💡Alright, so I've got this problem here: Given that ( a ) and ( b ) are unequal positive numbers, and ( a^2 - a + b^2 - b + ab = 0 ), I need to find the range of values for ( a + b ). The options are A: ( (0, dfrac{4}{3}) ), B: ( (1, dfrac{4}{3}) ), C: ( (0, dfrac{3}{2}) ), and D: ( (1, dfrac{3}{2}) ).Hmm, okay. Let me try to unpack this step by step.First, the equation given is ( a^2 - a + b^2 - b + ab = 0 ). I notice that this equation involves both ( a ) and ( b ), and it's quadratic in nature. Since ( a ) and ( b ) are positive and unequal, I need to find some relationship between them that can help me determine the range of ( a + b ).Maybe I can rearrange the equation to make it more manageable. Let's see:( a^2 + b^2 + ab - a - b = 0 ).Hmm, that looks a bit like the expansion of ( (a + b)^2 ), which is ( a^2 + 2ab + b^2 ). But in our equation, we have ( a^2 + b^2 + ab ) instead. So, it's similar but not the same.Let me write down the equation again:( a^2 + b^2 + ab = a + b ).Maybe I can express this in terms of ( a + b ). Let's denote ( S = a + b ) and ( P = ab ). Then, ( a^2 + b^2 = (a + b)^2 - 2ab = S^2 - 2P ). So, substituting back into the equation:( S^2 - 2P + P = S ).Simplifying that:( S^2 - P = S ).So, ( P = S^2 - S ).Okay, so now I have ( P ) in terms of ( S ). But ( P ) is also the product ( ab ), which is positive since both ( a ) and ( b ) are positive. So, ( P > 0 ). Therefore, ( S^2 - S > 0 ).Let me solve this inequality:( S^2 - S > 0 ).Factor:( S(S - 1) > 0 ).So, this inequality holds when either:1. ( S > 0 ) and ( S - 1 > 0 ) ⇒ ( S > 1 ), or2. ( S < 0 ) and ( S - 1 < 0 ) ⇒ ( S < 0 ).But since ( a ) and ( b ) are positive, ( S = a + b ) must also be positive. So, the second case where ( S < 0 ) is irrelevant here. Therefore, ( S > 1 ).So, ( a + b > 1 ). That gives me a lower bound for ( S ). Now, I need to find an upper bound.I remember that for two positive numbers, the arithmetic mean is greater than or equal to the geometric mean. So, ( dfrac{a + b}{2} geq sqrt{ab} ). Squaring both sides, ( left( dfrac{a + b}{2} right)^2 geq ab ).But I also have ( P = ab = S^2 - S ). So, substituting:( left( dfrac{S}{2} right)^2 geq S^2 - S ).Simplify:( dfrac{S^2}{4} geq S^2 - S ).Multiply both sides by 4 to eliminate the denominator:( S^2 geq 4S^2 - 4S ).Bring all terms to one side:( S^2 - 4S^2 + 4S geq 0 ).Simplify:( -3S^2 + 4S geq 0 ).Multiply both sides by -1 (remembering to reverse the inequality sign):( 3S^2 - 4S leq 0 ).Factor:( S(3S - 4) leq 0 ).So, this inequality holds when:1. ( S leq 0 ) and ( 3S - 4 geq 0 ) ⇒ Not possible since ( S > 1 ).2. ( S geq 0 ) and ( 3S - 4 leq 0 ) ⇒ ( S leq dfrac{4}{3} ).Since ( S > 1 ) from earlier, combining both results, we have:( 1 < S leq dfrac{4}{3} ).But wait, the problem states that ( a ) and ( b ) are unequal. So, we need to ensure that ( a neq b ). If ( a = b ), then the equation would have a different solution, but since they are unequal, we might need to adjust the upper bound slightly.Let me think about this. If ( a ) approaches ( b ), then ( S ) approaches ( 2a ), and ( P ) approaches ( a^2 ). But since ( a neq b ), ( S ) can't actually reach ( dfrac{4}{3} ) because that would require ( a = b ). Wait, is that the case?Let me test ( S = dfrac{4}{3} ). If ( S = dfrac{4}{3} ), then ( P = left( dfrac{4}{3} right)^2 - dfrac{4}{3} = dfrac{16}{9} - dfrac{12}{9} = dfrac{4}{9} ).So, ( a ) and ( b ) would satisfy ( a + b = dfrac{4}{3} ) and ( ab = dfrac{4}{9} ). Solving the quadratic equation ( x^2 - dfrac{4}{3}x + dfrac{4}{9} = 0 ), we get:( x = dfrac{dfrac{4}{3} pm sqrt{left( dfrac{4}{3} right)^2 - 4 cdot 1 cdot dfrac{4}{9}}}{2} ).Simplify the discriminant:( left( dfrac{16}{9} right) - dfrac{16}{9} = 0 ).So, ( x = dfrac{dfrac{4}{3}}{2} = dfrac{2}{3} ).Thus, both ( a ) and ( b ) would be ( dfrac{2}{3} ), which contradicts the condition that ( a ) and ( b ) are unequal. Therefore, ( S ) cannot actually reach ( dfrac{4}{3} ); it can only approach it. Hence, the upper bound is strictly less than ( dfrac{4}{3} ).Therefore, the range of ( a + b ) is ( (1, dfrac{4}{3}) ).So, the correct answer should be option B.**Final Answer**boxed{B}

