answer：Okay, so I have this problem about an arithmetic sequence. Let me try to understand it step by step. The problem says that the sum of the first n terms of an arithmetic sequence {a_n} is S_n. We are given that S_3 = 12 and S_6 = 42. We need to find the sum of the 10th, 11th, and 12th terms, which is a_{10} + a_{11} + a_{12}. The options are (A) 156, (B) 102, (C) 66, and (D) 48.First, I remember that in an arithmetic sequence, the sum of the first n terms can be calculated using the formula:S_n = n/2 * (2a_1 + (n - 1)d)where a_1 is the first term and d is the common difference.Given that S_3 = 12, let's plug in n = 3 into the formula:S_3 = 3/2 * (2a_1 + 2d) = 12Simplifying this:(3/2) * (2a_1 + 2d) = 12Multiply both sides by 2/3 to solve for (2a_1 + 2d):2a_1 + 2d = 8Divide both sides by 2:a_1 + d = 4 ...(1)Similarly, for S_6 = 42, plug in n = 6:S_6 = 6/2 * (2a_1 + 5d) = 42Simplify:3 * (2a_1 + 5d) = 42Divide both sides by 3:2a_1 + 5d = 14 ...(2)Now, we have two equations:1) a_1 + d = 42) 2a_1 + 5d = 14Let me solve these equations simultaneously. From equation (1), I can express a_1 in terms of d:a_1 = 4 - dSubstitute this into equation (2):2*(4 - d) + 5d = 14Simplify:8 - 2d + 5d = 14Combine like terms:8 + 3d = 14Subtract 8 from both sides:3d = 6Divide by 3:d = 2Now, substitute d = 2 back into equation (1):a_1 + 2 = 4So, a_1 = 2Alright, so the first term is 2 and the common difference is 2. Therefore, the arithmetic sequence is 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, and so on.Now, we need to find a_{10} + a_{11} + a_{12}. Let's find each term:a_{10} = a_1 + 9d = 2 + 9*2 = 2 + 18 = 20a_{11} = a_1 + 10d = 2 + 10*2 = 2 + 20 = 22a_{12} = a_1 + 11d = 2 + 11*2 = 2 + 22 = 24Adding them together:a_{10} + a_{11} + a_{12} = 20 + 22 + 24 = 66Wait, but let me think again. Is there a quicker way to find the sum of these three terms without calculating each individually? Maybe using the sum formula again?Yes, the sum of terms from a_{10} to a_{12} can be considered as the sum of the first 12 terms minus the sum of the first 9 terms. So, S_{12} - S_9.Let me calculate S_{12} and S_9 using the sum formula.First, S_{12}:S_{12} = 12/2 * (2a_1 + 11d) = 6*(2*2 + 11*2) = 6*(4 + 22) = 6*26 = 156Then, S_9:S_9 = 9/2 * (2a_1 + 8d) = (9/2)*(4 + 16) = (9/2)*20 = 9*10 = 90Therefore, S_{12} - S_9 = 156 - 90 = 66So, that confirms the earlier result. The sum is 66.Alternatively, I remember that in an arithmetic sequence, the sums of consecutive terms also form an arithmetic sequence. So, S_3, S_6 - S_3, S_9 - S_6, S_{12} - S_9, etc., form an arithmetic sequence.Given that S_3 = 12 and S_6 = 42, the sum from a_4 to a_6 is 42 - 12 = 30. So, the sums are 12, 30, ?, 66. Wait, but 12, 30, and then the next sum would be 30 + (30 - 12) = 48, and then 48 + 18 = 66. So, the sum from a_{10} to a_{12} is 66.This also gives the same answer. So, regardless of the method, the answer is 66.I think I've verified it enough times. So, I'm confident that the answer is 66.**Final Answer**boxed{C}

answer：First, I need to determine the simple interest (SI) by subtracting the principal amount from the total amount.Next, I'll use the simple interest formula to solve for the rate (R). The formula is:R = (SI × 100) / (P × T)I'll plug in the values: SI is Rs. 7,500, P is Rs. 12,000, and T is 7 years.Calculating this will give me the rate of interest.

