answer：Okay, so I have this problem here: "Given that √(a + 2) is the simplest quadratic root and can be combined with √12, the value of a is ______." Hmm, let me try to figure this out step by step.First, I need to understand what it means for √(a + 2) to be the simplest quadratic root. I think that means it's already simplified, so the expression inside the square root can't be broken down further into a product that includes a perfect square. So, the radicand (the number under the square root) should not have any square factors other than 1. That makes sense.Next, it says that √(a + 2) can be combined with √12. I remember that to combine square roots, they need to be like terms, meaning they have the same radicand. So, √(a + 2) and √12 must be like terms. That implies that √(a + 2) is equal to some multiple of √12, or maybe √12 can be simplified to have the same radicand as √(a + 2).Let me simplify √12 first. I know that 12 can be factored into 4 and 3, and 4 is a perfect square. So, √12 = √(4*3) = √4 * √3 = 2√3. So, √12 simplifies to 2√3. That means √(a + 2) must be a multiple of √3 as well because they can be combined.Since √(a + 2) is the simplest quadratic root, it can't be simplified further. So, the radicand (a + 2) must be 3 times some square number, but since it's already in simplest form, the coefficient in front of √3 must be 1. Wait, is that right?Wait, no. If √(a + 2) can be combined with √12, which is 2√3, then √(a + 2) must be a multiple of √3. So, √(a + 2) could be k√3, where k is some integer. But since √(a + 2) is the simplest quadratic root, k must be 1. Otherwise, if k is greater than 1, then √(a + 2) would not be in simplest form because you could factor out a square.Wait, let me think again. If √(a + 2) is in simplest form, then a + 2 must not have any square factors. So, a + 2 must be equal to 3 times a square number? Or is it just 3?Wait, no. Let's see. If √(a + 2) is to be combined with √12, which is 2√3, then √(a + 2) must be a multiple of √3. So, √(a + 2) = m√3, where m is an integer. Then, squaring both sides, we get a + 2 = m² * 3. So, a + 2 must be three times a perfect square.But since √(a + 2) is in simplest form, m must be 1 because if m were greater than 1, then √(a + 2) would not be in simplest form. For example, if m = 2, then √(a + 2) = 2√3, which is not in simplest form because it can be written as 2√3, but wait, actually, 2√3 is already simplified because 3 is square-free. Hmm, maybe I was wrong earlier.Wait, maybe the key is that √(a + 2) must be equal to √3, because if it's equal to 2√3, then it's not in simplest form? No, that doesn't make sense because 2√3 is simplified. The simplest form just means that the radicand has no perfect square factors, regardless of the coefficient.So, perhaps √(a + 2) can be any multiple of √3, but since it's given as the simplest quadratic root, it's just √3. So, a + 2 = 3, which would make a = 1. But wait, let me check.If a + 2 = 3, then a = 1. Then, √(1 + 2) = √3, which can be combined with √12 because √12 = 2√3. So, √3 and 2√3 are like terms and can be combined. That seems to fit.But wait, could a + 2 be another multiple of 3? Like 12? If a + 2 = 12, then a = 10. Then, √12 is 2√3, which is the same as √12. Wait, but √12 is already simplified to 2√3, so if a + 2 = 12, then √12 is 2√3, which is the same as √12. So, in that case, √12 can be combined with itself, but the problem says √(a + 2) can be combined with √12. So, if a + 2 = 12, then √(a + 2) is √12, which is 2√3, and that can be combined with √12, which is also 2√3. So, that also works.Wait, so does that mean a could be 1 or 10? Hmm, but the problem says "the value of a", implying a unique answer. Maybe I need to consider the simplest form.Wait, if a + 2 is 3, then √3 is already simplified, and it can be combined with √12, which is 2√3. So, that works. If a + 2 is 12, then √12 is 2√3, which is also simplified, and it can be combined with √12 as well. So, both a = 1 and a = 10 would satisfy the condition.But