💡Okay, so I have this problem: I need to find the smallest natural number by which 720 must be multiplied to get a perfect cube. Hmm, okay. I remember that a perfect cube is a number that can be written as some natural number raised to the power of three. So, for example, 8 is a perfect cube because it's 2 cubed, and 27 is 3 cubed, and so on.First, I think I need to understand what makes a number a perfect cube. From what I recall, in terms of prime factorization, a perfect cube has exponents in its prime factors that are multiples of three. So, if I can factorize 720 into its prime components, I can see what's missing to make each exponent a multiple of three.Alright, let's start by factorizing 720. I know that 720 is a commonly used number, especially in factorials and such. Let me break it down step by step.720 divided by 2 is 360.360 divided by 2 is 180.180 divided by 2 is 90.90 divided by 2 is 45.Okay, so that's four 2s. Now, 45 isn't divisible by 2, so I'll move to the next prime number, which is 3.45 divided by 3 is 15.15 divided by 3 is 5.And 5 divided by 3 doesn't work, so I move to the next prime, which is 5.5 divided by 5 is 1.So, putting it all together, the prime factors of 720 are 2^4 * 3^2 * 5^1. Let me write that down:720 = 2^4 * 3^2 * 5^1Now, for a number to be a perfect cube, each of the exponents in its prime factorization needs to be a multiple of 3. So, I need to adjust the exponents of 2, 3, and 5 so that they are all multiples of 3.Looking at each prime factor:- For 2: The exponent is 4. The next multiple of 3 after 4 is 6. So, I need 6 - 4 = 2 more 2s.- For 3: The exponent is 2. The next multiple of 3 after 2 is 3. So, I need 3 - 2 = 1 more 3.- For 5: The exponent is 1. The next multiple of 3 after 1 is 3. So, I need 3 - 1 = 2 more 5s.Therefore, to make 720 a perfect cube, I need to multiply it by 2^2 * 3^1 * 5^2.Let me calculate that:2^2 is 4,3^1 is 3,5^2 is 25.So, multiplying these together: 4 * 3 = 12, and 12 * 25 = 300.So, the smallest natural number I need to multiply 720 by is 300.But wait, let me double-check. If I multiply 720 by 300, I should get a perfect cube.720 * 300 = 216,000.Let me factorize 216,000 to see if it's a perfect cube.216,000 divided by 2 is 108,000.108,000 divided by 2 is 54,000.54,000 divided by 2 is 27,000.27,000 divided by 2 is 13,500.13,500 divided by 2 is 6,750.6,750 divided by 2 is 3,375.Okay, that's six 2s.3,375 divided by 3 is 1,125.1,125 divided by 3 is 375.375 divided by 3 is 125.Okay, that's three 3s.125 divided by 5 is 25.25 divided by 5 is 5.5 divided by 5 is 1.That's three 5s.So, 216,000 = 2^6 * 3^3 * 5^3.Now, since all the exponents are multiples of 3, this is indeed a perfect cube. Specifically, it's (2^2 * 3 * 5)^3, which is (4 * 3 * 5)^3 = (60)^3.So, 60 cubed is 216,000, which matches our calculation. Therefore, multiplying 720 by 300 gives us a perfect cube.I think that makes sense. I started by factorizing 720, figured out what exponents were needed to make it a cube, calculated the necessary multiplier, and then verified by multiplying and factorizing the result. Everything checks out, so I'm confident that 300 is the correct answer.

