answer：First, I need to determine the total number of regions created by the altitudes of the equilateral triangle. The altitudes divide the triangle into six equal sections.Next, I observe that two of these six sections are shaded. To find the probability that the spinner will land in a shaded region, I calculate the ratio of shaded sections to the total number of sections.This gives me the probability as (frac{2}{6}), which simplifies to (frac{1}{3}).Therefore, the probability that the spinner will come to rest in a shaded region is (boxed{frac{1}{3}}).

answer：First, I need to determine the number of letters in Christina's name. The name "Christina" has 9 distinct letters.Next, I calculate the total number of possible rearrangements of these letters using the factorial of the number of letters:[9! = 362,!880]Since Christina writes the rearrangements in two halves, I divide the total number by 2 to find the number of rearrangements in each half:[frac{362,!880}{2} = 181,!440]For the first half, she writes 12 rearrangements per minute. I calculate the time taken for the first half by dividing the number of rearrangements by the rate:[frac{181,!440}{12} = 15,!120 text{ minutes}]For the second half, she increases her rate to 18 rearrangements per minute. I calculate the time taken for the second half similarly:[frac{181,!440}{18} = 10,!080 text{ minutes}]To find the total time, I add the time taken for both halves:[15,!120 text{ minutes} + 10,!080 text{ minutes} = 25,!200 text{ minutes}]Finally, I convert the total time from minutes to hours by dividing by 60:[frac{25,!200}{60} = 420 text{ hours}]

answer：Okay, so I've got this sequence problem here. Let me try to figure it out step by step. The sequence is defined by ( a_1 = 1 ) and the recursive formula ( a_{n+1} = frac{a_n}{2a_n + 1} ). There are two parts to this problem. First, I need to find the general term formula for the sequence ( {b_n} ) where ( b_n = frac{1}{a_n} ). Second, I have to find the sum of the first ( n ) terms of the sequence ( {c_n} ) where ( c_n = a_n cdot a_{n+1} ). Let me start with the first part. So, ( b_n = frac{1}{a_n} ). That means each term ( b_n ) is just the reciprocal of ( a_n ). Maybe if I can find a relationship between ( b_{n+1} ) and ( b_n ), I can figure out a pattern or a formula.Given the recursive formula for ( a_{n+1} ), let's write that in terms of ( b_n ). We have:[ a_{n+1} = frac{a_n}{2a_n + 1} ]Taking reciprocal on both sides:[ frac{1}{a_{n+1}} = frac{2a_n + 1}{a_n} ]Simplify the right-hand side:[ frac{1}{a_{n+1}} = 2 + frac{1}{a_n} ]But ( frac{1}{a_{n+1}} = b_{n+1} ) and ( frac{1}{a_n} = b_n ), so substituting:[ b_{n+1} = 2 + b_n ]Oh, that's nice! So this tells me that each term ( b_{n+1} ) is equal to the previous term ( b_n ) plus 2. That means the sequence ( {b_n} ) is an arithmetic sequence with a common difference of 2.Now, since ( b_n ) is an arithmetic sequence, I can write its general term using the formula for the ( n )-th term of an arithmetic sequence:[ b_n = b_1 + (n - 1)d ]where ( d ) is the common difference.We know ( d = 2 ), and we can find ( b_1 ) since ( b_1 = frac{1}{a_1} ) and ( a_1 = 1 ). So:[ b_1 = frac{1}{1} = 1 ]Plugging into the formula:[ b_n = 1 + (n - 1) times 2 ][ b_n = 1 + 2n - 2 ][ b_n = 2n - 1 ]Okay, so that's the general term for ( b_n ). That wasn't too bad. Now, moving on to the second part.We need to find the sum of the first ( n ) terms of the sequence ( {c_n} ), where ( c_n = a_n cdot a_{n+1} ). First, let's express ( c_n ) in terms of ( b_n ). Since ( a_n = frac{1}{b_n} ) and ( a_{n+1} = frac{1}{b_{n+1}} ), then:[ c_n = a_n cdot a_{n+1} = frac{1}{b_n} cdot frac{1}{b_{n+1}} = frac{1}{b_n b_{n+1}} ]But we already know ( b_n = 2n - 1 ), so:[ c_n = frac{1}{(2n - 1)(2n + 1)} ]Hmm, that looks like it might be a telescoping series. Maybe I can use partial fractions to simplify this expression.Let me try to express ( c_n ) as a difference of two fractions. Let's assume:[ frac{1}{(2n - 1)(2n + 1)} = frac{A}{2n - 1} + frac{B}{2n + 1} ]Multiplying both sides by ( (2n - 1)(2n + 1) ):[ 1 = A(2n + 1) + B(2n - 1) ]Expanding the right-hand side:[ 1 = (2A + 2B)n + (A - B) ]Since this must hold for all ( n ), the coefficients of like terms must be equal. So, setting up equations for the coefficients:1. Coefficient of ( n ):[ 2A + 2B = 0 ]2. Constant term:[ A - B = 1 ]Let's solve these equations. From the first equation:[ 2A + 2B = 0 ]Divide both sides by 2:[ A + B = 0 ]So, ( A = -B )Plugging into the second equation:[ (-B) - B = 1 ][ -2B = 1 ][ B = -frac{1}{2} ]Then, since ( A = -B ):[ A = frac{1}{2} ]So, the partial fractions decomposition is:[ frac{1}{(2n - 1)(2n + 1)} = frac{1/2}{2n - 1} - frac{1/2}{2n + 1} ][ = frac{1}{2} left( frac{1}{2n - 1} - frac{1}{2n + 1} right) ]Therefore, ( c_n = frac{1}{2} left( frac{1}{2n - 1} - frac{1}{2n + 1} right) )Now, to find the sum ( S_n = c_1 + c_2 + dots + c_n ), let's write out the terms:[ S_n = frac{1}{2} left( left( frac{1}{1} - frac{1}{3} right) + left( frac{1}{3} - frac{1}{5} right) + left( frac{1}{5} - frac{1}{7} right) + dots + left( frac{1}{2n - 1} - frac{1}{2n + 1} right) right) ]I notice that this is a telescoping series, meaning that most terms cancel out. Let's see:- The first term inside the big parentheses is ( frac{1}{1} - frac{1}{3} )- The second term is ( frac{1}{3} - frac{1}{5} )- The third term is ( frac{1}{5} - frac{1}{7} )- ...- The last term is ( frac{1}{2n - 1} - frac{1}{2n + 1} )When we add all these up, the ( -frac{1}{3} ) from the first term cancels with the ( frac{1}{3} ) from the second term. Similarly, the ( -frac{1}{5} ) cancels with the ( frac{1}{5} ) from the third term, and so on. This pattern continues, and all the intermediate terms cancel out, leaving only the first term of the first expression and the last term of the last expression.So, after cancellation, we have:[ S_n = frac{1}{2} left( 1 - frac{1}{2n + 1} right) ]Let's simplify this expression:[ S_n = frac{1}{2} left( frac{(2n + 1) - 1}{2n + 1} right) ][ = frac{1}{2} left( frac{2n}{2n + 1} right) ][ = frac{n}{2n + 1} ]So, the sum of the first ( n ) terms of the sequence ( {c_n} ) is ( frac{n}{2n + 1} ).Let me just double-check my steps to make sure I didn't make any mistakes. Starting from the recursive formula, I correctly transformed it into a relationship between ( b_n ) terms, identified it as an arithmetic sequence, found the general term, then moved on to express ( c_n ) in terms of ( b_n ), used partial fractions, recognized the telescoping nature of the series, and performed the cancellation correctly. The final simplification also seems right. Yeah, I think that's solid. I don't see any errors in my reasoning.**Final Answer**1. The general term for ( {b_n} ) is ( boxed{2n - 1} ).2. The sum of the first ( n ) terms ( S_n ) is ( boxed{dfrac{n}{2n + 1}} ).

answer：First, I need to identify the number of trees yielding different amounts of nuts. According to the problem, there are ( x + 2 ) trees yielding 40 nuts per year, ( x ) trees yielding 120 nuts per year, and ( x - 2 ) trees yielding 180 nuts per year.Next, I'll calculate the total number of nuts produced by each group of trees:- For the trees yielding 40 nuts: ( 40 times (x + 2) )- For the trees yielding 120 nuts: ( 120 times x )- For the trees yielding 180 nuts: ( 180 times (x - 2) )Adding these together gives the total number of nuts produced by all trees.Then, I'll find the total number of trees by adding the number of trees in each group:( (x + 2) + x + (x - 2) )The average yield per tree is given as 100 nuts. I'll set up an equation where the total number of nuts divided by the total number of trees equals 100.Solving this equation will give me the value of ( x ).

