So, maybe I can express y in terms of x or vice versa. Let me look at the equations again:x = 2 cos θ y = 1 + cos 2θI notice that y is expressed in terms of cos 2θ. I recall that there's a double-angle identity for cosine: cos 2θ = 2 cos²θ - 1. Maybe I can use that to rewrite y in terms of cos θ, which is related to x.Let me substitute cos 2θ with 2 cos²θ - 1 in the equation for y:y = 1 + (2 cos²θ - 1) Simplify that: y = 1 + 2 cos²θ - 1 The 1 and -1 cancel out, so: y = 2 cos²θNow, from the equation for x, I have x = 2 cos θ. So, if I solve for cos θ, I get:cos θ = x / 2Now, let's substitute this into the equation for y. Since y = 2 cos²θ, and cos θ is x/2, then cos²θ is (x/2)² = x² / 4.So, substituting back:y = 2 * (x² / 4) Simplify that: y = (2x²) / 4 Which simplifies to: y = x² / 2Okay, so the Cartesian equation of the curve is y = (1/2)x². That looks like a parabola that opens upwards. Good, so now I know the curve is a parabola.Now, I need to find the focus of this parabola, which is point F. I remember that for a parabola in the form y = ax², the focus is at (0, 1/(4a)). Let me confirm that formula.Yes, for a parabola y = ax² + bx + c, the focus is at (-b/(2a), c + (1 - b²)/(4a)). But in this case, the equation is y = (1/2)x², so a = 1/2, b = 0, and c = 0. So, plugging into the formula:Focus x-coordinate: -b/(2a) = -0/(2*(1/2)) = 0 Focus y-coordinate: c + (1 - b²)/(4a) = 0 + (1 - 0)/(4*(1/2)) = 1 / 2So, the focus F is at (0, 1/2). Got it.Now, I have point M at (1/2, 0) and point F at (0, 1/2). I need to find the distance between these two points, |MF|.I remember the distance formula between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)² + (y2 - y1)²]. Let's apply that.So, plugging in the coordinates:Distance = sqrt[(0 - 1/2)² + (1/2 - 0)²] Simplify each term:(0 - 1/2)² = (-1/2)² = 1/4 (1/2 - 0)² = (1/2)² = 1/4So, adding them together: 1/4 + 1/4 = 2/4 = 1/2Then, take the square root of 1/2: sqrt(1/2) = (√2)/2So, the distance |MF| is √2 / 2.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Converted parametric equations to Cartesian: y = (1/2)x². That seems right because I used the double-angle identity correctly.2. Found the focus of the parabola. For y = ax², focus is at (0, 1/(4a)). Here, a = 1/2, so 1/(4*(1/2)) = 1/2. So, F is at (0, 1/2). That seems correct.3. Applied the distance formula between (1/2, 0) and (0, 1/2). Calculated each term, squared them, added, took the square root. Got √(1/2) which is √2 / 2. That seems correct.I think all the steps are right. So, the value of |MF| is √2 / 2.**Final Answer**The value of (|MF|) is boxed{dfrac{sqrt{2}}{2}}.

