💡Okay, so I have this algebra problem here: I need to find the coefficient of x in the expression 5(x - 6) + 6(8 - 3x² + 7x) - 9(4x - 3). Hmm, let's see. I remember that to find the coefficient of a term, I need to simplify the expression and then look at the number in front of x. First, I think I should expand each part of the expression separately. That means I'll distribute the numbers outside the parentheses into each term inside. Let me start with the first part: 5(x - 6). So, I'll multiply 5 by x and then 5 by -6. That should give me 5x - 30. Okay, so the coefficient of x here is 5.Next, I'll move on to the second part: 6(8 - 3x² + 7x). I need to distribute the 6 across all three terms inside the parentheses. So, 6 times 8 is 48, 6 times -3x² is -18x², and 6 times 7x is 42x. So, this part becomes 48 - 18x² + 42x. The coefficient of x here is 42.Now, the third part is -9(4x - 3). I have to be careful with the negative sign here. So, I'll distribute -9 to both terms inside the parentheses. That means -9 times 4x is -36x, and -9 times -3 is +27. So, this part simplifies to -36x + 27. The coefficient of x here is -36.Alright, so now I have all the parts expanded. Let me write them out together:5x - 30 + 48 - 18x² + 42x - 36x + 27.Now, I need to combine like terms. Like terms are terms that have the same variable raised to the same power. So, first, I'll look for the x² term. There's only one: -18x². Then, I'll look for the x terms: 5x, 42x, and -36x. Finally, the constant terms are -30, 48, and 27.Let's combine the x terms first. 5x + 42x is 47x, and then subtracting 36x gives me 47x - 36x, which is 11x. So, the coefficient of x is 11.Just to make sure I didn't make a mistake, let me double-check my calculations. Starting with the first part: 5(x - 6) gives 5x - 30. Correct. Then, 6(8 - 3x² + 7x) gives 48 - 18x² + 42x. Correct. Finally, -9(4x - 3) gives -36x + 27. Correct.Combining all the x terms: 5x + 42x - 36x. 5 + 42 is 47, and 47 - 36 is indeed 11. So, the coefficient of x is 11.I think that's it. I don't see any other x terms or any mistakes in my calculations. So, the coefficient of x in the given expression is 11.

💡Okay, so I have this problem where point P is given as (sin 40°, 1 + cos 40°), and it's on the terminal side of an acute angle α. I need to find the measure of α. The options are 10°, 20°, 70°, and 80°. Hmm, let's break this down step by step.First, I remember that any point on the terminal side of an angle in standard position can be used to find the tangent of that angle. The tangent is just the y-coordinate divided by the x-coordinate. So, in this case, tan α should be equal to (1 + cos 40°) divided by sin 40°. That makes sense because the point P is (x, y), so x is sin 40° and y is 1 + cos 40°.So, tan α = (1 + cos 40°) / sin 40°. Now, I need to simplify this expression. I recall some trigonometric identities that might help here. Specifically, the double-angle identities. Let me see... There's an identity for 1 + cos θ, which is 2 cos²(θ/2). Similarly, sin θ can be written as 2 sin(θ/2) cos(θ/2). Maybe I can use these to simplify the expression.Let's apply that. If I let θ = 40°, then 1 + cos 40° becomes 2 cos²(20°), and sin 40° becomes 2 sin(20°) cos(20°). So substituting these into the expression for tan α, I get:tan α = (2 cos²(20°)) / (2 sin(20°) cos(20°))I can cancel out the 2 in the numerator and denominator, and also one cos(20°) from the numerator and denominator. That leaves me with:tan α = cos(20°) / sin(20°)Wait a second, cos(20°)/sin(20°) is the same as cot(20°), right? Because cotangent is the reciprocal of tangent, so cot θ = cos θ / sin θ. So, tan α = cot(20°).But I need to find α, and I know that tan α = cot(20°). I remember that cot θ is equal to tan(90° - θ). So, cot(20°) is equal to tan(70°). Therefore, tan α = tan(70°).Since α is an acute angle, it must be between 0° and 90°, so the only solution is α = 70°. That makes sense because 70° is one of the answer choices, specifically option C.Let me just double-check my steps to make sure I didn't make any mistakes. I started by recognizing that tan α is y/x, which is (1 + cos 40°)/sin 40°. Then I used the double-angle identities to rewrite 1 + cos 40° and sin 40°, which simplified the expression to cot(20°). Then, using the co-function identity, I converted cot(20°) to tan(70°), leading me to conclude that α is 70°. Everything seems to check out.I think it's also helpful to visualize this. If I plot the point P(sin 40°, 1 + cos 40°), it should lie in the first quadrant since both sin 40° and cos 40° are positive. The angle α is the angle between the positive x-axis and the line connecting the origin to point P. By calculating the tangent, I'm essentially finding the slope of that line, which corresponds to the angle α.Another way to think about it is using the unit circle. The coordinates of point P can be related to the unit circle, but since the coordinates aren't necessarily on the unit circle (because sin 40° and 1 + cos 40° might not satisfy x² + y² = 1), I might need to consider scaling. However, since we're only interested in the angle, the scaling factor doesn't affect the tangent value, so it's okay to use the coordinates directly.I also considered whether there could be another angle in the first quadrant that has the same tangent value, but since α is acute, it has to be between 0° and 90°, and tan(70°) is unique in that range. So, there's no ambiguity here.Just to be thorough, let me plug in the values numerically to see if it makes sense. Let's calculate tan(70°) and see if it matches (1 + cos 40°)/sin 40°.First, calculate sin 40° and cos 40°:sin 40° ≈ 0.6428cos 40° ≈ 0.7660So, 1 + cos 40° ≈ 1 + 0.7660 = 1.7660Then, (1 + cos 40°)/sin 40° ≈ 1.7660 / 0.6428 ≈ 2.747Now, tan(70°) ≈ 2.747, which matches exactly. So, this numerical check confirms that α is indeed 70°.I think that solidifies the conclusion. I don't see any other angles in the given options that would satisfy this condition. 10°, 20°, and 80° are all either too small or too large compared to 70°, and their tangent values wouldn't match the calculated slope.So, after carefully working through the problem, simplifying using trigonometric identities, and verifying with numerical calculations, I'm confident that the measure of α is 70°, which is option C.

💡Okay, so I have this polynomial, right? It's ( 3x^3 + bx + 8 ). The problem says that this polynomial has a factor of the form ( x^2 + px + 2 ). I need to find the value of ( b ). Hmm, okay, let me think about how to approach this.First, I remember that if a polynomial has a factor, then it can be expressed as the product of that factor and another polynomial. Since the given polynomial is a cubic (degree 3) and the factor is a quadratic (degree 2), the other factor must be a linear polynomial (degree 1). So, I can write:[3x^3 + bx + 8 = (x^2 + px + 2)(text{something})]And since the something is linear, it should look like ( ax + c ), where ( a ) and ( c ) are constants that I need to find. But wait, the leading term on the left side is ( 3x^3 ), and the leading term on the right side would be ( x^2 times ax = ax^3 ). To make these match, ( a ) must be 3. So, the linear factor is ( 3x + c ). That makes sense.So now, I can write:[3x^3 + bx + 8 = (x^2 + px + 2)(3x + c)]Okay, next step is to expand the right-hand side and then compare coefficients with the left-hand side. Let's do that.First, expand ( (x^2 + px + 2)(3x + c) ):Multiply each term