answer：Okay, so I have this function f(x) = e^{sin x} + e^{-sin x}, and I need to figure out which of the statements A, B, C, or D is incorrect. Let me go through each option one by one.Starting with option A: It says that f(x) is an even function on ℝ. Hmm, an even function satisfies f(-x) = f(x) for all x. Let me check that. So, f(-x) would be e^{sin(-x)} + e^{-sin(-x)}. But sin(-x) is equal to -sin x, right? So substituting that in, f(-x) becomes e^{-sin x} + e^{sin x}, which is the same as f(x). So yes, f(x) is even. So A is correct.Moving on to option B: It states that π is a period of f(x). A period means that f(x + π) = f(x) for all x. Let's compute f(x + π). So, sin(x + π) is equal to -sin x because sine has a period of 2π and shifting by π flips the sign. Therefore, f(x + π) = e^{sin(x + π)} + e^{-sin(x + π)} = e^{-sin x} + e^{sin x}, which is again f(x). So, π is indeed a period. So B is correct.Now, option C: π is a point of local minimum for f(x). To check this, I need to analyze the behavior of f(x) around x = π. Maybe I should compute the derivative of f(x) to find critical points and determine if π is a minimum.So, f(x) = e^{sin x} + e^{-sin x}. Let's find f'(x). The derivative of e^{sin x} is e^{sin x} * cos x, and the derivative of e^{-sin x} is -e^{-sin x} * cos x. So, f'(x) = cos x (e^{sin x} - e^{-sin x}).Now, let's evaluate f'(x) around x = π. When x is slightly less than π, say x = π - ε where ε is a small positive number, sin x is sin(π - ε) = sin ε ≈ ε, so e^{sin x} ≈ e^{ε} and e^{-sin x} ≈ e^{-ε}. Therefore, e^{sin x} - e^{-sin x} ≈ e^{ε} - e^{-ε} ≈ 2ε (since for small ε, e^{ε} ≈ 1 + ε and e^{-ε} ≈ 1 - ε, so their difference is approximately 2ε). Also, cos x at x = π - ε is cos(π - ε) = -cos ε ≈ -1. Therefore, f'(x) ≈ (-1)(2ε) = -2ε, which is negative.When x is slightly more than π, say x = π + ε, sin x = sin(π + ε) = -sin ε ≈ -ε. So, e^{sin x} ≈ e^{-ε} and e^{-sin x} ≈ e^{ε}. Thus, e^{sin x} - e^{-sin x} ≈ e^{-ε} - e^{ε} ≈ -2ε. Cos x at x = π + ε is cos(π + ε) = -cos ε ≈ -1. Therefore, f'(x) ≈ (-1)(-2ε) = 2ε, which is positive.So, the derivative changes from negative to positive as x increases through π, which means that x = π is indeed a local minimum. So, option C is correct.Finally, option D: f(x) is decreasing on the interval (0, π/2). Let's check the derivative on this interval. For x in (0, π/2), sin x is positive and increasing from 0 to 1. Therefore, e^{sin x} is increasing and e^{-sin x} is decreasing. However, let's look at the derivative f'(x) = cos x (e^{sin x} - e^{-sin x}).In the interval (0, π/2), cos x is positive because cosine is positive in the first quadrant. Now, e^{sin x} - e^{-sin x} is also positive because sin x is positive, making e^{sin x} > e^{-sin x}. Therefore, f'(x) is positive in this interval, which means f(x) is increasing, not decreasing. So, option D is incorrect.Wait, let me double-check that. If f'(x) is positive, then f(x) is increasing. So, on (0, π/2), f(x) is increasing, which contradicts option D's statement that it's decreasing. So, yes, D is incorrect.So, summarizing:- A is correct.- B is correct.- C is correct.- D is incorrect.Therefore, the incorrect statement is D.**Final Answer**The incorrect statement is boxed{D}.