the problem says "the simplest quadratic root". Maybe "simplest" implies that the radicand is as small as possible? So, 3 is smaller than 12, so a = 1 would be the answer.Alternatively, maybe the problem is expecting a unique answer, so perhaps a + 2 must be equal to 3, making a = 1.Wait, let me think again. The problem says √(a + 2) is the simplest quadratic root and can be combined with √12. So, √(a + 2) must be a simplified radical that is like √12. Since √12 simplifies to 2√3, √(a + 2) must be a multiple of √3. So, √(a + 2) = k√3, where k is an integer. Then, a + 2 = k² * 3.But since √(a + 2) is in simplest form, k must be 1 because if k is greater than 1, then √(a + 2) would not be in simplest form. Wait, no, that's not necessarily true. For example, 2√3 is already in simplest form because 3 is square-free. So, k can be any integer, but the radicand must not have square factors. So, a + 2 must be 3 times a perfect square.But since the problem says "the simplest quadratic root", maybe it's implying that √(a + 2) is as simple as possible, meaning the radicand is square-free. So, if a + 2 is 3, then it's square-free. If a + 2 is 12, then it's not square-free because 12 = 4*3, which has a square factor. So, maybe a + 2 must be 3, making a = 1.Yes, that makes sense. So, the simplest quadratic root would have a square-free radicand. Therefore, a + 2 must be 3, so a = 1.Let me check: If a = 1, then √(1 + 2) = √3. √3 can be combined with √12 because √12 = 2√3. So, √3 and 2√3 are like terms, so they can be combined. That fits.If a = 10, then √(10 + 2) = √12 = 2√3. 2√3 can be combined with √12, which is also 2√3, so that also works. But √12 is not in simplest form because 12 has a square factor. Wait, no, √12 simplifies to 2√3, which is in simplest form. So, √12 is already simplified, so a + 2 = 12 is also acceptable.But the problem says √(a + 2) is the simplest quadratic root. So, if a + 2 = 12, then √(a + 2) is √12, which simplifies to 2√3. So, √12 is not in simplest form, but 2√3 is. Wait, so maybe √(a + 2) must be in simplest form, meaning a + 2 must be square-free. Therefore, a + 2 = 3, so a = 1.Yes, that seems to be the case. So, the answer is a = 1.Wait, let me make sure. If a + 2 is 3, then √3 is in simplest form. If a + 2 is 12, then √12 is not in simplest form because 12 has a square factor. So, the problem says √(a + 2) is the simplest quadratic root, so a + 2 must be square-free. Therefore, a + 2 = 3, so a = 1.Yes, that makes sense. So, the value of a is 1.

answer：First, I will simplify the expression ((2 - a)(2 + a) - 2a(a + 3) + 3a^2).I recognize that ((2 - a)(2 + a)) is a difference of squares, which simplifies to (4 - a^2).Next, I'll expand (-2a(a + 3)) to get (-2a^2 - 6a).Now, I'll combine all the terms:[4 - a^2 - 2a^2 - 6a + 3a^2]Combining like terms, the (a^2) terms cancel out, leaving:[4 - 6a]Finally, I'll substitute (a = -frac{1}{3}) into the simplified expression:[4 - 6left(-frac{1}{3}right) = 4 + 2 = 6]

answer：Okay, so I have this limit to compute:lim _{n rightarrow infty}left(frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}right)^{1-2 n}Hmm, it's an expression raised to a power that depends on n, and we're taking the limit as n approaches infinity. I remember that for limits of the form (1 + a/n)^{bn}, we can use the exponential function, but this doesn't look exactly like that. Let me think.First, maybe I should simplify the fraction inside the limit. Both the numerator and the denominator are quadratic polynomials in n. Let me write them out:Numerator: 4n^2 + 4n - 1Denominator: 4n^2 + 2n + 3Since both have the same degree, maybe I can divide numerator and denominator by n^2 to simplify. Let's try that:frac{4n^2 + 4n - 1}{4n^2 + 2n + 3} = frac{4 + frac{4}{n} - frac{1}{n^2}}{4 + frac{2}{n} + frac{3}{n^2}}As n approaches infinity, the terms with 1/n and 1/n^2 will go to zero. So, the fraction simplifies to 4/4 = 1. But wait, the base is approaching 1, and the exponent is approaching negative infinity. So, we