in the first polynomial by each term in the second polynomial:1. ( x^2 times 3x = 3x^3 )2. ( x^2 times c = c x^2 )3. ( px times 3x = 3p x^2 )4. ( px times c = c p x )5. ( 2 times 3x = 6x )6. ( 2 times c = 2c )Now, combine like terms:- The ( x^3 ) term: ( 3x^3 )- The ( x^2 ) terms: ( c x^2 + 3p x^2 = (c + 3p) x^2 )- The ( x ) terms: ( c p x + 6x = (c p + 6) x )- The constant term: ( 2c )So, putting it all together, the expanded form is:[3x^3 + (c + 3p) x^2 + (c p + 6) x + 2c]Now, this should be equal to the original polynomial ( 3x^3 + bx + 8 ). So, let's set the coefficients equal to each other.First, the coefficient of ( x^3 ) is 3 on both sides, so that's already matching.Next, the coefficient of ( x^2 ) on the left side is 0 (since there's no ( x^2 ) term), and on the right side, it's ( c + 3p ). So:[c + 3p = 0 quad text{(1)}]Then, the coefficient of ( x ) on the left side is ( b ), and on the right side, it's ( c p + 6 ). So:[c p + 6 = b quad text{(2)}]Finally, the constant term on the left side is 8, and on the right side, it's ( 2c ). So:[2c = 8 quad text{(3)}]Alright, now I have three equations:1. ( c + 3p = 0 )2. ( c p + 6 = b )3. ( 2c = 8 )Let's solve these step by step.Starting with equation (3):[2c = 8 implies c = frac{8}{2} = 4]So, ( c = 4 ). Now, plug this value into equation (1):[4 + 3p = 0 implies 3p = -4 implies p = -frac{4}{3}]Got it, ( p = -frac{4}{3} ). Now, let's find ( b ) using equation (2):[b = c p + 6 = 4 times left(-frac{4}{3}right) + 6]Calculating that:[4 times left(-frac{4}{3}right) = -frac{16}{3}][-frac{16}{3} + 6 = -frac{16}{3} + frac{18}{3} = frac{2}{3}]So, ( b = frac{2}{3} ).Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting with ( c = 4 ), then ( p = -4/3 ). Then, plugging into ( b = c p + 6 ):[4 times (-4/3) = -16/3][-16/3 + 6 = -16/3 + 18/3 = 2/3]Yes, that seems correct. So, ( b = frac{2}{3} ).Just to be thorough, let me verify by plugging everything back into the original equation.So, the quadratic factor is ( x^2 + px + 2 = x^2 - frac{4}{3}x + 2 ), and the linear factor is ( 3x + 4 ). Let's multiply them together:[(x^2 - frac{4}{3}x + 2)(3x + 4)]Multiply term by term:1. ( x^2 times 3x = 3x^3 )2. ( x^2 times 4 = 4x^2 )3. ( -frac{4}{3}x times 3x = -4x^2 )4. ( -frac{4}{3}x times 4 = -frac{16}{3}x )5. ( 2 times 3x = 6x )6. ( 2 times 4 = 8 )Now, combine like terms:- ( 3x^3 )- ( 4x^2 - 4x^2 = 0 ) (so no ( x^2 ) term)- ( -frac{16}{3}x + 6x = -frac{16}{3}x + frac{18}{3}x = frac{2}{3}x )- ( 8 )So, the product is:[3x^3 + frac{2}{3}x + 8]Which matches the original polynomial ( 3x^3 + bx + 8 ) with ( b = frac{2}{3} ). So, that checks out.I think I'm confident that ( b = frac{2}{3} ) is the correct answer.**Final Answer**The value of ( b ) is boxed{dfrac{2}{3}}.

💡Okay, so I'm trying to solve this problem where I need to find the smallest positive integer ( K ) such that in the sequence defined by ( a_1 = K ) and the rules:- If ( a_n ) is even, then ( a_{n+1} = a_n - 1 ).- If ( a_n ) is odd, then ( a_{n+1} = frac{a_n - 1}{2} ),the term ( a_{2005} ) is the first term that equals zero. Hmm, that's a bit complex. Let me try to break it down step by step.First, I need to understand how the sequence behaves. Let's consider some small values of ( K ) and see what happens.Suppose ( K = 1 ):- ( a_1 = 1 ) (odd)- ( a_2 = frac{1 - 1}{2} = 0 )So, ( a_2 = 0 ). But we need ( a_{2005} ) to be the first zero. So, ( K = 1 ) is too small.Next, ( K = 2 ):- ( a_1 = 2 ) (even)- ( a_2 = 2 - 1 = 1 ) (odd)- ( a_3 = frac{1 - 1}{2} = 0 )Again, ( a_3 = 0 ). Still too early.How about ( K = 3 ):- ( a_1 = 3 ) (odd)- ( a_2 = frac{3 - 1}{2} = 1 ) (odd)- ( a_3 = frac{1 - 1}{2} = 0 )Still, ( a_3 = 0 ). Not enough.Wait, maybe I need to find a ( K ) such that the sequence takes exactly 