answer：Okay, so I have this problem where I need to find the magnitude of the vector (overrightarrow{a} - overrightarrow{b}). The given information is that the angle between vectors (overrightarrow{a}) and (overrightarrow{b}) is (30^circ), the magnitude of (overrightarrow{a}) is (sqrt{3}), and the magnitude of (overrightarrow{b}) is 2. Hmm, let me think about how to approach this.First, I remember that when dealing with vectors and their magnitudes, the dot product formula is often useful. The dot product of two vectors can be expressed in terms of their magnitudes and the cosine of the angle between them. The formula is:[overrightarrow{a} cdot overrightarrow{b} = |overrightarrow{a}| |overrightarrow{b}| cos theta]Where (theta) is the angle between the vectors. In this case, (theta = 30^circ), (|overrightarrow{a}| = sqrt{3}), and (|overrightarrow{b}| = 2). Let me plug these values into the formula.Calculating the dot product:[overrightarrow{a} cdot overrightarrow{b} = sqrt{3} times 2 times cos 30^circ]I know that (cos 30^circ = frac{sqrt{3}}{2}), so substituting that in:[overrightarrow{a} cdot overrightarrow{b} = sqrt{3} times 2 times frac{sqrt{3}}{2}]Simplifying this expression, the 2 in the numerator and denominator cancel out, and (sqrt{3} times sqrt{3} = 3). So, the dot product simplifies to:[overrightarrow{a} cdot overrightarrow{b} = 3]Alright, so now I have the dot product of (overrightarrow{a}) and (overrightarrow{b}). The next step is to find the magnitude of (overrightarrow{a} - overrightarrow{b}). I recall that the magnitude of the difference of two vectors can be found using the formula:[|overrightarrow{a} - overrightarrow{b}| = sqrt{|overrightarrow{a}|^2 + |overrightarrow{b}|^2 - 2(overrightarrow{a} cdot overrightarrow{b})}]This formula comes from the law of cosines, which is used to find the length of a side of a triangle when two sides and the included angle are known. In this case, the vectors (overrightarrow{a}) and (overrightarrow{b}) form a triangle with the vector (overrightarrow{a} - overrightarrow{b}), and the angle between (overrightarrow{a}) and (overrightarrow{b}) is (30^circ).Let me substitute the known values into this formula. First, calculate (|overrightarrow{a}|^2):[|overrightarrow{a}|^2 = (sqrt{3})^2 = 3]Next, calculate (|overrightarrow{b}|^2):[|overrightarrow{b}|^2 = 2^2 = 4]Now, plug these into the formula along with the dot product we found earlier:[|overrightarrow{a} - overrightarrow{b}| = sqrt{3 + 4 - 2 times 3}]Simplifying inside the square root:[|overrightarrow{a} - overrightarrow{b}| = sqrt{7 - 6}]Wait, that can't be right. Let me double-check my calculations. I have:[|overrightarrow{a}|^2 = 3, quad |overrightarrow{b}|^2 = 4, quad overrightarrow{a} cdot overrightarrow{b} = 3]So, substituting into the formula:[|overrightarrow{a} - overrightarrow{b}| = sqrt{3 + 4 - 2 times 3} = sqrt{7 - 6} = sqrt{1} = 1]Hmm, that seems too simple. Did I make a mistake somewhere? Let me go back through the steps.First, the dot product calculation:[overrightarrow{a} cdot overrightarrow{b} = sqrt{3} times 2 times cos 30^circ = sqrt{3} times 2 times frac{sqrt{3}}{2} = 3]That seems correct. Then, the magnitudes squared:[|overrightarrow{a}|^2 = 3, quad |overrightarrow{b}|^2 = 4]Yes, that's right. Then, plugging into the formula:[sqrt{3 + 4 - 2 times 3} = sqrt{7 - 6} = sqrt{1} = 1]Wait a minute, that would mean the magnitude of (overrightarrow{a} - overrightarrow{b}) is 1. But intuitively, if the angle between them is 30 degrees, and their magnitudes are (sqrt{3}) and 2, I would expect the magnitude of their difference to be something a bit more complex, not just 1. Maybe I missed a step or made an error in the formula.Let me double-check the formula for the magnitude of the difference of two vectors. The formula is:[|overrightarrow{a} - overrightarrow{b}| = sqrt{|overrightarrow{a}|^2 + |overrightarrow{b}|^2 - 2|overrightarrow{a}||overrightarrow{b}| cos theta}]Wait, no, that's not quite right. The formula should actually be:[|overrightarrow{a} - overrightarrow{b}|^2 = |overrightarrow{a}|^2 + |overrightarrow{b}|^2 - 2|overrightarrow{a}||overrightarrow{b}| cos theta]So, taking the square root of that gives the magnitude. But in my earlier calculation, I used the dot product directly, which is equal to (|overrightarrow{a}||overrightarrow{b}| cos theta). So, substituting the dot product into the formula is correct.Wait, but if I use the formula with the dot product, it's:[|overrightarrow{a} - overrightarrow{b}| = sqrt{|overrightarrow{a}|^2 + |overrightarrow{b}|^2 - 