answer：Okay, so I have this problem here: "If the sum of the absolute values of all coefficients in the expansion of ((2x - frac{1}{sqrt[3]{x}})^n) is 81, then the constant term is ______." Hmm, let me try to figure this out step by step.First, I remember that the sum of the absolute values of all coefficients in a polynomial expansion can be found by plugging in (x = 1). That makes sense because substituting 1 for (x) essentially removes the variable part, leaving just the coefficients. So, if I set (x = 1) in the expression ((2x - frac{1}{sqrt[3]{x}})^n), it should give me the sum of the absolute values of the coefficients.Let me write that down:[(2(1) - frac{1}{sqrt[3]{1}})^n = (2 - 1)^n = 1^n = 1]Wait, that can't be right because the problem says the sum is 81, not 1. Did I do something wrong? Oh, I see! The problem mentions the sum of the absolute values of all coefficients, but in my substitution, I didn't take the absolute values. Let me correct that.Since we're dealing with absolute values, I should consider the absolute values of each term inside the parentheses before raising them to the power of (n). So, the expression becomes:[(|2x| + |frac{1}{sqrt[3]{x}}|)^n]But actually, since we're substituting (x = 1), the absolute values of the coefficients are just the absolute values of the constants. So, it simplifies to:[(|2| + |frac{1}{sqrt[3]{1}}|)^n = (2 + 1)^n = 3^n]Ah, that makes more sense! So, the sum of the absolute values of all coefficients is (3^n), and this is equal to 81. Therefore, I can set up the equation:[3^n = 81]I know that (81) is (3^4), so:[3^n = 3^4 implies n = 4]Great, so (n = 4). Now, I need to find the constant term in the expansion of ((2x - frac{1}{sqrt[3]{x}})^4).To find the constant term, I should recall the binomial expansion formula. The general term in the expansion of ((a + b)^n) is given by:[T_{k+1} = binom{n}{k} a^{n - k} b^k]In this case, (a = 2x) and (b = -frac{1}{sqrt[3]{x}}). So, the general term is:[T_{k+1} = binom{4}{k} (2x)^{4 - k} left(-frac{1}{sqrt[3]{x}}right)^k]Simplifying this term, let's break it down:First, (binom{4}{k}) is the binomial coefficient.Then, ((2x)^{4 - k}) can be written as (2^{4 - k} x^{4 - k}).Next, (left(-frac{1}{sqrt[3]{x}}right)^k) can be rewritten as ((-1)^k cdot left(x^{-1/3}right)^k = (-1)^k x^{-k/3}).Putting it all together, the general term becomes:[T_{k+1} = binom{4}{k} cdot 2^{4 - k} cdot (-1)^k cdot x^{4 - k - k/3}]Simplify the exponent of (x):[4 - k - frac{k}{3} = 4 - frac{3k}{3} - frac{k}{3} = 4 - frac{4k}{3}]We want the term where the exponent of (x) is zero because that's the constant term. So, set the exponent equal to zero:[4 - frac{4k}{3} = 0]Solving for (k):[4 = frac{4k}{3} implies 4 cdot 3 = 4k implies 