💡Okay, so I have this problem about a sequence {a_n} where the sum of the first n terms is given by S_n = (a_n)/2 + 1/(a_n) - 1, and each a_n is positive. I need to find the first three terms, a1, a2, a3, and then guess a general formula for a_n. After that, I have to prove that this formula is correct.Alright, let's start with part (1). I need to find a1, a2, a3. Since S_n is the sum of the first n terms, S1 is just a1. So, for n=1, S1 = a1. According to the formula, S1 = (a1)/2 + 1/(a1) - 1. So, setting them equal:a1 = (a1)/2 + 1/(a1) - 1.Hmm, let's solve this equation for a1. Let's subtract (a1)/2 from both sides:a1 - (a1)/2 = 1/(a1) - 1.Simplify the left side: (a1)/2 = 1/(a1) - 1.Multiply both sides by 2a1 to eliminate denominators:a1 * 2a1 * (a1)/2 = a1 * 2a1 * (1/(a1) - 1).Wait, that seems complicated. Maybe a better approach is to multiply both sides by a1 first:(a1)/2 * a1 = (1/(a1) - 1) * a1.So, (a1^2)/2 = 1 - a1.Multiply both sides by 2:a1^2 = 2 - 2a1.Bring all terms to one side:a1^2 + 2a1 - 2 = 0.Now, this is a quadratic equation. Let's use the quadratic formula:a1 = [-2 ± sqrt(4 + 8)] / 2 = [-2 ± sqrt(12)] / 2 = [-2 ± 2*sqrt(3)] / 2 = -1 ± sqrt(3).Since a1 is positive, we take the positive solution:a1 = -1 + sqrt(3) ≈ 0.732.Okay, so a1 is sqrt(3) - 1. Got that.Now, moving on to a2. S2 is the sum of a1 and a2. So, S2 = a1 + a2. According to the given formula, S2 = (a2)/2 + 1/(a2) - 1.So, we have:a1 + a2 = (a2)/2 + 1/(a2) - 1.We already know a1 is sqrt(3) - 1, so let's plug that in:(sqrt(3) - 1) + a2 = (a2)/2 + 1/(a2) - 1.Let's simplify the left side:sqrt(3) - 1 + a2.The right side is (a2)/2 + 1/(a2) - 1.So, set them equal:sqrt(3) - 1 + a2 = (a2)/2 + 1/(a2) - 1.Let's subtract (a2)/2 from both sides:sqrt(3) - 1 + (a2)/2 = 1/(a2) - 1.Add 1 to both sides:sqrt(3) + (a2)/2 = 1/(a2).Multiply both sides by a2:sqrt(3)*a2 + (a2)^2 / 2 = 1.Multiply both sides by 2 to eliminate the fraction:2*sqrt(3)*a2 + (a2)^2 = 2.Rearrange:(a2)^2 + 2*sqrt(3)*a2 - 2 = 0.Again, a quadratic equation. Let's solve for a2:a2 = [-2*sqrt(3) ± sqrt{(2*sqrt(3))^2 + 8}]/2.Calculate discriminant:(2*sqrt(3))^2 = 4*3 = 12, so sqrt(12 + 8) = sqrt(20) = 2*sqrt(5).Thus,a2 = [-2*sqrt(3) ± 2*sqrt(5)] / 2 = [-sqrt(3) ± sqrt(5)].Since a2 is positive, we take the positive solution:a2 = -sqrt(3) + sqrt(5) ≈ 1.268.So, a2 is sqrt(5) - sqrt(3).Hmm, interesting. So, a1 is sqrt(3) - 1, a2 is sqrt(5) - sqrt(3). I see a pattern here. Maybe a3 is sqrt(7) - sqrt(5). Let's check.For a3, S3 = a1 + a2 + a3. According to the formula, S3 = (a3)/2 + 1/(a3) - 1.So,a1 + a2 + a3 = (a3)/2 + 1/(a3) - 1.We know a1 = sqrt(3) - 1, a2 = sqrt(5) - sqrt(3). So, a1 + a2 = (sqrt(3) - 1) + (sqrt(5) - sqrt(3)) = sqrt(5) - 1.Thus, the equation becomes:sqrt(5) - 1 + a3 = (a3)/2 + 1/(a3) - 1.Simplify:sqrt(5) - 1 + a3 = (a3)/2 + 1/(a3) - 1.Subtract (a3)/2 from both sides:sqrt(5) - 1 + (a3)/2 = 1/(a3) - 1.Add 1 to both sides:sqrt(5) + (a3)/2 = 1/(a3).Multiply both sides by a3:sqrt(5)*a3 + (a3)^2 / 2 = 1.Multiply both sides by 2:2*sqrt(5)*a3 + (a3)^2 = 2.Rearrange:(a3)^2 + 2*sqrt(5)*a3 - 2 = 0.Quadratic equation again. Solve for a3:a3 = [-2*sqrt(5) ± sqrt{(2*sqrt(5))^2 + 8}]/2.Discriminant:(2*sqrt(5))^2 = 4*5 = 20, so sqrt(20 + 8) = sqrt(28) = 2*sqrt(7).Thus,a3 = [-2*sqrt(5) ± 2*sqrt(7)] / 2 = [-sqrt(5) ± sqrt(7)].Since a3 is positive, take the positive solution:a3 = -sqrt(5) + sqrt(7) ≈ 1.268.So, a3 is sqrt(7) - sqrt(5).So, the pattern seems to be that each term a_n is sqrt(2n + 1) - sqrt(2n - 1). Let's test this conjecture.For n=1: sqrt(3) - sqrt(1) = sqrt(3) - 1, which matches a1.For n=2: sqrt(5) - sqrt(3), which matches a2.For n=3: sqrt(7) - sqrt(5), which matches a3.So, the conjecture is that a_n = sqrt(2n + 1) - sqrt(2n - 1).Now, for part (2), we need to prove this formula is correct for all n.One way to do this is by mathematical induction.Base case: n=1. We already saw that a1 = sqrt(3) - 1, which fits the formula. So, base case holds.Inductive step: Assume that for some k >=1, a_k = sqrt(2k + 1) - sqrt(2k - 1). We need to show that a_{k+1} = sqrt(2(k+1) + 1) - sqrt(2(k+1) - 1) = sqrt(2k + 3) - sqrt(2k + 1).From the given formula, S_{k+1} = (a_{k+1})/2 + 1/(a_{k+1}) - 1.But S_{k+1} is also equal to S_k + a_{k+1}. From the inductive hypothesis, S_k = (a_k)/2 + 1/(a_k) - 1.So,S_{k+1} = S_k + a_{k+1} = (a_k)/2 + 1/(a_k) - 1 + a_{k+1}.But S_{k+1} is also equal to (a_{k+1})/2 + 1/(a_{k+1}) - 1.Therefore,(a_k)/2 + 1/(a_k) - 1 + a_{k+1} = (a_{k+1})/2 + 1/(a_{k+1}) - 1.Let's simplify this equation.First, subtract (a_{k+1})/2 from both sides:(a_k)/2 + 1/(a_k) - 1 + (a_{k+1})/2 = 1/(a_{k+1}) - 1.Add 1 to both sides:(a_k)/2 + 1/(a_k) + (a_{k+1})/2 = 1/(a_{k+1}).Multiply both sides by 2a_{k+1} to eliminate denominators:a_{k+1}*a_k + 2 + (a_{k+1})^2 = 2.Wait, let's double-check that step.Wait, if I have:(a_k)/2 + 1/(a_k) + (a_{k+1})/2 = 1/(a_{k+1}).Multiply both sides by 2a_{k+1}:2a_{k+1}*(a_k)/2 + 2a_{k+1}*(1/(a_k)) + 2a_{k+1}*(a_{k+1})/2 = 2a_{k+1}*(1/(a_{k+1})).Simplify each term:First term: a_{k+1}*a_k.Second term: 2a_{k+1}/(a_k).Third term: (a_{k+1})^2.Right side: 2.So, the equation becomes:a_{k+1}*a_k + 2a_{k+1}/(a_k) + (a_{k+1})^2 = 2.Hmm, this seems a bit messy. Maybe there's a better approach.Alternatively, let's express a_{k+1} in terms of S_{k+1} and S_k.We have S_{k+1} = S_k + a_{k+1}.From the given formula:S_{k+1} = (a_{k+1})/2 + 1/(a_{k+1}) - 1.And S_k = (a_k)/2 + 1/(a_k) - 1.So,S_{k+1} = S_k + a_{k+1} = (a_k)/2 + 1/(a_k) - 1 + a_{k+1}.Set equal to the other expression for S_{k+1}:(a_{k+1})/2 + 1/(a_{k+1}) - 1 = (a_k)/2 + 1/(a_k) - 1 + a_{k+1}.Subtract (a_{k+1})/2 from both sides:1/(a_{k+1}) - 1 = (a_k)/2 + 1/(a_k) - 1 + (a_{k+1})/2.Wait, that might not be helpful. Let's try rearranging terms.Bring all terms to one side:(a_{k+1})/2 + 1/(a_{k+1}) - 1 - (a_k)/2 - 1/(a_k) + 1 - a_{k+1} = 0.Simplify:(a_{k+1})/2 - a_{k+1} + 1/(a_{k+1}) - (a_k)/2 - 1/(a_k) = 0.Combine like terms:(-a_{k+1}/2) + 