answer：Okay, so I need to prove that the sum of the first ( n ) odd numbers equals ( n^2 ). Hmm, let me think about how to approach this. Maybe I can start by writing out the first few odd numbers and see if I can spot a pattern.The first few odd numbers are 1, 3, 5, 7, 9, and so on. So, if I take the sum of the first ( n ) odd numbers, let's see what that looks like for small values of ( n ).For ( n = 1 ), the sum is just 1, and ( 1^2 = 1 ). Okay, that works.For ( n = 2 ), the sum is 1 + 3 = 4, and ( 2^2 = 4 ). Still matches.For ( n = 3 ), the sum is 1 + 3 + 5 = 9, and ( 3^2 = 9 ). Hmm, this is interesting. It seems like the pattern holds for these small numbers.Maybe I can try ( n = 4 ). The sum would be 1 + 3 + 5 + 7 = 16, and ( 4^2 = 16 ). Wow, it's consistent so far.But how do I prove this for any ( n )? I guess I need a more general approach. Maybe mathematical induction would work here. I remember that induction involves two main steps: the base case and the inductive step.First, the base case. Let's verify it for ( n = 1 ). As I did earlier, the sum is 1, and ( 1^2 = 1 ). So, the base case holds.Now, for the inductive step, I need to assume that the formula holds for some arbitrary positive integer ( k ). That is, I'll assume that the sum of the first ( k ) odd numbers is ( k^2 ). This is called the inductive hypothesis.So, assuming that ( 1 + 3 + 5 + dots + (2k - 1) = k^2 ), I need to show that the sum of the first ( k + 1 ) odd numbers is ( (k + 1)^2 ).Let's write out the sum for ( k + 1 ):( 1 + 3 + 5 + dots + (2k - 1) + (2(k + 1) - 1) )Simplifying the last term, ( 2(k + 1) - 1 = 2k + 2 - 1 = 2k + 1 ).So, the sum becomes:( [1 + 3 + 5 + dots + (2k - 1)] + (2k + 1) )But from the inductive hypothesis, we know that the sum inside the brackets is ( k^2 ). So, substituting that in:( k^2 + (2k + 1) )Now, let's simplify this expression:( k^2 + 2k + 1 = (k + 1)^2 )So, we've shown that if the sum of the first ( k ) odd numbers is ( k^2 ), then the sum of the first ( k + 1 ) odd numbers is ( (k + 1)^2 ). This completes the inductive step.Since both the base case and the inductive step have been verified, by the principle of mathematical induction, the formula holds for all positive integers ( n ).Wait, but is there another way to think about this? Maybe a more visual or geometric interpretation? I recall that squares can be represented as dots arranged in a square grid. If I think of each odd number as adding a layer to the square, then the total number of dots would be the square of the side length.For example, starting with 1 dot (which is ( 1^2 )), adding 3 dots forms a 2x2 square (which is ( 2^2 )), adding 5 dots forms a 3x3 square (which is ( 3^2 )), and so on. This seems to align with the idea that each odd number adds a new layer to the square, increasing the total count by that odd number.So, this geometric interpretation also supports the idea that the sum of the first ( n ) odd numbers is ( n^2 ).Another thought: maybe using algebraic manipulation without induction. Let's see. The ( n )-th odd number can be expressed as ( 2n - 1 ). So, the sum of the first ( n ) odd numbers is:( sum_{i=1}^{n} (2i - 1) )Let's expand this sum:( 2sum_{i=1}^{n} i - sum_{i=1}^{n} 1 )We know that ( sum_{i=1}^{n} i = frac{n(n + 1)}{2} ) and ( sum_{i=1}^{n} 1 = n ). Substituting these in:( 2 cdot frac{n(n + 1)}{2} - n = n(n + 1) - n = n^2 + n - n = n^2 )So, this algebraic approach also confirms that the sum is ( n^2 ).Hmm, so both induction and algebraic methods confirm the result. It seems pretty solid.I wonder if there are any other ways to think about this. Maybe using sequences or series properties? Let's see. The sequence of odd numbers is an arithmetic sequence where the first term ( a_1 = 1 ) and the common difference ( d = 2 ). The ( n )-th term of an arithmetic sequence is given by ( a_n = a_1 + (n - 1)d ), which in this case is ( 1 + (n - 1) cdot 2 = 2n - 1 ), which matches what we had earlier.The sum of the first ( n ) terms of an arithmetic sequence is ( S_n = frac{n}{2}(a_1 + a_n) ). Plugging in the values:( S_n = frac{n}{2}(1 + (2n - 1)) = frac{n}{2}(2n) = n^2 )So, using the formula for the sum of an arithmetic series also gives us ( n^2 ). That's another confirmation.It's interesting how multiple approaches lead to the same conclusion. It makes me more confident that the result is correct.I guess the key takeaway here is that the sum of the first ( n ) odd numbers forms a perfect square, and this can be shown through various methods like induction, algebraic manipulation, and properties of arithmetic sequences.Another perspective could be using generating functions or combinatorial arguments, but I think for now, the methods I've considered are sufficient to establish the proof.In summary, whether through induction, algebra, or arithmetic series properties, the sum of the first ( n ) odd numbers indeed equals ( n^2 ). This makes sense intuitively as well, considering the way squares grow by adding successive odd numbers.I think I've covered several angles to approach this proof, and each one reinforces the conclusion. It's a neat result that connects number theory with geometry and algebra.