2004 steps to reach zero for the first time. So, the sequence should not reach zero before the 2005th term.This seems like a problem that can be approached by working backwards. Instead of starting from ( K ) and moving forward, maybe I can start from zero and build up the sequence in reverse.If I think about it, each term before zero must have been such that applying the reverse operation leads to the next term. Let me define the reverse operations:- If ( a_{n+1} = a_n - 1 ) (which happens when ( a_n ) is even), then to reverse it, ( a_n = a_{n+1} + 1 ).- If ( a_{n+1} = frac{a_n - 1}{2} ) (which happens when ( a_n ) is odd), then to reverse it, ( a_n = 2a_{n+1} + 1 ).So, starting from ( a_{2005} = 0 ), I can work backwards to find ( a_{2004} ), ( a_{2003} ), and so on, up to ( a_1 = K ).Let me try to see the pattern here. Since each step can either add 1 or double and add 1, depending on whether the previous term was even or odd.Wait, but when working backwards, I don't know whether the previous term was even or odd. So, I might have multiple possibilities at each step. That complicates things.But since we want the smallest ( K ), maybe we can choose the operations that result in the smallest possible numbers at each step.Alternatively, perhaps there's a pattern or a formula that can help us determine ( K ) without having to compute each term individually.Let me think about the operations again. When moving forward:- If the term is even, subtract 1.- If the term is odd, subtract 1 and then divide by 2.This reminds me a bit of the operations in the Collatz conjecture, but it's slightly different.Alternatively, maybe I can model this as a binary tree, where each term branches into two possibilities when working backwards: one where the previous term was even, and one where it was odd.But with 2004 steps, that's a lot of branching. It might not be feasible to compute all possibilities.Wait, maybe I can represent the sequence in terms of binary operations. Let's see.If I consider the operations in reverse:- If the next term is ( a_{n+1} ), then the previous term could be either ( a_n = a_{n+1} + 1 ) (if ( a_n ) was even), or ( a_n = 2a_{n+1} + 1 ) (if ( a_n ) was odd).So, each step back can be represented as either adding 1 or doubling and adding 1.To get the smallest ( K ), I probably want to minimize the number of times I double and add 1, as that operation increases the number more significantly.But I need to ensure that the sequence doesn't reach zero before the 2005th term. So, I need to structure the operations such that zero is only reached at step 2005.Alternatively, maybe I can think of the sequence as a binary number, where each operation corresponds to a bit.Wait, let's try to model this.Suppose we represent ( K ) in binary. Each operation either subtracts 1 (if even) or subtracts 1 and divides by 2 (if odd). But I'm not sure if that directly helps. Maybe another approach.Let me consider the number of steps required to reach zero. Each time we subtract 1 when even, which is similar to counting down, but when odd, we do a more drastic operation.Wait, perhaps I can model this as a binary counter, where each subtraction of 1 is a decrement, and the other operation is a kind of division.Alternatively, maybe I can think of the sequence as a path in a binary tree, where each node has two children: one for the even operation and one for the odd operation.But again, with 2004 steps, it's a huge tree.Wait, maybe I can find a pattern or a formula for ( K ) in terms of the number of steps.Let me try to see what happens with smaller numbers of steps.Suppose I want ( a_2 = 0 ). Then ( a_1 ) must be 1, because:- If ( a_1 ) is odd, then ( a_2 = frac{1 - 1}{2} = 0 ).So, ( K = 1 ).If I want ( a_3 = 0 ), then ( a_2 ) must be 1, and ( a_1 ) must be 2 or 3.Wait, let's see:- If ( a_2 = 1 ), then ( a_1 ) could be 2 (since 2 is even, so ( a_2 = 2 - 1 = 1 )), or ( a_1 ) could be 3 (since 3 is odd, so ( a_2 = frac{3 - 1}{2} = 1 )).So, the smallest ( K ) is 2.Wait, but earlier I thought ( K = 1 ) gives ( a_2 = 0 ), which is smaller. So, to have ( a_3 = 0 ), the smallest ( K ) is 2.Similarly, for ( a_4 = 0 ), let's see:- ( a_4 = 0 ), so ( a_3 = 1 ).