1/(a_{k+1}) - (a_k)/2 - 1/(a_k) = 0.Multiply both sides by 2a_{k+1}a_k to eliminate denominators:- a_{k+1}^2 a_k + 2a_k - a_{k+1} a_k^2 - 2a_{k+1} = 0.This seems complicated. Maybe there's a better way.Wait, let's consider the expression for a_n. If a_n = sqrt(2n + 1) - sqrt(2n - 1), then let's compute S_n.S_n = sum_{k=1}^n a_k = sum_{k=1}^n [sqrt(2k + 1) - sqrt(2k - 1)].This is a telescoping series. Let's write out the terms:For k=1: sqrt(3) - sqrt(1).For k=2: sqrt(5) - sqrt(3).For k=3: sqrt(7) - sqrt(5)....For k=n: sqrt(2n + 1) - sqrt(2n - 1).When we add them up, most terms cancel:S_n = sqrt(2n + 1) - sqrt(1) = sqrt(2n + 1) - 1.So, S_n = sqrt(2n + 1) - 1.Now, let's check if this S_n satisfies the given formula S_n = (a_n)/2 + 1/(a_n) - 1.We have S_n = sqrt(2n + 1) - 1.Compute (a_n)/2 + 1/(a_n) - 1.Given a_n = sqrt(2n + 1) - sqrt(2n - 1).Compute (a_n)/2:[sqrt(2n + 1) - sqrt(2n - 1)] / 2.Compute 1/(a_n):1 / [sqrt(2n + 1) - sqrt(2n - 1)].Multiply numerator and denominator by [sqrt(2n + 1) + sqrt(2n - 1)]:[sqrt(2n + 1) + sqrt(2n - 1)] / [ (sqrt(2n + 1))^2 - (sqrt(2n - 1))^2 ] = [sqrt(2n + 1) + sqrt(2n - 1)] / [ (2n + 1) - (2n - 1) ] = [sqrt(2n + 1) + sqrt(2n - 1)] / 2.So, 1/(a_n) = [sqrt(2n + 1) + sqrt(2n - 1)] / 2.Now, add (a_n)/2 + 1/(a_n):[sqrt(2n + 1) - sqrt(2n - 1)] / 2 + [sqrt(2n + 1) + sqrt(2n - 1)] / 2 = [2*sqrt(2n + 1)] / 2 = sqrt(2n + 1).Subtract 1:sqrt(2n + 1) - 1.Which matches S_n. Therefore, the formula holds.Thus, by induction, the conjecture is correct.

💡Okay, so I have this inequality to solve: -4x - 8 > 0. Hmm, I remember that solving inequalities is somewhat similar to solving equations, but there are some rules I need to keep in mind, especially when dealing with negative numbers. Let me try to work through this step by step.First, I want to isolate the variable x on one side of the inequality. The inequality is -4x - 8 > 0. To get rid of that -8, I can add 8 to both sides. That should help me get closer to having x by itself.So, adding 8 to both sides:-4x - 8 + 8 > 0 + 8Simplifying that:-4x > 8Alright, now I have -4x > 8. The next step is to get x by itself, so I need to divide both sides by -4. But wait, I remember that when you divide or multiply both sides of an inequality by a negative number, the direction of the inequality sign flips. That's an important rule to apply here.Dividing both sides by -4:x < 8 / (-4)Simplifying that:x < -2Okay, so after following those steps, I end up with x < -2. Let me double-check to make sure I didn't make any mistakes. Starting from the original inequality:-4x - 8 > 0Adding 8 to both sides:-4x > 8Dividing by -4 and flipping the inequality:x < -2That seems correct. I think I applied the rules properly, especially remembering to flip the inequality when dividing by a negative number. So, the solution set should be all real numbers less than -2.Looking back at the options provided:A: x > -2B: x < -2C: x > 2D: x < 2The correct answer should be option B: x < -2.