💡First, I need to determine the number of leap years and regular years within the 210-year span from 1812 to 2022. Leap years occur every 4 years, but years divisible by 100 are not leap years unless they are also divisible by 400. In this case, the year 1900 is a non-leap year. Therefore, there are 51 leap years and 159 regular years.Next, I calculate the total number of days to move back. Each regular year contributes 1 day backward modulo 7, and each leap year contributes 2 days backward modulo 7. This results in a total of 261 days to move back.Reducing 261 modulo 7 gives 3, meaning I need to move 3 days backward from Monday. Counting backward from Monday, the days are Sunday (1), Saturday (2), and Friday (3). Therefore, Charles Dickens was born on a Friday.

💡Okay, so I have this problem where point A is at (-3, 2), and F is the focus of the parabola given by the equation y² = -4x. I need to find the coordinates of point P on the parabola such that the sum of the distances |PA| + |PF| is minimized. Hmm, okay, let's break this down step by step.First, I remember that for a parabola, the definition involves a focus and a directrix. The standard form of a parabola that opens to the left is y² = -4ax, where 'a' is the distance from the vertex to the focus. Comparing this with the given equation y² = -4x, it looks like 4a = 4, so a = 1. That means the focus F is at (-a, 0) which is (-1, 0). Got that down.Now, the directrix of a parabola is a line perpendicular to the axis of symmetry. Since this parabola opens to the left, the axis of symmetry is the x-axis, so the directrix should be a vertical line. The directrix is located at x = a, which in this case is x = 1. So, the directrix is the line x = 1.I remember that one of the properties of a parabola is that any point on the parabola is equidistant from the focus and the directrix. So, for any point P on the parabola, the distance from P to F is equal to the distance from P to the directrix. Let me denote this distance as d. So, |PF| = d.The problem asks for the minimum value of |PA| + |PF|. Since |PF| is equal to d, we can rewrite the expression as |PA| + d. So, we need to minimize |PA| + d.Wait, so if |PA| + d is the expression we need to minimize, and d is the distance from P to the directrix, which is x = 1, then maybe we can interpret this as |PA| + distance from P to x=1. Hmm, how can we approach this?I think this might relate to reflecting points across the directrix or something like that. Let me recall that in optimization problems involving distances, sometimes reflecting a point across a line can help find the shortest path.In this case, since the directrix is x = 1, maybe we can reflect point A across the directrix and then find the distance from the reflection to the focus? Or perhaps the other way around.Let me think. If I reflect point A across the directrix x = 1, what would be the coordinates of the reflection? The directrix is a vertical line, so reflecting across it would invert the x-coordinate relative to the line x = 1.Point A is at (-3, 2). The distance from A to the directrix x = 1 is |(-3) - 1| = 4 units. So, reflecting A across x = 1 would place it 4 units on the other side of the directrix. So, starting from x = 1, moving 4 units to the right (since A is 4 units to the left), the x-coordinate becomes 1 + 4 = 5. The y-coordinate remains the same, so the reflection point A' would be at (5, 2).Now, if I consider the expression |PA| + |PF|, and since |PF| = d (distance from P to directrix), which is equal to the distance from P to F, maybe I can relate this to the reflection.Wait, actually, since |PF| is equal to the distance from P to the directrix, which is a vertical line x = 1, maybe I can express |PA| + |PF| as |PA| + distance from P to x=1.But how does that help me? Hmm.Alternatively, since |PF| is equal to the distance from P to the directrix, which is x=1, and |PA| is the distance from P to A, maybe I can think of |PA| + |PF| as |PA| + distance from P to x=1.But I'm not sure if that directly helps. Maybe another approach.Wait, I remember that in optimization problems, especially involving reflections, the minimal path often involves reflecting a point and then finding a straight line distance.So, if I reflect point A across the directrix, getting point A' at (5, 2), then the minimal path from A to F via a point P on the parabola would correspond to the straight line from A' to F intersecting the parabola at point P.Is that right? Let me think.Because for any point P on the parabola, |PF| = distance from P to directrix. So, |PA| + |PF| is equal to |PA| + distance from P to directrix.But if I reflect A across the directrix to get A', then the distance from P to A is equal to the distance from P to A' minus twice the distance from A to the directrix? Hmm, maybe not.Wait, no. If I reflect A over the directrix, then the distance from P to A is equal to the distance from P to A' minus twice the distance from A to the directrix? Hmm, maybe not exactly.Alternatively, perhaps |PA| + |PF| is equal to |PA| + distance from P to directrix, which is equal to |PA| + |PA'|? No, that doesn't seem right.Wait, maybe I need to think differently. Since |PF| is equal to the distance from P to the directrix, which is x=1, then |PA| + |PF| is |PA| + distance from P to x=1.But how can I minimize this sum?Alternatively, perhaps I can think of |PA| + |PF| as |PA| + |PF|, and since |PF| is fixed for points on the parabola, but actually, no, |PF| varies as P moves along the parabola.Wait, no, |PF| is equal to the distance from P to the directrix, which is fixed for each P, but varies depending on where P is.Wait, maybe I can use calculus here. Let me parameterize point P on the parabola and then express |PA| + |PF| as a function, then find its minimum.Given that the parabola is y² = -4x, I can parameterize P as (x, y) where y² = -4x. So, x = -y²/4.So, coordinates of P are (-y²/4, y).Then, |PA| is the distance from P to A(-3, 2). So, |PA| = sqrt[ (-y²/4 + 3)² + (y - 2)² ]Similarly, |PF| is the distance from P to F(-1, 0). So, |PF| = sqrt[ (-y²/4 + 1)² + (y - 0)² ]So, the expression to minimize is sqrt[ (-y²/4 + 3)² + (y - 2)² ] + sqrt[ (-y²/4 + 1)² + y² ]Hmm, that seems complicated. Maybe there's a better way.Wait, earlier I thought about reflecting point A over the directrix. Let me try that again.The directrix is x = 1. Reflecting point A(-3, 2) over x = 1.The reflection of a point (x, y) over the vertical line x = a is (2a - x, y). So, here, a = 1, so reflection of A(-3, 2) is (2*1 - (-3), 2) = (2 + 3, 2) = (5, 2). So, A' is (5, 2).Now, if I consider the distance from A' to F(-1, 0), that would be sqrt[ (5 - (-1))² + (2 - 0)² ] = sqrt[6² + 2²] = sqrt[36 + 4] = sqrt[40] = 2*sqrt(10).But how does this help me?Wait, since |PA| + |PF| = |PA| + |PF|, and |PF| is equal to the distance from P to directrix, which is x=1.But if I reflect A over