answer：First, identify the number of boys and girls in the school. There are 841 boys and 34 girls.To find out how many more boys there are than girls, subtract the number of girls from the number of boys.Perform the subtraction: 841 minus 34 equals 807.Therefore, there are 807 more boys than girls in the school.

answer：Okay, so I have this problem about a factory producing units of a product, and I need to figure out how many units they should produce to maximize their total profit. Let me try to break this down step by step.First, the problem gives me the total cost function, which is c(x) = 1200 + (2/75)x³, where x is the number of units produced, and the cost is in ten thousand yuan. That means if they produce x units, their total cost is 1200 ten thousand yuan plus (2/75) times x cubed. Okay, got that.Then, it says that the unit price P (in ten thousand yuan) and the number of units x satisfy the equation P² = k/x. So, the square of the unit price is inversely proportional to the number of units produced, with k being the constant of proportionality. Interesting. I need to find k, probably using the given information.It also tells me that the unit price for producing 100 units is 50 ten thousand yuan. So, when x = 100, P = 50. Let me plug that into the equation P² = k/x to find k.Calculating that: 50² = k / 100. So, 2500 = k / 100. Therefore, k = 2500 * 100 = 250,000. Wait, 2500 times 100 is 250,000. Hmm, that seems right. So, k is 250,000.Now, with k known, I can write the unit price P as P = sqrt(k / x). Plugging in k, that becomes P = sqrt(250,000 / x). Let me simplify that. The square root of 250,000 is 500, so P = 500 / sqrt(x). Okay, so the unit price is 500 divided by the square root of x. Got it.Next, I need to find the total profit. Profit is generally total revenue minus total cost. Total revenue is the number of units sold times the unit price, so that's x * P. So, total revenue R(x) = x * (500 / sqrt(x)) = 500 * sqrt(x). That simplifies nicely.Total cost is given as c(x) = 1200 + (2/75)x³. So, total profit L(x) is R(x) - c(x), which is 500*sqrt(x) - 1200 - (2/75)x³. Okay, so L(x) = 500x^(1/2) - 1200 - (2/75)x³.To find the maximum profit, I need to find the critical points of L(x). That means taking the derivative of L(x) with respect to x and setting it equal to zero.Let's compute L'(x). The derivative of 500x^(1/2) is 500*(1/2)x^(-1/2) = 250x^(-1/2). The derivative of -1200 is zero. The derivative of -(2/75)x³ is -(2/75)*3x² = -(6/75)x², which simplifies to -(2/25)x². So, putting it all together, L'(x) = 250x^(-1/2) - (2/25)x².Now, set L'(x) equal to zero to find the critical points:250x^(-1/2) - (2/25)x² = 0.Let me rewrite this equation:250 / sqrt(x) = (2/25)x².I can multiply both sides by sqrt(x) to eliminate the denominator:250 = (2/25)x² * sqrt(x).Simplify the right side: x² * sqrt(x) is x^(5/2). So, 250 = (2/25)x^(5/2).Multiply both sides by 25 to get rid of the denominator:250 * 25 = 2x^(5/2).250 * 25 is 6250, so:6250 = 2x^(5/2).Divide both sides by 2:3125 = x^(5/2).Now, to solve for x, I can raise both sides to the power of 2/5:x = 3125^(2/5).Hmm, let me compute that. 3125 is 5^5 because 5^5 = 3125. So, 3125^(2/5) is (5^5)^(2/5) = 5^(5*(2/5)) = 5^2 = 25.So, x = 25. That means producing 25 units will give a critical point. Now, I need to check if this is a maximum.To confirm it's a maximum, I can check the second derivative or analyze the behavior of the first derivative around x = 25.Let me compute the second derivative L''(x). The first derivative was L'(x) = 250x^(-1/2) - (2/25)x².The derivative of 250x^(-1/2) is 250*(-1/2)x^(-3/2) = -125x^(-3/2). The derivative of -(2/25)x² is -(4/25)x.So, L''(x) = -125x^(-3/2) - (4/25)x.At x = 25, let's compute L''(25):First term: -125*(25)^(-3/2). 25^(3/2) is (25^(1/2))^3 = 5^3 = 125. So, 25^(-3/2) is 1/125. Therefore, -125*(1/125) = -1.Second term: -(4/25)*25 = -4.So, L''(25) = -1 - 4 = -5, which is negative. Since the second derivative is negative, the function is concave down at x = 25, meaning this critical point is indeed a local maximum.Therefore, the total profit is maximized when x = 25 units are produced.Wait, just to make sure I didn't make any mistakes in my calculations. Let me go through the steps again.1. Found k by plugging in x = 100 and P = 50 into P² = k/x: 50² = k/100 => k = 250,000. Correct.2. Expressed P as 500 / sqrt(x). Correct.3. Calculated total revenue as x * P = 500*sqrt(x). Correct.4. Subtracted total cost to get profit: L(x) = 500x^(1/2) - 1200 - (2/75)x³. Correct.5. Took derivative: L'(x) = 250x^(-1/2) - (2/25)x². Correct.6. Set derivative equal to zero: 250 / sqrt(x) = (2/25)x². Correct.7. Multiplied both sides by sqrt(x): 250 = (2/25)x^(5/2). Correct.8. Multiplied both sides by 25: 6250 = 2x^(5/2). Correct.9. Divided by 2: 3125 = x^(5/2). Correct.10. Took both sides to the power of 2/5: x = 25. Correct.11. Checked second derivative: L''(x) = -125x^(-3/2) - (4/25)x. At x=25, it's -1 -4 = -5. Correct.So, all steps seem to check out. Therefore, I'm confident that x = 25 is the number of units that should be produced to maximize the total profit.**Final Answer**The factory should produce boxed{25} units to maximize the total profit.