12 = 4k implies k = 3]So, the constant term corresponds to (k = 3). Let's plug (k = 3) back into the general term:[T_{4} = binom{4}{3} cdot 2^{4 - 3} cdot (-1)^3 cdot x^{4 - 3 - 3/3}]Simplify each part:- (binom{4}{3} = 4)- (2^{4 - 3} = 2^1 = 2)- ((-1)^3 = -1)- The exponent of (x) is (4 - 3 - 1 = 0), so (x^0 = 1)Putting it all together:[T_{4} = 4 cdot 2 cdot (-1) cdot 1 = 4 cdot 2 cdot (-1) = 8 cdot (-1) = -8]So, the constant term is (-8).Wait, let me double-check my steps to make sure I didn't make a mistake. First, I found (n = 4) correctly by setting the sum of absolute coefficients to 81, which is (3^4). Then, using the binomial expansion, I identified the general term and set the exponent of (x) to zero to find the constant term. I found (k = 3) and substituted back into the term, calculating each part step by step. It all seems to check out.Just to be thorough, let me write out the entire expansion for (n = 4) and see if the constant term is indeed (-8).The expansion of ((2x - frac{1}{sqrt[3]{x}})^4) is:[sum_{k=0}^{4} binom{4}{k} (2x)^{4 - k} left(-frac{1}{sqrt[3]{x}}right)^k]Calculating each term:1. (k = 0): [ binom{4}{0} (2x)^4 left(-frac{1}{sqrt[3]{x}}right)^0 = 1 cdot 16x^4 cdot 1 = 16x^4 ] 2. (k = 1): [ binom{4}{1} (2x)^3 left(-frac{1}{sqrt[3]{x}}right)^1 = 4 cdot 8x^3 cdot (-x^{-1/3}) = 4 cdot 8 cdot (-1) x^{3 - 1/3} = -32x^{8/3} ] 3. (k = 2): [ binom{4}{2} (2x)^2 left(-frac{1}{sqrt[3]{x}}right)^2 = 6 cdot 4x^2 cdot x^{-2/3} = 6 cdot 4 x^{2 - 2/3} = 24x^{4/3} ] 4. (k = 3): [ binom{4}{3} (2x)^1 left(-frac{1}{sqrt[3]{x}}right)^3 = 4 cdot 2x cdot (-x^{-1}) = 4 cdot 2 cdot (-1) x^{1 - 1} = -8x^0 = -8 ] 5. (k = 4): [ binom{4}{4} (2x)^0 left(-frac{1}{sqrt[3]{x}}right)^4 = 1 cdot 1 cdot x^{-4/3} = x^{-4/3} ] So, the expansion is:[16x^4 - 32x^{8/3} + 24x^{4/3} - 8 + x^{-4/3}]Looking at this, the constant term is indeed (-8). All the other terms have exponents of (x) that are not zero, so they aren't constants. Therefore, my earlier calculation was correct.I think I've got it right. The constant term is (-8).**Final Answer**The constant term is boxed{-8}.

answer：First, I need to determine the total amount of money Chad earned or received throughout the year. He made 600 from mowing yards, received 250 for his birthday or holidays, earned 150 by selling old video games, and another 150 from doing odd jobs. Adding these amounts together gives:600 + 250 + 150 + 150 = 1150So, Chad's total earnings for the year are 1150.Next, I need to find out what percentage of this total amount Chad saved. He saved 460.To calculate the percentage saved, I divide the amount saved by the total earnings and then multiply by 100:(460 / 1150) * 100 = 0.4 * 100 = 40%Therefore, Chad saves 40% of his earnings.