have a 1^(-infty) form, which is an indeterminate form. That means I can't just directly substitute; I need to manipulate it further.I remember that when dealing with limits of the form (1 + f(n))^{g(n)}, if f(n) approaches 0 and g(n) approaches infinity, we can use the approximation (1 + f(n))^{g(n)} approx e^{f(n)g(n)}. Maybe I can rewrite the expression in this form.Let me set:frac{4n^2 + 4n -1}{4n^2 + 2n +3} = 1 + frac{(4n^2 +4n -1) - (4n^2 +2n +3)}{4n^2 +2n +3}Calculating the numerator of the fraction:(4n^2 +4n -1) - (4n^2 +2n +3) = (0n^2) + (2n) -4 = 2n -4So, the expression becomes:1 + frac{2n -4}{4n^2 +2n +3}Therefore, the original limit can be written as:lim_{n to infty} left(1 + frac{2n -4}{4n^2 +2n +3}right)^{1 - 2n}Now, let me denote:a_n = frac{2n -4}{4n^2 +2n +3}So, the expression is (1 + a_n)^{1 - 2n}. As n approaches infinity, a_n approaches 0 because the denominator is quadratic and the numerator is linear. So, we can use the approximation:(1 + a_n)^{1 - 2n} approx e^{a_n (1 - 2n)}So, let's compute the exponent:a_n (1 - 2n) = frac{2n -4}{4n^2 +2n +3} times (1 - 2n)Let me compute this product:First, multiply numerator and denominator:Numerator: (2n -4)(1 - 2n) = 2n(1) + 2n(-2n) -4(1) -4(-2n) = 2n -4n^2 -4 +8n = (-4n^2) + (2n +8n) -4 = -4n^2 +10n -4Denominator: 4n^2 +2n +3So, the exponent becomes:frac{-4n^2 +10n -4}{4n^2 +2n +3}Again, as n approaches infinity, both numerator and denominator are dominated by their quadratic terms. So, let's factor out n^2 from numerator and denominator:Numerator: -4n^2(1 - frac{10}{4n} + frac{4}{4n^2}) = -4n^2(1 - frac{5}{2n} + frac{1}{n^2})Denominator: 4n^2(1 + frac{2}{4n} + frac{3}{4n^2}) = 4n^2(1 + frac{1}{2n} + frac{3}{4n^2})So, the exponent becomes:frac{-4n^2(1 - frac{5}{2n} + frac{1}{n^2})}{4n^2(1 + frac{1}{2n} + frac{3}{4n^2})} = frac{-4n^2}{4n^2} times frac{1 - frac{5}{2n} + frac{1}{n^2}}{1 + frac{1}{2n} + frac{3}{4n^2}} = -1 times frac{1 - frac{5}{2n} + frac{1}{n^2}}{1 + frac{1}{2n} + frac{3}{4n^2}}As n approaches infinity, the terms with 1/n and 1/n^2 go to zero, so the fraction simplifies to:-1 times frac{1 - 0 + 0}{1 + 0 + 0} = -1Therefore, the exponent tends to -1. So, the original limit is approximately:e^{-1} = frac{1}{e}But wait, let me double-check. I approximated (1 + a_n)^{1 - 2n} as e^{a_n(1 - 2n)}. Is this a valid approximation?Yes, because as n approaches infinity, a_n approaches 0, and the exponent (1 - 2n) approaches negative infinity. So, the expression is of the form (1 + 0)^{-infty}, which is indeterminate, but the exponential approximation is a standard technique here.Alternatively, I can use logarithms to compute the limit. Let me try that method to confirm.Let me denote:L = lim_{n to infty} left(frac{4n^2 +4n -1}{4n^2 +2n +3}right)^{1 - 2n}Take the natural logarithm of both sides:ln L = lim_{n to infty} (1 - 2n) lnleft(frac{4n^2 +4n -1}{4n^2 +2n +3}right)Now, let me compute the logarithm term:lnleft(frac{4n^2 +4n -1}{4n^2 +2n +3}right) = lnleft(1 + frac{2n -4}{4n^2 +2n +3}right)Again, as n approaches infinity, the fraction frac{2n -4}{4n^2 +2n +3} approaches 0. So, we can use the approximation ln(1 + x) approx x for small x.Thus,lnleft(1 + frac{2n -4}{4n^2 +2n +3}right) approx frac{2n -4}{4n^2 +2n +3}Therefore,ln L approx lim_{n to infty} (1 - 2n) times frac{2n -4}{4n^2 +2n +3}Which is the same exponent as before:lim_{n to infty} frac{(1 - 2n)(2n -4)}{4n^2 +2n +3}Which we already computed as -1. So,ln L = -1 implies L = e^{-1} = frac{1}{e}Okay, so both methods give the same result. Therefore, the limit is 1/e.But wait, just to be thorough, let me compute the exponent more carefully.We had:frac{-4n^2 +10n -4}{4n^2 +2n +3}Let me divide numerator and denominator by n^2:frac{-4 + frac{10}{n} - frac{4}{n^2}}{4 + frac{2}{n} + frac{3}{n^2}}As n approaches infinity, this becomes:frac{-4 + 0 - 0}{4 + 0 + 0} = frac{-4}{4} = -1So yes, the exponent tends to -1, so the limit is e^{-1}.Therefore, I'm confident that the limit is 1/e.**Final Answer**The limit is boxed{dfrac{1}{e}}.