the directrix to get A', then the distance |PA| is equal to the distance from P to A, which is the same as the distance from P to A' minus twice the distance from A to the directrix? Hmm, maybe not.Wait, actually, reflecting A over the directrix gives A', and then the distance from P to A is equal to the distance from P to A' because reflection preserves distances. Wait, no, that's not correct.Wait, no, reflection across a line preserves distances in the sense that the distance from P to A is equal to the distance from P to A' only if P is on the line of reflection, which is the directrix. But P is on the parabola, not necessarily on the directrix.Hmm, maybe I'm overcomplicating.Wait, another idea: since |PF| is equal to the distance from P to the directrix, which is x=1, then |PA| + |PF| is equal to |PA| + distance from P to x=1.But the distance from P to x=1 is just |x_P - 1|, since it's a vertical line. Since P is on the parabola y² = -4x, x_P = -y²/4. So, distance from P to x=1 is | -y²/4 - 1 | = | - (y² + 4)/4 | = (y² + 4)/4, since distance is positive.So, |PA| + |PF| = |PA| + (y² + 4)/4.But |PA| is sqrt[ (x_P + 3)^2 + (y - 2)^2 ] = sqrt[ ( (-y²/4) + 3 )^2 + (y - 2)^2 ]So, the expression becomes sqrt[ ( (-y²/4 + 3 )^2 + (y - 2)^2 ) ] + (y² + 4)/4.This seems complicated to minimize with respect to y. Maybe calculus is needed.Let me denote f(y) = sqrt[ ( (-y²/4 + 3 )^2 + (y - 2)^2 ) ] + (y² + 4)/4.To find the minimum, take derivative f’(y), set it to zero.But this might get messy. Let me see if I can simplify the expression inside the square root.First, compute (-y²/4 + 3)^2:= ( (-y²/4 + 3) )^2= (3 - y²/4)^2= 9 - (3)(y²/2) + y⁴/16= 9 - (3y²)/2 + y⁴/16Then, compute (y - 2)^2:= y² - 4y + 4So, adding these together:9 - (3y²)/2 + y⁴/16 + y² - 4y + 4Combine like terms:y⁴/16 + (-3y²/2 + y²) + (-4y) + (9 + 4)Simplify:y⁴/16 - (3y²/2 - y²) = y⁴/16 - y²/2-4y remains9 + 4 = 13So, the expression inside the sqrt becomes:y⁴/16 - y²/2 - 4y + 13So, f(y) = sqrt(y⁴/16 - y²/2 - 4y + 13) + (y² + 4)/4This is still complicated, but maybe manageable.Let me denote g(y) = sqrt(y⁴/16 - y²/2 - 4y + 13)and h(y) = (y² + 4)/4So, f(y) = g(y) + h(y)Compute derivative f’(y) = g’(y) + h’(y)First, compute h’(y):h(y) = (y² + 4)/4 = (1/4)y² + 1h’(y) = (1/4)(2y) = y/2Now, compute g’(y):g(y) = sqrt(y⁴/16 - y²/2 - 4y + 13) = (y⁴/16 - y²/2 - 4y + 13)^(1/2)g’(y) = (1/2)(y⁴/16 - y²/2 - 4y + 13)^(-1/2) * ( derivative of inside )Derivative of inside:d/dy [ y⁴/16 - y²/2 - 4y + 13 ] = (4y³)/16 - (2y)/2 - 4 + 0 = y³/4 - y - 4So, g’(y) = (1/2)(y⁴/16 - y²/2 - 4y + 13)^(-1/2) * (y³/4 - y - 4)Thus, f’(y) = [ (1/2)(y³/4 - y - 4) ] / sqrt(y⁴/16 - y²/2 - 4y + 13) + y/2Set f’(y) = 0:[ (1/2)(y³/4 - y - 4) ] / sqrt(...) + y/2 = 0Multiply both sides by 2:(y³/4 - y - 4)/sqrt(...) + y = 0Bring y to the other side:(y³/4 - y - 4)/sqrt(...) = -yMultiply both sides by sqrt(...):y³/4 - y - 4 = -y * sqrt(y⁴/16 - y²/2 - 4y + 13)This is getting really complicated. Maybe squaring both sides would help, but it might introduce extraneous solutions.Let me denote S = sqrt(y⁴/16 - y²/2 - 4y + 13)Then, equation becomes:y³/4 - y - 4 = -y * SBring all terms to left:y³/4 - y - 4 + y * S = 0Hmm, not sure. Maybe square both sides:(y³/4 - y - 4)^2 = y² * (y⁴/16 - y²/2 - 4y + 13)Let me compute left side:(y³/4 - y - 4)^2Let me denote A = y³/4, B = -y, C = -4So, (A + B + C)^2 = A² + B² + C² + 2AB + 2AC + 2BCCompute each term:A² = (y³/4)^2 = y⁶/16B² = (-y)^2 = y²C² = (-4)^2 = 