2(overrightarrow{a} cdot overrightarrow{b})}]Which is what I did. So, substituting:[sqrt{3 + 4 - 2 times 3} = sqrt{7 - 6} = sqrt{1} = 1]So, according to this, the magnitude is indeed 1. But let me think geometrically. If the angle between them is 30 degrees, and their magnitudes are (sqrt{3}) and 2, is it possible that their difference has a magnitude of 1?Let me draw a rough sketch in my mind. Imagine vector (overrightarrow{a}) with length (sqrt{3}) and vector (overrightarrow{b}) with length 2, forming a 30-degree angle. The vector (overrightarrow{a} - overrightarrow{b}) would be the vector from the tip of (overrightarrow{b}) to the tip of (overrightarrow{a}) if they are placed tail to tail. The triangle formed by (overrightarrow{a}), (overrightarrow{b}), and (overrightarrow{a} - overrightarrow{b}) would have sides of length (sqrt{3}), 2, and 1, with the angle opposite the side of length 1 being 30 degrees.Wait, that doesn't seem to fit the triangle inequality. The sum of the lengths of any two sides must be greater than the length of the remaining side. Here, (sqrt{3} approx 1.732), so:[1.732 + 1 = 2.732 > 2][1.732 + 2 = 3.732 > 1][2 + 1 = 3 > 1.732]So, the triangle inequality holds. But does a triangle with sides 1, 2, and approximately 1.732 with an angle of 30 degrees opposite the side of length 1 make sense?Using the law of sines, the ratio of the sides to the sines of their opposite angles should be equal. So:[frac{1}{sin 30^circ} = frac{2}{sin alpha} = frac{sqrt{3}}{sin beta}]Where (alpha) and (beta) are the other two angles. Calculating:[frac{1}{0.5} = 2 = frac{2}{sin alpha} implies sin alpha = 1 implies alpha = 90^circ]Then, the third angle (beta = 180^circ - 30^circ - 90^circ = 60^circ).So, the triangle would have angles of 30°, 60°, and 90°, with sides opposite to these angles being 1, (sqrt{3}), and 2, respectively. That makes sense because it's a 30-60-90 triangle, which is a well-known right-angled triangle with sides in the ratio 1 : (sqrt{3}) : 2.Therefore, the magnitude of (overrightarrow{a} - overrightarrow{b}) being 1 is correct. It forms a right-angled triangle with the given vectors, which aligns with the calculations.Wait, but in the initial problem, the angle between (overrightarrow{a}) and (overrightarrow{b}) is 30 degrees, not 90 degrees. How does that reconcile with the triangle having a right angle?Ah, I see. The angle between (overrightarrow{a}) and (overrightarrow{b}) is 30 degrees, but when we form the triangle with (overrightarrow{a}), (overrightarrow{b}), and (overrightarrow{a} - overrightarrow{b}), the angle opposite the side (overrightarrow{a} - overrightarrow{b}) is 30 degrees, and the triangle itself is right-angled at the angle opposite the side of length 1. So, the triangle is right-angled, but the angle between the original vectors is 30 degrees. That makes sense because the triangle's angles are determined by the vectors' magnitudes and the angle between them.So, after carefully going through the calculations and verifying with the law of sines and the properties of a 30-60-90 triangle, I can confirm that the magnitude of (overrightarrow{a} - overrightarrow{b}) is indeed 1.But just to be thorough, let me recast the problem using coordinate systems to see if I arrive at the same result.Let's place vector (overrightarrow{a}) along the positive x-axis for simplicity. So, the coordinates of (overrightarrow{a}) would be:[overrightarrow{a} = (|overrightarrow{a}|, 0) = (sqrt{3}, 0)]Now, vector (overrightarrow{b}) makes a 30-degree angle with (overrightarrow{a}). Therefore, the coordinates of (overrightarrow{b}) can be expressed as:[overrightarrow{b} = (|overrightarrow{b}| cos 30^circ, |overrightarrow{b}| sin 30^circ) = (2 times frac{sqrt{3}}{2}, 2 times frac{1}{2}) = (sqrt{3}, 1)]Now, to find (overrightarrow{a} - overrightarrow{b}), we subtract the corresponding components:[overrightarrow{a} - overrightarrow{b} = (sqrt{3} - sqrt{3}, 0 - 1) = (0, -1)]So, the vector (overrightarrow{a} - overrightarrow{b}) is (0, -1). The magnitude of this vector is:[|overrightarrow{a} - overrightarrow{b}| = sqrt{0^2 + (-1)^2} = sqrt{0 + 1} = sqrt{1} = 1]This confirms the earlier result. Therefore, the magnitude of (overrightarrow{a} - overrightarrow{b}) is indeed 1.I think I've thoroughly checked this problem from different angles—using the dot product formula, verifying with the law of sines, and even breaking it down into coordinate components. All methods lead to the same conclusion that the magnitude is 1. So, I'm confident that this is the correct answer.