- ( a_3 = 1 ) implies ( a_2 = 2 ) or ( a_2 = 3 ).- If ( a_2 = 2 ), then ( a_1 = 3 ) (since 2 is even, ( a_2 = 2 - 1 = 1 ), but wait, ( a_2 = 2 ) would lead to ( a_3 = 1 ), but ( a_3 = 1 ) leads to ( a_4 = 0 ). So, ( a_1 ) could be 3 or 4.Wait, let's compute:If ( a_1 = 3 ):- ( a_1 = 3 ) (odd)- ( a_2 = frac{3 - 1}{2} = 1 )- ( a_3 = frac{1 - 1}{2} = 0 )But we wanted ( a_4 = 0 ), so this is too early.If ( a_1 = 4 ):- ( a_1 = 4 ) (even)- ( a_2 = 4 - 1 = 3 ) (odd)- ( a_3 = frac{3 - 1}{2} = 1 )- ( a_4 = frac{1 - 1}{2} = 0 )So, ( a_4 = 0 ). Therefore, ( K = 4 ) is the smallest for ( a_4 = 0 ).Wait, but earlier, ( K = 2 ) gives ( a_3 = 0 ), and ( K = 4 ) gives ( a_4 = 0 ). So, the pattern seems like ( K ) doubles each time we increase the step by 2.Wait, let's see:- ( a_2 = 0 ): ( K = 1 )- ( a_3 = 0 ): ( K = 2 )- ( a_4 = 0 ): ( K = 4 )- ( a_5 = 0 ): Let's compute.For ( a_5 = 0 ), ( a_4 = 1 ), so ( a_3 = 2 ) or ( a_3 = 3 ).If ( a_3 = 2 ), then ( a_2 = 3 ) (since 2 is even, ( a_3 = 2 - 1 = 1 ), but that would make ( a_4 = 0 ), which is too early). Wait, no, ( a_3 = 2 ) implies ( a_2 = 3 ) (since ( a_3 = 2 ) is even, so ( a_2 = 2 + 1 = 3 ) in reverse). Then ( a_2 = 3 ) implies ( a_1 = 4 ) or ( a_1 = 7 ).Wait, this is getting complicated. Maybe I need a better approach.Alternatively, perhaps I can model this as a binary number where each step corresponds to a bit, and the operations correspond to certain transformations.Wait, another idea: since each time we have two choices when working backwards (either add 1 or double and add 1), and we want the smallest ( K ), we should choose the smallest possible operation at each step.But to ensure that the sequence doesn't reach zero before step 2005, we need to structure the operations such that zero is only reached at step 2005.Wait, maybe the minimal ( K ) is ( 2^{2004} - 2 ). But that seems too large.Wait, let's think recursively. Let me define ( f(n) ) as the minimal ( K ) such that ( a_n = 0 ) and all previous terms are positive.Then, ( f(1) = 1 ) (since ( a_1 = 1 ) leads to ( a_2 = 0 )).For ( f(2) ), we need ( a_2 = 0 ), so ( a_1 ) must be 2 (since ( a_1 = 2 ) leads to ( a_2 = 1 ), then ( a_3 = 0 )). Wait, no, ( a_1 = 2 ) leads to ( a_2 = 1 ), which leads to ( a_3 = 0 ). So, ( f(3) = 2 ).Wait, maybe I'm getting confused. Let me try to define ( f(n) ) as the minimal ( K ) such that ( a_n = 0 ) and ( a_{n-1} ) is the first term to reach 1.Wait, perhaps it's better to think in terms of the number of steps required to reach zero.Each time we have an odd term, we do a more drastic operation, which might reduce the number more quickly. So, to make the sequence last longer, we want to minimize the number of times we have odd terms.Alternatively, to maximize the number of steps, we want as many even terms as possible, because subtracting 1 each time takes more steps.Wait, but we're looking for the minimal ( K ) such that it takes exactly 2004 steps to reach zero. So, perhaps the minimal ( K ) is related to the number of steps in a way that each step corresponds to a binary digit.Wait, another approach: let's model the sequence as a binary number where each step corresponds to a bit, and the operations correspond to certain transformations.Wait, perhaps I can represent ( K ) in binary and see how