💡First, I need to determine how much money Peter spent on the soda. He started with 2 and left with 0.50, so the amount spent is 2 - 0.50 = 1.50.Next, I know that Peter bought 6 ounces of soda. To find the cost per ounce, I divide the total amount spent by the number of ounces.So, the cost per ounce is 1.50 divided by 6 ounces, which equals 0.25 per ounce.

💡First, I recognize that the total length of the iron wire can be considered as 1 unit.The wire is first used up to a length of 2/9 of its total length. This means that after the first usage, the remaining length of the wire is 1 - 2/9 = 7/9.Next, the wire is used up to an additional length of 3/9 of its total length. Subtracting this from the remaining length gives 7/9 - 3/9 = 4/9.Therefore, the fraction of the iron wire that is left is 4/9.

💡Alright, so I've got this problem here that asks about sufficient but not necessary conditions for ( a > b ). There are four options: A, B, C, and D. I need to figure out which one is a sufficient condition but not a necessary one. Let me break this down step by step.First, let's recall what a sufficient condition means. A sufficient condition is something that, if it's true, guarantees that the statement is true. However, it doesn't have to be the only way for the statement to be true. In other words, if condition X is sufficient for Y, then X being true means Y is definitely true, but Y could also be true even if X isn't.On the other hand, a necessary condition is something that must be true for the statement to be true, but it doesn't guarantee it. So, if condition X is necessary for Y, then Y can't be true unless X is true, but X being true doesn't necessarily mean Y is true.Okay, so the question is asking for a condition that is sufficient but not necessary for ( a > b ). That means we're looking for a condition that, if it's met, will definitely make ( a > b ) true, but ( a > b ) can still be true even if this condition isn't met.Let's look at each option one by one.**Option A: ( dfrac{1}{a} < dfrac{1}{b} )**Hmm, this one is about the reciprocals of ( a ) and ( b ). I know that if both ( a ) and ( b ) are positive, then ( dfrac{1}{a} < dfrac{1}{b} ) implies ( a > b ). But what if ( a ) and ( b ) are negative? Let's test some numbers.Suppose ( a = 2 ) and ( b = 1 ). Then ( dfrac{1}{2} < dfrac{1}{1} ), which is true, and ( a > b ) is also true. That works.But what if ( a = -1 ) and ( b = -2 )? Then ( dfrac{1}{-1} = -1 ) and ( dfrac{1}{-2} = -0.5 ). So ( -1 < -0.5 ) is true, but ( a = -1 ) is actually greater than ( b = -2 ). Wait, so in this case, ( a > b ) is still true, but the reciprocal condition also holds. Hmm, maybe I need a different example.Let me try ( a = 1 ) and ( b = -1 ). Then ( dfrac{1}{1} = 1 ) and ( dfrac{1}{-1} = -1 ). So ( 1 < -1 ) is false. But ( a = 1 > b = -1 ) is true. So in this case, ( a > b ) is true, but the reciprocal condition is false. That means the reciprocal condition isn't necessary because ( a > b ) can be true without the reciprocal condition being true.But is the reciprocal condition sufficient? Let's see. If ( dfrac{1}{a} < dfrac{1}{b} ), does that always mean ( a > b )? Well, in the first example with positive numbers, yes. But what if one is positive and the other is