answer：To find the complex number ( z ) that satisfies the equation ( iz = 2 + 4i ), I start by isolating ( z ).First, I divide both sides of the equation by ( i ):[z = frac{2 + 4i}{i}]To simplify this expression, I multiply the numerator and the denominator by ( i ) to eliminate the imaginary unit in the denominator:[z = frac{(2 + 4i)i}{i cdot i} = frac{2i + 4i^2}{i^2}]Since ( i^2 = -1 ), the equation becomes:[z = frac{2i + 4(-1)}{-1} = frac{-4 + 2i}{-1}]Finally, I simplify the fraction:[z = 4 - 2i]The complex number ( z = 4 - 2i ) corresponds to the point ( (4, -2) ) in the complex plane.

answer：Okay, so I need to find the inverse of this matrix:[begin{pmatrix} 5 & 10 -3 & -6 end{pmatrix}]Hmm, I remember that to find the inverse of a 2x2 matrix, there's a formula involving the determinant. Let me try to recall how that works. I think the formula is something like:[A^{-1} = frac{1}{text{det}(A)} begin{pmatrix} d & -b -c & a end{pmatrix}]where the original matrix ( A ) is:[begin{pmatrix} a & b c & d end{pmatrix}]So, first, I need to calculate the determinant of the matrix. The determinant for a 2x2 matrix is ( ad - bc ). Let me plug in the values from my matrix.Here, ( a = 5 ), ( b = 10 ), ( c = -3 ), and ( d = -6 ). So, the determinant ( text{det}(A) ) should be:[text{det}(A) = (5)(-6) - (10)(-3)]Calculating that:[(5)(-6) = -30][(10)(-3) = -30][text{det}(A) = -30 - (-30) = -30 + 30 = 0]Wait, the determinant is zero? That means the matrix is singular, right? And if the determinant is zero, the matrix doesn't have an inverse. So, does that mean I can't find an inverse for this matrix?But the problem says if the inverse doesn't exist, I should provide the zero matrix. Okay, so I guess I just need to write down the zero matrix instead of the inverse.Just to make sure I didn't make a mistake in calculating the determinant, let me double-check:[text{det}(A) = (5)(-6) - (10)(-3) = -30 - (-30) = 0]Yep, that's correct. So, the determinant is definitely zero, which means the inverse doesn't exist. Therefore, the answer should be the zero matrix.I think that's all there is to it. Since the determinant is zero, the matrix doesn't have an inverse, so I'll provide the zero matrix as instructed.

answer：First, I need to determine the number of seconds in half a year. There are 182.5 days in half a year, and each day has 24 hours. Each hour has 60 minutes, and each minute has 60 seconds.So, the total number of seconds in half a year is calculated as:182.5 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute = 15,768,000 seconds.Next, I will use the speed of light, which is approximately 299,792 kilometers per second, to find out how far light travels in half a year. I multiply the speed of light by the total number of seconds in half a year:299,792 km/s × 15,768,000 s = 4,725,817,856,000 kilometers.To express this distance in scientific notation, I approximate it as:4,725,817,856,000 km ≈ 4.73 × 10¹² kilometers.Therefore, the distance light travels in half a year is approximately 4.73 × 10¹² kilometers.