answer：Alright, let's try to figure out how many people attended the ball. The problem says there are fewer than 50 people in total, and it's a mix of ladies and gentlemen. During the first dance, a quarter of the ladies weren't invited to dance, and two-sevenths of the gentlemen didn't invite anyone. Hmm, okay, let's break this down step by step.First, let's denote the number of ladies as L and the number of gentlemen as G. So, the total number of people is L + G, and we know that L + G < 50.Now, during the first dance, a quarter of the ladies weren't invited to dance. That means three-quarters of the ladies were dancing. So, the number of ladies who danced is (3/4)L. Similarly, two-sevenths of the gentlemen didn't invite anyone, which means five-sevenths of the gentlemen did invite someone. So, the number of gentlemen who danced is (5/7)G.Since each dance involves one gentleman and one lady, the number of dancing ladies must equal the number of dancing gentlemen. Otherwise, there would be an imbalance, and some people wouldn't have partners. So, we can set up the equation:(3/4)L = (5/7)GThis equation should help us relate the number of ladies and gentlemen. Let's write that down:(3/4)L = (5/7)GTo solve this, let's cross-multiply to eliminate the fractions. Multiplying both sides by 28 (which is the least common multiple of 4 and 7) to make the numbers easier:28*(3/4)L = 28*(5/7)GSimplifying both sides:7*3 L = 4*5 G21L = 20GSo, 21L = 20G. Let's write this as:21L = 20GWhich can be rearranged to:L = (20/21)GHmm, so the number of ladies is (20/21) times the number of gentlemen. Since the number of people must be whole numbers, both L and G must be integers. Therefore, G must be a multiple of 21 to make L an integer because 20 and 21 are coprime (they have no common factors other than 1). Similarly, L must be a multiple of 20.Let's denote G as 21k, where k is a positive integer. Then, L would be 20k.So, G = 21k and L = 20k.Therefore, the total number of people is L + G = 20k + 21k = 41k.But we know that the total number of people is fewer than 50, so:41k < 50Let's solve for k:k < 50/41k < approximately 1.2195Since k must be a positive integer, the only possible value is k = 1.Therefore, G = 21*1 = 21 and L = 20*1 = 20.So, the total number of people is 21 + 20 = 41.Wait, let me verify this because sometimes when dealing with fractions, things can get tricky. Let's check if the numbers make sense.Number of ladies not dancing: 1/4 of 20 is 5. So, 20 - 5 = 15 ladies danced.Number of gentlemen not inviting anyone: 2/7 of 21 is 6. So, 21 - 6 = 15 gentlemen danced.Yes, 15 dancing ladies and 15 dancing gentlemen. That matches up perfectly, so everyone who wanted to dance had a partner.Also, 21 + 20 is 41, which is less than 50, so that fits the condition.Is there another possible value for k? If k=2, then total people would be 82, which is way over 50. So, k=1 is the only solution.Therefore, the total number of people who attended the ball is 41.**Final Answer**The total number of people who attended the ball is boxed{41}.