the operations affect the bits.For example, when ( a_n ) is even, subtracting 1 is equivalent to flipping the least significant bit from 0 to 1 and subtracting 1, but actually, subtracting 1 from an even number just decreases it by 1, which in binary is flipping the last bit from 0 to 1 and subtracting 1, but that might not be directly helpful.When ( a_n ) is odd, subtracting 1 and dividing by 2 is equivalent to right-shifting the binary representation after subtracting 1.Wait, let's take an example. Suppose ( a_n = 5 ) (which is 101 in binary). Subtracting 1 gives 4 (100), then dividing by 2 gives 2 (10). So, ( a_{n+1} = 2 ).Similarly, if ( a_n = 6 ) (110), subtracting 1 gives 5 (101), which is odd, so ( a_{n+1} = frac{5 - 1}{2} = 2 ).Wait, so the operation when ( a_n ) is odd is equivalent to subtracting 1 and then dividing by 2, which in binary is like right-shifting after subtracting 1.Hmm, maybe I can model the sequence as a binary number where each operation corresponds to certain bit manipulations.But I'm not sure if that directly helps. Maybe another approach.Let me consider the sequence in reverse. Starting from 0, and building up to ( K ) in 2004 steps.At each step, I can either add 1 or double and add 1. To get the smallest ( K ), I should choose the operations that result in the smallest possible numbers.But since we need to reach zero at step 2005, we need to ensure that in the reverse process, each step leads us closer to ( K ) without reaching zero before.Wait, actually, in the reverse process, starting from 0, each step can be:- ( a_n = a_{n+1} + 1 ) (if the next term was even)- ( a_n = 2a_{n+1} + 1 ) (if the next term was odd)To minimize ( K ), we should choose the smallest possible ( a_n ) at each step. So, we should prefer adding 1 over doubling and adding 1 whenever possible.But we need to ensure that the sequence doesn't reach zero before step 2005. So, we need to structure the operations such that zero is only reached at step 2005.Wait, but in the reverse process, starting from 0, each step can be either adding 1 or doubling and adding 1. To minimize ( K ), we should choose the smallest possible operations, which would be adding 1 as much as possible.But if we only add 1 each time, we would get ( K = 2004 ), but that's probably not the case because sometimes we have to double and add 1 to avoid reaching zero too early.Wait, maybe I need to alternate between adding 1 and doubling and adding 1 in a way that ensures that zero is only reached at step 2005.Alternatively, perhaps the minimal ( K ) is ( 2^{1002} - 2 ). Wait, why?Let me think about the number of times we have to double. Since each time we double, we can cover two steps, perhaps.Wait, let's consider that in the reverse process, each time we double and add 1, we're effectively covering two steps in the forward process.So, if we have 2004 steps, we can cover them with 1002 doublings and 1002 additions.Wait, but I'm not sure. Let me try to see.If I start from 0, and in each pair of steps, I do:- Step 1: add 1 (to get 1)- Step 2: double and add 1 (to get 3)Then, in the forward process, starting from 3:- 3 is odd, so ( a_2 = frac{3 - 1}{2} = 1 )- 1 is odd, so ( a_3 = 0 )So, that's two steps in the forward process.Similarly, in the reverse process, starting from 0, two steps give us 3.So, each pair of steps in the reverse process corresponds to two steps in the forward process.Therefore, for 2004 steps in the forward process, we need 1002 pairs of steps in the reverse process.Each pair consists of adding 1 and then doubling and adding 1.So, starting from 0:- After 1 step: 1- After 2 steps: 3- After 3 steps: 4- After 4 steps: 9- After 5 steps: 10- After 6 steps: 21- ...Wait, I see a pattern here. Each pair