💡Okay, so I have this problem here: In the plane rectangular coordinate system xOy, the coordinates of the point A(1,4) symmetric to the axis of symmetry of the parabola y = a(x + 2)^2 are ______.Hmm, I need to find the symmetric point of A(1,4) with respect to the axis of symmetry of the given parabola. Let me break this down step by step.First, I remember that the axis of symmetry of a parabola in the form y = a(x - h)^2 + k is the vertical line x = h. In this case, the equation is y = a(x + 2)^2. So, comparing it to the standard form, h is -2 because (x + 2) is the same as (x - (-2)). Therefore, the axis of symmetry is x = -2.Alright, so the axis of symmetry is the vertical line x = -2. Now, I need to find the point that is symmetric to A(1,4) with respect to this line.To find the symmetric point, I need to determine how far point A is from the axis of symmetry and then place the symmetric point the same distance on the other side of the axis.Let me visualize this. The axis of symmetry is at x = -2, and point A is at x = 1. So, the distance between x = 1 and x = -2 is |1 - (-2)| = |1 + 2| = 3 units. That means point A is 3 units to the right of the axis of symmetry.Therefore, the symmetric point should be 3 units to the left of the axis of symmetry. So, starting from x = -2, if I move 3 units to the left, I subtract 3 from -2: -2 - 3 = -5. So, the x-coordinate of the symmetric point is -5.The y-coordinate remains the same because reflection over a vertical line doesn't change the y-value. So, the y-coordinate is still 4.Putting it all together, the symmetric point should be (-5, 4).Wait, let me double-check my calculations to make sure I didn't make a mistake. The distance from A(1,4) to x = -2 is indeed 3 units. Subtracting 3 from -2 gives -5, and the y-coordinate stays 4. Yep, that seems correct.I think that's it. The symmetric point is (-5, 4).

💡First, I need to determine how much money Fred earned over the weekend by washing cars.I know that Fred had 19 last week and now has 40 after washing cars.To find out how much he earned, I subtract the initial amount from the final amount.So, 40 minus 19 equals 21.Therefore, Fred earned 21 over the weekend.

💡First, I need to determine the effect of the measurement errors on the sides of the rectangle. One side is measured with a 10% excess, meaning the measured length is 110% of the actual length. The other side is measured with a 5% deficit, so the measured width is 95% of the actual width.Next, I'll calculate the measured area using these incorrect dimensions. The measured area is the product of the measured length and measured width.To find the error in the area, I'll compare the measured area to the actual area. The error is the difference between the measured area and the actual area.Finally, to express this error as a percentage, I'll divide the error by the actual area and multiply by 100. This will give me the percentage error in the area calculation.

💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. First, part (I) says that the sequence {a_n} is an arithmetic sequence with a non-zero common difference. So, that means each term increases by a fixed amount, right? Let's call that common difference 'd'. It also says that S_6 = 60. I remember that S_n is the sum of the first n terms of an arithmetic sequence. The formula for the sum of the first n terms is S_n = n/2 * [2a_1 + (n-1)d], where a_1 is the first term. So, for S_6, that would be 6/2 * [2a_1 + 5d] = 60. Let me write that down:6/2 * [2a_1 + 5d] = 60 Simplifying that, 3*(2a_1 + 5d) = 60 So, 2a_1 + 5d = 20. Okay, that's one equation.Next, it says that a_1, a_6, a_{21} form a geometric sequence. Hmm, a geometric sequence means that each term is multiplied by a common ratio. So, if a_1, a_6, a_{21} are in geometric sequence, then a_6 / a_1 = a_{21} / a_6. That ratio should be the same.Let me express a_6 and a_{21} in terms of a_1 and d. Since it's an arithmetic sequence, a_n = a_1 + (n-1)d. So:a_6 = a_1 + 5d a_{21} = a_1 + 20dSo, plugging into the geometric sequence condition:(a_1 + 5d) / a_1 = (a_1 + 20d) / (a_1 + 5d)Let me write that as:(a_1 + 5d)^2 = a_1(a_1 + 20d)Expanding both sides:Left side: (a_1 + 5d)^2 = a_1^2 + 10a_1d + 25d^2 Right side: a_1(a_1 + 20d) = a_1^2 + 20a_1dSo, setting them equal:a_1^2 + 10a_1d + 25d^2 = a_1^2 + 20a_1dSubtract a_1^2 from both sides:10a_1d + 25d^2 = 20a_1dSubtract 10a_1d from both sides:25d^2 = 10a_1dDivide both sides by d (since d is non-zero):25d = 10a_1 So, 5d = 2a_1 Therefore, a_1 = (5/2)dOkay, so now I have two equations:1. 2a_1 + 5d = 20 2. a_1 = (5/2)dLet me substitute equation 2 into equation 1:2*(5/2)d + 5d = 20 Simplify: 5d + 5d = 20 So, 10d = 20 Thus, d = 2Now, plug d = 2 into equation 2:a_1 = (5/2)*2 = 5So, the first term is 5 and the common difference is 2. Therefore, the general term formula for {a_n} is:a_n = a_1 + (n - 1)d = 5 + (n - 1)*2 = 5 + 2n - 2 = 2n + 3Wait, let me check that:a_1 = 2*1 + 3 = 5, correct a_2 = 2*2 + 3 = 7, which is 5 + 2, correct a_6 = 2*6 + 3 = 15, which should be a_1 + 5d = 5 + 10 = 15, correct a_{21} = 2*21 + 3 = 45, which should be a_1 + 20d = 5 + 40 = 45, correctGood, that seems consistent.Now, moving on to part (II). It says that sequence {b_n} satisfies b_{n+1} - b_n = a_n for n in natural numbers, and b_1 = 3. We need to find the general term formula for {b_n}.Hmm, so {b_n} is a sequence where the difference between consecutive terms is given by the arithmetic sequence {a_n}. So, to find b_n, we can think of it as the sum of the a_n terms up to n-1, plus b_1.Wait, let me think. Since b_{n+1} - b_n = a_n, then for each n, b_{n} = b_{n-1} + a_{n-1}. So, recursively, b_n = b_1 + sum_{k=1}^{n-1} a_k.So, b_n = 3 + sum_{k=1}^{n-1} a_k.But since a_k is an arithmetic sequence with a_1 = 5 and d = 2, the sum of the first m terms is S_m = m/2*(2a_1 + (m - 1)d).So, sum_{k=1}^{n-1} a_k = S_{n-1} = (n - 1)/2*(2*5 + (n - 2)*2) = (n - 1)/2*(10 + 2n - 4) = (n - 1)/2*(2n + 6) = (n - 1)(n + 3)Therefore, b_n = 3 + (n - 1)(n + 3) = 3 + n^2 + 3n - n - 3 = n^2 + 2nWait, let me compute that step by step:(n - 1)(n + 3) = n(n + 3) - 1(n + 3) = n^2 + 3n - n - 3 = n^2 + 2n - 3So, b_n = 3 + n^2 + 2n - 3 = n^2 + 2nSimplify: b_n = n(n + 2)Let me check for n=1: b_1 = 1*(1 + 2) = 3, correct n=2: b_2 = b_1 + a_1 = 3 + 5 = 8, and formula gives 2*(2 + 2) = 8, correct n=3: b_3 = b_2 + a_2 = 8 + 7 = 15, formula gives 3*(3 + 2) = 15, correct n=4: b_4 = 15 + 9 = 24, formula gives 4*6=24, correctLooks good. So, the general term for {b_n} is n(n + 2).**Final Answer**(I) The general term formula for the sequence is boxed{2n + 3}.(II) The general term formula for the sequence is boxed{n(n + 2)}.

💡First, I need to calculate the total area of the rectangular park in square feet. The park is 250 feet wide and 350 feet long, so the area is 250 multiplied by 350, which equals 87,500 square feet.Next, there's a smaller square area within the park that is 50 feet on each side. The area of this square is 50 multiplied by 50, which is 2,500 square feet.To find the remaining area of the park excluding this smaller square, I subtract the area of the square from the total area of the park: 87,500 square feet minus 2,500 square feet equals 85,000 square feet.Since the ant density is given per square inch for the main area, I need to convert the remaining area from square feet to square inches. There are 144 square inches in a square foot, so 85,000 square feet multiplied by 144 equals 12,240,000 square inches.Now, I can calculate the number of ants in the main area by multiplying the ant density (4 ants per square inch) by the area in square inches: 4 multiplied by 12,240,000 equals 48,960,000 ants.For the smaller square area, the ant density is 6 ants per square foot. So, I multiply 6 by 2,500 square feet to get 15,000 ants.Finally, I add the number of ants in both areas to find the total number of ants in the park: 48,960,000 ants plus 15,000 ants equals 48,975,000 ants. This number is closest to 50 million.

💡First, I need to evaluate the expression ( z = i^2 + i^3 ).I know that ( i^2 = -1 ) and ( i^3 = -i ).Substituting these values into the expression, I get:[z = -1 + (-i) = -1 - i]This means the complex number ( z ) has a real part of (-1) and an imaginary part of (-1).On the complex plane, the real part corresponds to the x-coordinate, and the imaginary part corresponds to the y-coordinate. Therefore, the point representing ( z ) is at ((-1, -1)).Since both the x and y coordinates are negative, the point lies in the third quadrant of the complex plane.