162AB = 2*(y³/4)*(-y) = 2*(-y⁴/4) = -y⁴/22AC = 2*(y³/4)*(-4) = 2*(-y³) = -2y³2BC = 2*(-y)*(-4) = 8ySo, left side:y⁶/16 + y² + 16 - y⁴/2 - 2y³ + 8yRight side:y²*(y⁴/16 - y²/2 - 4y + 13) = y⁶/16 - y⁴/2 - 4y³ + 13y²So, set left = right:y⁶/16 + y² + 16 - y⁴/2 - 2y³ + 8y = y⁶/16 - y⁴/2 - 4y³ + 13y²Subtract right side from both sides:(y⁶/16 - y⁶/16) + (y² - 13y²) + (16) + (-y⁴/2 + y⁴/2) + (-2y³ + 4y³) + 8y = 0Simplify term by term:0 + (-12y²) + 16 + 0 + 2y³ + 8y = 0So, equation becomes:2y³ - 12y² + 8y + 16 = 0Divide both sides by 2:y³ - 6y² + 4y + 8 = 0Now, we have a cubic equation: y³ - 6y² + 4y + 8 = 0Let me try to factor this. Maybe rational roots? Possible rational roots are ±1, ±2, ±4, ±8.Test y=1: 1 - 6 + 4 + 8 = 7 ≠ 0y=2: 8 - 24 + 8 + 8 = 0. Oh, y=2 is a root.So, factor out (y - 2):Using polynomial division or synthetic division.Divide y³ - 6y² + 4y + 8 by (y - 2):Coefficients: 1 | -6 | 4 | 8Bring down 1.Multiply by 2: 1*2=2. Add to next coefficient: -6 + 2 = -4Multiply by 2: -4*2 = -8. Add to next coefficient: 4 + (-8) = -4Multiply by 2: -4*2 = -8. Add to last coefficient: 8 + (-8) = 0So, the cubic factors as (y - 2)(y² - 4y - 4) = 0Thus, roots are y=2 and solutions to y² - 4y - 4 = 0.Solve y² - 4y - 4 = 0:Using quadratic formula: y = [4 ± sqrt(16 + 16)] / 2 = [4 ± sqrt(32)] / 2 = [4 ± 4*sqrt(2)] / 2 = 2 ± 2*sqrt(2)So, roots are y=2, y=2 + 2√2, y=2 - 2√2.Now, let's check which of these are valid solutions.Remember that when we squared both sides, we might have introduced extraneous roots. So, we need to verify each solution in the original equation.First, y=2.Compute left side: y³/4 - y - 4 = (8)/4 - 2 - 4 = 2 - 2 - 4 = -4Compute right side: -y * S = -2 * sqrt(y⁴/16 - y²/2 - 4y + 13)Compute inside sqrt:y=2: y⁴=16, so 16/16=1; y²=4, so -4/2=-2; -4y=-8; +13.So, inside sqrt: 1 - 2 - 8 + 13 = 4. So sqrt(4)=2.Thus, right side: -2*2 = -4So, left side = right side: -4 = -4. So, y=2 is a valid solution.Now, y=2 + 2√2.Compute left side: y³/4 - y - 4First, compute y³:y = 2 + 2√2y³ = (2 + 2√2)^3Let me compute that:First, compute (a + b)^3 = a³ + 3a²b + 3ab² + b³Where a=2, b=2√2.a³ = 83a²b = 3*(4)*(2√2) = 24√23ab² = 3*(2)*(8) = 48b³ = (2√2)^3 = 16√2So, y³ = 8 + 24√2 + 48 + 16√2 = (8 + 48) + (24√2 + 16√2) = 56 + 40√2Thus, y³/4 = (56 + 40√2)/4 = 14 + 10√2Then, y³/4 - y - 4 = (14 + 10√2) - (2 + 2√2) - 4 = (14 - 2 - 4) + (10√2 - 2√2) = 8 + 8√2Compute right side: -y * SFirst, compute S = sqrt(y⁴/16 - y²/2 - 4y + 13)Compute y⁴:y = 2 + 2√2y² = (2 + 2√2)^2 = 4 + 8√2 + 8 = 12 + 8√2y⁴ = (y²)^2 = (12 + 8√2)^2 = 144 + 192√2 + 128 = 272 + 192√2So, y⁴/16 = (272 + 192√2)/16 = 17 + 12√2y²/2 = (12 + 8√2)/2 = 6 + 4√2-4y = -4*(2 + 2√2) = -8 - 8√2So, inside sqrt:17 + 12√2 - 6 - 4√2 - 8 - 8√2 + 13Combine like terms:Constants: 17 - 6 - 8 + 13 = 16√2 terms: 12√2 - 4√2 - 8√2 = 0√2So, inside sqrt is 16. So, S = sqrt(16) = 4Thus, right side: -y * S = -(2 + 2√2)*4 = -8 - 8√2Compare to left side: 8 + 8√2So, left side = 8 + 8√2, right side = -8 - 8√2They are not equal. So, y=2 + 2√2 is an extraneous solution.Similarly, check y=2 - 2√2.Compute left side: y³/4 - y - 4Compute y³:y = 2 - 2√2y³ = (2 - 2√2)^3Again, using (a - b)^3 = a³ - 3a²b + 3ab² - b³a=2, b=2√2a³=8-3a²b = -3*(4)*(2√2) = -24√23ab² = 3*(2)*(8) = 48-b³ = -(16√2)So, y³ = 8 - 24√2 + 48 - 16√2 = (8 + 48) + (-24√2 -16√2) = 56 - 40√2Thus, y³/4 = (56 - 40√2)/4 = 14 - 10√2Then, y³/4 - y - 4 = (14 - 10√2) - (2 - 2√2) - 4 = (14 - 2 - 4) + (-10√2 + 2√2) = 8 - 8√2Compute right side: -y * SFirst, compute S = sqrt(y⁴/16 - y²/2 - 4y + 13)Compute y⁴:y = 2 - 2√2y² = (2 - 2√2)^2 = 4 - 8√2 + 8 = 12 - 8√2y⁴ = (y²)^2 = (12 - 8√2)^2 = 144 - 192√2 + 128 = 272 - 192√2y⁴/16 = (272 - 192√2)/16 = 17 - 12√2y²/2 = (12 - 8√2)/2 = 6 - 4√2-4y = -4*(2 - 2√2) = -8 + 8√2So, inside sqrt:17 - 12√2 - 6 + 4√2 - 8 + 8√2 + 13Combine like terms:Constants: 17 - 6 - 8 + 13 = 16√2 terms: -12√2 + 4√2 + 8√2 = 0√2So, inside sqrt is 16. Thus, S = 4Thus, right side: -y * S = -(2 - 2√2)*4 = -8 + 8√2Compare to left side: 8 - 8√2So, left side = 8 - 8√2, right side = -8 + 8√2These are not equal. So, y=2 - 2√2 is also extraneous.Therefore, the only valid solution is y=2.So, when y=2, what is x?From the parabola equation y² = -4x, so x = -y²/4 = -(4)/4 = -1.Thus, point P is (-1, 2).Wait, but let me verify this.Compute |PA| + |PF| when P is (-1, 2).Compute |PA|: distance from (-1, 2) to (-3, 2):sqrt[ (-1 + 3)^2 + (2 - 2)^2 ] = sqrt[ (2)^2 + 0 ] = 2Compute |PF|: distance from (-1, 2) to (-1, 0):sqrt[ (-1 +1)^2 + (2 - 0)^2 ] = sqrt[0 + 4] = 2So, |PA| + |PF| = 2 + 2 = 4Is this the minimum?Let me check another point on the parabola, say the vertex at (0,0).Compute |PA|: distance from (0,0) to (-3,2):sqrt[ (0 + 3)^2 + (0 - 2)^2 ] = sqrt[9 + 4] = sqrt(13) ≈ 3.605Compute |PF|: distance from (0,0) to (-1,0):sqrt[ (0 +1)^2 + 0 ] = 1So, |PA| + |PF| ≈ 3.605 + 1 ≈ 4.605, which is more than 4.Another point, say y=0, which is the vertex, we already checked.What about y=4? Wait, y=4 would give x = -16/4 = -4. So, P=(-4,4)Compute |PA|: distance from (-4,4) to (-3,2):sqrt[ (-4 +3)^2 + (4 -2)^2 ] = sqrt[1 + 4] = sqrt(5) ≈ 2.236Compute |PF|: distance from (-4,4) to (-1,0):sqrt[ (-4 +1)^2 + (4 -0)^2 ] = sqrt[9 + 16] = sqrt(25) = 5So, |PA| + |PF| ≈ 2.236 + 5 ≈ 7.236, which is much larger.What about y=1? x = -1/4.Compute |PA|: distance from (-1/4,1) to (-3,2):sqrt[ (-1/4 +3)^2 + (1 -2)^2 ] = sqrt[ (11/4)^2 + (-1)^2 ] = sqrt[121/16 + 1] = sqrt[137/16] ≈ 3.122Compute |PF|: distance from (-1/4,1) to (-1,0):sqrt[ (-1/4 +1)^2 + (1 -0)^2 ] = sqrt[ (3/4)^2 + 1 ] = sqrt[9/16 + 16/16] = sqrt[25/16] = 5/4 = 1.25So, |PA| + |PF| ≈ 3.122 + 1.25 ≈ 4.372, which is still more than 4.So, it seems that when y=2, the sum |PA| + |PF| is minimized at 4.Therefore, the coordinates of P are (-1, 2).**Final Answer**The coordinates of point P are boxed{(-1, 2)}.

💡Alright, so I have this problem with two sequences, {a_n} and {b_n}, defined by some initial values and recurrence relations. I need to find their general terms and then use those to figure out the smallest n such that the sum of S_k + T_k from k=1 to n exceeds 2017. First, let's tackle part (1). The sequences are defined as:a₁ = 3, b₁ = 1And for any positive integer n,a_{n+1} = a_n + b_n + sqrt(a_n² - a_n b_n + b_n²)b_{n+1} = a_n + b_n - sqrt(a_n² - a_n b_n + b_n²)Hmm, okay. These recurrence relations look a bit complicated, but maybe there's a pattern or a way to simplify them.I notice that both a_{n+1} and b_{n+1} involve the same square root term. Let me denote that term as something else to make it easier. Let's say:sqrt(a_n² - a_n b_n + b_n²) = s_nThen, the recurrence relations become:a_{n+1} = a_n + b_n + s_nb_{n+1} = a_n + b_n - s_nInteresting. So, if I add these two equations together, the s_n terms will cancel out:a_{n+1} + b_{n+1} = (a_n + b_n + s_n) + (a_n + b_n - s_n) = 2(a_n + b_n)So, this tells me that the sum of a_{n+1} and b_{n+1} is twice the sum of a_n and b_n. That looks like a geometric