of steps seems to be multiplying by 2 and adding 1.Wait, let's see:- After 2 steps: 3 = 2^2 - 1- After 4 steps: 9 = 2^3 + 1Wait, no, 9 is 2^3 + 1, but 3 is 2^2 - 1.Wait, maybe it's better to see that each pair of steps in reverse corresponds to a multiplication by 2 and adding 1.Wait, let's think recursively. Let me define ( f(n) ) as the minimal ( K ) such that ( a_n = 0 ).Then, ( f(1) = 1 ) (since ( a_1 = 1 ) leads to ( a_2 = 0 )).For ( f(2) ), we need ( a_2 = 0 ), so ( a_1 ) must be 2 (since ( a_1 = 2 ) leads to ( a_2 = 1 ), then ( a_3 = 0 )). Wait, no, ( a_1 = 2 ) leads to ( a_2 = 1 ), which leads to ( a_3 = 0 ). So, ( f(3) = 2 ).Wait, maybe I'm getting confused again. Let me try to see the pattern.From the earlier example, after 2 reverse steps, we get 3, which is ( 2^2 - 1 ).After 4 reverse steps, we get 9, which is ( 2^3 + 1 ).Wait, maybe it's ( 2^{n/2 + 1} - 1 ) for even ( n ).Wait, let's test:For ( n = 2 ), ( 2^{2/2 + 1} - 1 = 2^{2} - 1 = 3 ). That matches.For ( n = 4 ), ( 2^{4/2 + 1} - 1 = 2^{3} - 1 = 7 ). But earlier, after 4 reverse steps, I got 9. Hmm, that doesn't match.Wait, maybe my initial assumption is wrong.Alternatively, perhaps the minimal ( K ) is ( 2^{1003} - 2 ).Wait, let's see:If we have 2004 steps in the forward process, which is 1002 pairs of steps.Each pair of steps in the forward process corresponds to one doubling in the reverse process.So, starting from 0, after 1002 doublings and 1002 additions, we get ( K = 2^{1002} times 1 + ... ). Hmm, not sure.Wait, another approach: in the reverse process, each time we choose to add 1, we're effectively increasing the number by 1, and each time we choose to double and add 1, we're increasing it more.To minimize ( K ), we should use as many adds as possible, but we have to ensure that we don't reach zero too early.Wait, but in the reverse process, starting from 0, each step can be:- ( a_n = a_{n+1} + 1 ) (if the next term was even)- ( a_n = 2a_{n+1} + 1 ) (if the next term was odd)To minimize ( K ), we should choose the smallest possible ( a_n ) at each step, which would be adding 1 as much as possible.But we have to ensure that the sequence doesn't reach zero before step 2005. So, we need to structure the operations such that zero is only reached at step 2005.Wait, but in the reverse process, starting from 0, each step can be either adding 1 or doubling and adding 1. To minimize ( K ), we should choose the smallest possible operations, which would be adding 1 as much as possible.But if we only add 1 each time, we would get ( K = 2004 ), but that's probably not the case because sometimes we have to double and add 1 to avoid reaching zero too early.Wait, maybe I need to alternate between adding 1 and doubling and adding 1 in a way that ensures that zero is only reached at step 2005.Alternatively, perhaps the minimal ( K ) is ( 2^{1003} - 2 ). Let me check.If ( K = 2^{1003} - 2 ), then in the forward process, it would take 2004 steps to reach zero.Wait, let's see:Starting from ( K = 2^{1003} - 2 ), which is even.- ( a_1 = 2^{1003} - 2 ) (even)- ( a_2 = a_1 - 1 = 2^{1003} - 3 ) (odd)- ( a_3 = frac{a_2 - 1}{2} = frac{2^{1003} - 4}{2} = 2^{1002} - 2 ) (even)- ( a_4 = a_3 - 1 = 2^{1002} - 3 ) (odd)- ( a_5 = frac{a_4 - 1}{2} = frac{2^{1002} - 4}{2} = 2^{1001} - 2 ) (even)- ...Continuing this pattern, each pair of steps reduces the exponent by 1 and subtracts 2.So, after 2 steps, we go from ( 2^{1003} - 2 ) to ( 2^{1002} - 2 ).After 4 steps, we go to ( 2^{1001} - 2 ).Continuing this way, after 2004 steps, we would reach ( 2^{1} - 2 = 0 ).Yes, that makes sense. So, ( K = 2^{1003} - 2 ) would take exactly 2004 steps to reach zero, making ( a_{2005} = 0 ) the first zero.Therefore, the smallest ( K ) is ( 2^{1003} - 2 ).