negative?Suppose ( a = 1 ) and ( b = -1 ). Then ( dfrac{1}{1} = 1 ) and ( dfrac{1}{-1} = -1 ). So ( 1 < -1 ) is false. But if I switch them, ( a = -1 ) and ( b = 1 ), then ( dfrac{1}{-1} = -1 ) and ( dfrac{1}{1} = 1 ). So ( -1 < 1 ) is true, but ( a = -1 ) is not greater than ( b = 1 ). So in this case, the reciprocal condition is true, but ( a > b ) is false. Therefore, the reciprocal condition isn't sufficient because it can be true even when ( a > b ) is false.So, Option A is not a sufficient condition because it can sometimes lead to ( a > b ) being false, and it's also not necessary because ( a > b ) can be true without it. So A is out.**Option B: ( a > b - 1 )**Okay, so if ( a > b - 1 ), does that guarantee ( a > b )? Let's test with some numbers.Suppose ( a = 3 ) and ( b = 2 ). Then ( a > b - 1 ) becomes ( 3 > 1 ), which is true, and ( a > b ) is also true. That works.But what if ( a = 1.5 ) and ( b = 2 )? Then ( a > b - 1 ) becomes ( 1.5 > 1 ), which is true, but ( a = 1.5 ) is not greater than ( b = 2 ). So in this case, the condition is true, but ( a > b ) is false. Therefore, ( a > b - 1 ) is not a sufficient condition because it can be true without ( a > b ) being true.Is it a necessary condition? Let's see. If ( a > b ), does that mean ( a > b - 1 )? Yes, because if ( a > b ), then subtracting 1 from ( b ) only makes ( b - 1 ) smaller, so ( a ) is definitely greater than ( b - 1 ). So B is a necessary condition, but not a sufficient one. Since the question asks for a sufficient but not necessary condition, B is not the answer.**Option C: ( a^2 > b^2 )**Alright, so if ( a^2 > b^2 ), does that mean ( a > b )? Let's test with some numbers.First, both positive: ( a = 3 ), ( b = 2 ). Then ( 9 > 4 ), and ( 3 > 2 ). That works.But what if ( a = -3 ) and ( b = 2 )? Then ( (-3)^2 = 9 > 4 ), so ( a^2 > b^2 ) is true, but ( a = -3 ) is not greater than ( b = 2 ). So in this case, the condition is true, but ( a > b ) is false. Therefore, ( a^2 > b^2 ) is not a sufficient condition because it can be true without ( a > b ) being true.Is it a necessary condition? Let's see. If ( a > b ), does that mean ( a^2 > b^2 )? Not necessarily. For example, if ( a = 2 ) and ( b = 1 ), then ( a > b ) and ( a^2 > b^2 ). But if ( a = -1 ) and ( b = -2 ), then ( a = -1 > b = -2 ), but ( a^2 = 1 ) and ( b^2 = 4 ), so ( a^2 < b^2 ). Therefore, ( a > b ) doesn't necessarily lead to ( a^2 > b^2 ). So C is neither sufficient nor necessary. So C is out.**Option D: ( a > b + 1 )**Okay, let's see. If ( a > b + 1 ), does that guarantee ( a > b )? Well, if ( a ) is greater than ( b + 1 ), then it's definitely greater than ( b ) itself because ( b + 1 > b ). So yes, ( a > b + 1 ) implies ( a > b ). Therefore, it's a sufficient condition.Is it a necessary condition? That is, does ( a > b ) necessarily mean ( a > b + 1 )? No, because ( a ) could be just slightly bigger than ( b ), like ( a = b + 0.5 ). In that case, ( a > b ) is true, but ( a > b + 1 ) is false. So D is not a necessary condition.Therefore, D is a sufficient but not necessary condition for ( a > b ).To summarize:- A: Not sufficient, not necessary.- B: Necessary, not sufficient.- C: Neither sufficient nor necessary.- D: Sufficient, not necessary.So the answer should be D.