answer：Okay, so I have this arithmetic sequence problem here. Let me try to figure it out step by step. The problem says:"In the arithmetic sequence {a_n}, if 2(a₁ + a₄ + a₇) + 3(a₉ + a₁₁) = 24, then the sum of the first 13 terms of this sequence equals ______."Alright, so I need to find the sum of the first 13 terms. Hmm, arithmetic sequences have a common difference, right? So, let's recall some properties of arithmetic sequences.First, the general term of an arithmetic sequence is given by:a_n = a₁ + (n - 1)dwhere a₁ is the first term and d is the common difference.Also, the sum of the first n terms, S_n, is given by:S_n = n/2 [2a₁ + (n - 1)d]or sometimes written as:S_n = n(a₁ + a_n)/2Since I need to find S₁₃, maybe I can express everything in terms of a₁ and d, and then solve for them using the given equation.Let me write down the given equation:2(a₁ + a₄ + a₇) + 3(a₉ + a₁₁) = 24Let me substitute the expressions for each term using the general formula.First, a₁ is just a₁.a₄ = a₁ + (4 - 1)d = a₁ + 3da₇ = a₁ + (7 - 1)d = a₁ + 6dSimilarly, a₉ = a₁ + (9 - 1)d = a₁ + 8da₁₁ = a₁ + (11 - 1)d = a₁ + 10dSo, let's substitute these into the equation:2(a₁ + (a₁ + 3d) + (a₁ + 6d)) + 3((a₁ + 8d) + (a₁ + 10d)) = 24Now, let's simplify each part step by step.First, compute the terms inside the first set of parentheses:a₁ + (a₁ + 3d) + (a₁ + 6d) = a₁ + a₁ + 3d + a₁ + 6dCombine like terms:= 3a₁ + 9dThen, multiply by 2:2*(3a₁ + 9d) = 6a₁ + 18dNow, compute the terms inside the second set of parentheses:(a₁ + 8d) + (a₁ + 10d) = a₁ + 8d + a₁ + 10dCombine like terms:= 2a₁ + 18dMultiply by 3:3*(2a₁ + 18d) = 6a₁ + 54dNow, add these two results together:6a₁ + 18d + 6a₁ + 54d = (6a₁ + 6a₁) + (18d + 54d) = 12a₁ + 72dSo, the equation becomes:12a₁ + 72d = 24Hmm, let's simplify this equation. I can factor out 12:12(a₁ + 6d) = 24Divide both sides by 12:a₁ + 6d = 2Interesting. So, a₁ + 6d = 2.Wait, let me think. In the arithmetic sequence, a₁ + 6d is actually a₇, right? Because a₇ = a₁ + 6d.So, that tells me that a₇ = 2. Hmm, that might be useful.But let's see, I need to find the sum of the first 13 terms, which is S₁₃.Using the sum formula:S₁₃ = 13/2 [2a₁ + (13 - 1)d] = 13/2 [2a₁ + 12d]Simplify inside the brackets:2a₁ + 12d = 2(a₁ + 6d)But from earlier, we know that a₁ + 6d = 2, so:2(a₁ + 6d) = 2*2 = 4Therefore, S₁₃ = 13/2 * 4 = 13 * 2 = 26Wait, so the sum of the first 13 terms is 26? That seems straightforward.Let me verify my steps to make sure I didn't make a mistake.1. Expressed each term a₄, a₇, a₉, a₁₁ in terms of a₁ and d. That seems correct.2. Substituted into the given equation:2(a₁ + a₄ + a₇) + 3(a₉ + a₁₁) = 243. Simplified each part:- a₁ + a₄ + a₇ = 3a₁ + 9d- a₉ + a₁₁ = 2a₁ + 18d4. Multiplied and added:2*(3a₁ + 9d) = 6a₁ + 18d3*(2a₁ + 18d) = 6a₁ + 54dTotal: 12a₁ + 72d = 245. Factored out 12: 12(a₁ + 6d) = 24 => a₁ + 6d = 26. Recognized that a₁ + 6d is a₇, so a₇ = 2.7. Then, for S₁₃, used the formula:S₁₃ = 13/2 [2a₁ + 12d] = 13/2 * 2(a₁ + 6d) = 13/2 * 4 = 26Yes, that seems consistent.Alternatively, another way to think about the sum S₁₃ is that it's equal to 13 times the average of the first and thirteenth terms. Since in an arithmetic sequence, the average of the first and nth term is equal to the average of all terms.So, S₁₃ = 13*(a₁ + a₁₃)/2But a₁₃ = a₁ + 12d, so:S₁₃ = 13*(2a₁ + 12d)/2 = 13*(a₁ + 6d)But from earlier, a₁ + 6d = 2, so:S₁₃ = 13*2 = 26Same result. So, that's a good check.Therefore, I'm confident that the sum of the first 13 terms is 26.**Final Answer**The sum of the first 13 terms is boxed{26}.

answer：First, I need to convert all measurements to the same unit to make calculations easier. I'll convert meters to feet since the final answer needs to be in feet.Next, I'll break down Henry's movements into their respective directions:- Walking 9 meters north.- Turning east and walking 24 feet.- Turning south and walking 9 meters plus 32 feet.After converting the meters to feet, I'll calculate the net north-south displacement by subtracting the southward distance from the northward distance. Then, I'll determine the eastward displacement.Finally, I'll use the Pythagorean theorem to find the straight-line distance from Henry's starting point to his final position.