💡Okay, so I have this problem where point P is the intersection of an ellipse C₁ and a hyperbola C₂, both sharing the same foci F₁ and F₂. The eccentricities of the ellipse and hyperbola are e₁ and e₂ respectively. The condition given is that the vectors PF₁ and PF₂ are perpendicular, meaning their dot product is zero. I need to find the value of 1/e₁² + 1/e₂².First, I should recall some properties of ellipses and hyperbolas. Both conic sections have foci, and their eccentricities relate to how "stretched" they are. For an ellipse, the eccentricity e₁ is less than 1, and for a hyperbola, e₂ is greater than 1.Since both the ellipse and hyperbola share the same foci, they must have the same distance between the foci, which is 2c, where c is the distance from the center to each focus. For the ellipse, the major axis length is 2a, and for the hyperbola, the transverse axis length is 2m. The eccentricity of the ellipse is e₁ = c/a, and for the hyperbola, e₂ = c/m.Given that vectors PF₁ and PF₂ are perpendicular, the triangle formed by points P, F₁, and F₂ is a right triangle with the right angle at P. So, by the Pythagorean theorem, |PF₁|² + |PF₂|² = |F₁F₂|². Since F₁ and F₂ are 2c apart, |F₁F₂| = 2c, so |F₁F₂|² = (2c)² = 4c².Therefore, |PF₁|² + |PF₂|² = 4c².Now, I need to relate |PF₁| and |PF₂| to the properties of the ellipse and hyperbola.For the ellipse, the sum of distances from any point on the ellipse to the two foci is constant and equal to 2a. So, |PF₁| + |PF₂| = 2a.For the hyperbola, the difference of distances from any point on the hyperbola to the two foci is constant and equal to 2m. So, |PF₁| - |PF₂| = 2m (assuming P is on the right branch of the hyperbola, so |PF₁| > |PF₂|).Let me denote |PF₁| as d₁ and |PF₂| as d₂. Then, from the ellipse, we have:d₁ + d₂ = 2a ...(1)From the hyperbola, we have:d₁ - d₂ = 2m ...(2)And from the perpendicularity condition, we have:d₁² + d₂² = 4c² ...(3)I can solve equations (1) and (2) to find expressions for d₁ and d₂.Adding equations (1) and (2):(d₁ + d₂) + (d₁ - d₂) = 2a + 2m ⇒ 2d₁ = 2(a + m) ⇒ d₁ = a + mSubtracting equation (2) from equation (1):(d₁ + d₂) - (d₁ - d₂) = 2a - 2m ⇒ 2d₂ = 2(a - m) ⇒ d₂ = a - mSo, d₁ = a + m and d₂ = a - m.Now, substitute d₁ and d₂ into equation (3):(a + m)² + (a - m)² = 4c²Expanding both squares:(a² + 2am + m²) + (a² - 2am + m²) = 4c²Simplify:a² + 2am + m² + a² - 2am + m² = 4c²Combine like terms:2a² + 2m² = 4c²Divide both sides by 2:a² + m² = 2c²So, a² + m² = 2c².Now, recall that for the ellipse, e₁ = c/a, so c = a e₁.Similarly, for the hyperbola, e₂ = c/m, so c = m e₂.Therefore, from c = a e₁ and c = m e₂, we can write a = c / e₁ and m = c / e₂.Substitute a and m into the equation a² + m² = 2c²:(c / e₁)² + (c / e₂)² = 2c²Factor out c²:c² (1 / e₁² + 1 / e₂²) = 2c²Divide both sides by c² (assuming c ≠ 0, which it isn't since it's a focus distance):1 / e₁² + 1 / e₂² = 2Therefore, the value of 1/e₁² + 1/e₂² is 2.**Final Answer**boxed{B}

💡Okay, so I've got this equation here: (n - 2)x^{|n - 1|} + 5 = 0. And I need to find the value of n such that this equation is linear in one variable. Hmm, let me think about what that means.First off, a linear equation in one variable typically looks like ax + b = 0, where a and b are constants, and the exponent of x is 1. So, for this equation to be linear, the exponent of x must be 1, right? That means |n - 1| should equal 1 because the exponent is |n - 1|.Alright, so if |n - 1| = 1, that gives me two possible equations to solve: n - 1 = 1 and n - 1 = -1. Let's solve those.First, n - 1 = 1. Adding 1 to both sides, I get n = 2. Okay, that's one possibility.Second, n - 1 = -1. Adding 1 to both sides here, I get n = 0. So, n could be either 2 or 0.But wait, I need to make sure that the equation is indeed linear. So, let's check both possibilities.If n = 2, then the equation becomes (2 - 2)x^{|2 - 1|} + 5 = 0. Simplifying that, (0)x^{1} + 5 = 0, which is just 0 + 5 = 0, or 5 = 0. That doesn't make sense because 5 isn't equal to 0. So, n = 2 can't be a valid solution because it makes the coefficient of x zero, which would eliminate the x term entirely, leaving us with a false statement.Now, let's check n = 0. Plugging that into the equation, we get (0 - 2)x^{|0 - 1|} + 5 = 0. Simplifying, (-2)x^{1} + 5 = 0, which is -2x + 5 = 0. That's a perfectly good linear equation because the exponent of x is 1, and the coefficient (-2) isn't zero. So, this works.Therefore, the only valid solution is n = 0.I think that's it. I considered both cases where the exponent equals 1, checked if they result in a valid linear equation, and only n = 0 works. So, yeah, n must be 0.