answer：Okay, so I have this problem where I need to convert the rectangular coordinates (3, -3) to polar coordinates. I remember that polar coordinates are represented as (r, θ), where r is the distance from the origin to the point, and θ is the angle made with the positive x-axis. First, I think I need to find r. From what I recall, the formula for r is the square root of (x squared plus y squared). So, plugging in the values, that would be sqrt(3^2 + (-3)^2). Let me compute that: 3 squared is 9, and (-3) squared is also 9. Adding those together gives 18. So, r is sqrt(18). Hmm, sqrt(18) can be simplified because 18 is 9 times 2, and the square root of 9 is 3. So, sqrt(18) is 3*sqrt(2). Got that, so r is 3√2.Now, onto finding θ. I know that θ is the angle formed with the positive x-axis, and it's measured in radians. The formula for θ is arctangent of y over x, so θ = arctan(y/x). Here, y is -3 and x is 3, so that would be arctan(-3/3) which simplifies to arctan(-1). Wait, arctan(-1) is a value I should remember. I think arctan(1) is π/4 because tan(π/4) is 1. But since we have a negative value, arctan(-1) would be -π/4. However, angles in polar coordinates are typically given between 0 and 2π, so I need to adjust this negative angle to fit within that range.Since the point (3, -3) is in the fourth quadrant (because x is positive and y is negative), the angle θ should be between 3π/2 and 2π. To convert -π/4 to a positive angle, I can add 2π to it. So, -π/4 + 2π equals (8π/4 - π/4) which is 7π/4. Let me double-check that. If θ is 7π/4, then cos(7π/4) should be √2/2 and sin(7π/4) should be -√2/2. Multiplying these by r, which is 3√2, should give me back the original coordinates. Calculating x: r*cos(θ) = 3√2*(√2/2) = 3*(2)/2 = 3. That matches the original x-coordinate. Calculating y: r*sin(θ) = 3√2*(-√2/2) = 3*(-2)/2 = -3. That also matches the original y-coordinate. So, it seems like my calculations are correct. Therefore, the polar coordinates for the point (3, -3) are (3√2, 7π/4).

answer：Alright, so I've got this problem here about a quadratic inequality. The solution set is given as (-3, 2), and I need to figure out which of the statements A, B, C, D are correct. Let me try to break this down step by step.First, the inequality is ax² + bx + c > 0, and the solution is between -3 and 2. That means the quadratic equation ax² + bx + c = 0 has roots at x = -3 and x = 2. So, I can write the quadratic in its factored form as a(x + 3)(x - 2) = 0. Expanding this, I get a(x² + x - 6) = 0, which simplifies to ax² + bx + c = 0. Comparing coefficients, I can see that b = a and c = -6a.Now, since the solution set is (-3, 2), the quadratic must open downwards. That means the coefficient 'a' must be negative. So, statement A: a < 0 is definitely true.Moving on to statement B: The solution set of the inequality bx - c > 0 is x < 6. Let's substitute b and c with the expressions in terms of a. So, bx - c becomes a*x - (-6a) = a*x + 6a. The inequality is a*x + 6a > 0. Since a is negative, I can factor it out: a(x + 6) > 0. Dividing both sides by a (which is negative, so the inequality sign flips), we get x + 6 < 0, which simplifies to x < -6. But statement B says the solution is x < 6, which is not correct. So, statement B is false.Next, statement C: a - b + c > 0. Let's substitute b and c again. So, a - b + c becomes a - a + (-6a) = -6a. Since a is negative, -6a is positive. Therefore, -6a > 0, which means statement C is true.Finally, statement D: The solution set of the inequality cx² - bx + a < 0 is (-∞, -1/2) ∪ (1/3, +∞). Let's substitute c, b, and a. So, cx² - bx + a becomes -6a*x² - a*x + a. Factoring out 'a' (which is negative), we get a(-6x² - x + 1) < 0. Since a is negative, the inequality flips when we divide both sides by 'a', giving us -6x² - x + 1 > 0. Multiplying both sides by -1 (and flipping the inequality again), we get 6x² + x - 1 < 0. To solve this quadratic inequality, I'll find the roots of 6x² + x - 1 = 0. Using the quadratic formula: x = [-1 ± sqrt(1 + 24)] / 12 = [-1 ± 5]/12. So, the roots are x = (4)/12 = 1/3 and x = (-6)/12 = -1/2. Since the coefficient of x² is positive, the quadratic opens upwards. Therefore, the inequality 6x² + x - 1 < 0 is satisfied between the roots: -1/2 < x < 1/3. But statement D says the solution is (-∞, -1/2) ∪ (1/3, +∞), which is the opposite. So, statement D is false.In summary, statements A and C are correct.