progression.Let me define c_n = a_n + b_n. Then, the recurrence becomes:c_{n+1} = 2c_nWith c₁ = a₁ + b₁ = 3 + 1 = 4So, c_n is a geometric sequence with ratio 2 and first term 4. Therefore, c_n = 4 * 2^{n-1} = 2^{n+1}Okay, so that's the sum of a_n and b_n. Now, what about their difference? Let's see.From the recurrence relations:a_{n+1} - b_{n+1} = (a_n + b_n + s_n) - (a_n + b_n - s_n) = 2s_nSo,a_{n+1} - b_{n+1} = 2s_nBut s_n is sqrt(a_n² - a_n b_n + b_n²). Let's see if we can find a pattern or a recurrence for this difference.Let me denote d_n = a_n - b_n. Then, from above:d_{n+1} = 2s_nBut s_n = sqrt(a_n² - a_n b_n + b_n²). Let me see if I can express s_n in terms of d_n and c_n.We have:a_n² - a_n b_n + b_n² = (a_n - b_n)^2 + a_n b_nWait, let me check that:(a_n - b_n)^2 = a_n² - 2a_n b_n + b_n²So, a_n² - a_n b_n + b_n² = (a_n - b_n)^2 + a_n b_nHmm, not sure if that helps. Alternatively, maybe express in terms of c_n and d_n.We know that c_n = a_n + b_n and d_n = a_n - b_n.So, a_n = (c_n + d_n)/2 and b_n = (c_n - d_n)/2.Let me compute a_n² - a_n b_n + b_n² using these expressions.First, a_n²:a_n² = [(c_n + d_n)/2]^2 = (c_n² + 2c_n d_n + d_n²)/4Similarly, b_n² = [(c_n - d_n)/2]^2 = (c_n² - 2c_n d_n + d_n²)/4Now, a_n b_n:a_n b_n = [(c_n + d_n)/2] * [(c_n - d_n)/2] = (c_n² - d_n²)/4So, a_n² - a_n b_n + b_n²:= [ (c_n² + 2c_n d_n + d_n²)/4 ] - [ (c_n² - d_n²)/4 ] + [ (c_n² - 2c_n d_n + d_n²)/4 ]Let me compute each term:First term: (c_n² + 2c_n d_n + d_n²)/4Second term: - (c_n² - d_n²)/4Third term: (c_n² - 2c_n d_n + d_n²)/4Now, adding them together:= [ (c_n² + 2c_n d_n + d_n²) - (c_n² - d_n²) + (c_n² - 2c_n d_n + d_n²) ] / 4Let me compute the numerator:= c_n² + 2c_n d_n + d_n² - c_n² + d_n² + c_n² - 2c_n d_n + d_n²Simplify term by term:c_n² - c_n² + c_n² = c_n²2c_n d_n - 2c_n d_n = 0d_n² + d_n² + d_n² = 3d_n²So, numerator is c_n² + 3d_n²Therefore,a_n² - a_n b_n + b_n² = (c_n² + 3d_n²)/4So, s_n = sqrt( (c_n² + 3d_n²)/4 ) = (1/2) sqrt(c_n² + 3d_n²)But from earlier, we have d_{n+1} = 2s_n, so:d_{n+1} = 2 * (1/2) sqrt(c_n² + 3d_n²) = sqrt(c_n² + 3d_n²)Hmm, that's a bit complicated, but maybe we can find a pattern.We know that c_n = 2^{n+1}, so c_n² = (2^{n+1})² = 4^{n+1}So, c_n² + 3d_n² = 4^{n+1} + 3d_n²Therefore,d_{n+1} = sqrt(4^{n+1} + 3d_n²)This seems recursive, but maybe we can find a pattern for d_n.Let's compute the first few terms to see if we can spot a pattern.Given a₁ = 3, b₁ = 1, so d₁ = a₁ - b₁ = 3 - 1 = 2c₁ = 4Now, compute d₂:d₂ = sqrt(c₁² + 3d₁²) = sqrt(16 + 3*4) = sqrt(16 + 12) = sqrt(28) ≈ 5.2915Wait, that's not an integer. Hmm, but a_n and b_n are defined with square roots, so maybe they are not integers? But the initial terms are integers. Let me check:a₂ = a₁ + b₁ + sqrt(a₁² - a₁ b₁ + b₁²) = 3 + 1 + sqrt(9 - 3 + 1) = 4 + sqrt(7) ≈ 4 + 2.6458 ≈ 6.6458Similarly, b₂ = 4 - sqrt(7) ≈ 1.3542So, a₂ and b₂ are not integers. Hmm, that complicates things because in part (2), we have to take the floor function of a_i and b_i, which are not integers. So, their floors would be integers, but a_i and b_i themselves are not.Wait, but in part (1), it just says "Find the general terms of the sequences {a_n} and {b_n}." It doesn't specify whether they are integers or not. So, maybe they are real numbers, but their floors are integers.But let's see if we can find a pattern for d_n.We have:d₁ = 2d₂ = sqrt(4² + 3*(2)²) = sqrt(16 + 12) = sqrt(28) = 2*sqrt(7)d₃ = sqrt(4³ + 3*(2*sqrt(7))²) = sqrt(64 + 3*4*7) = sqrt(64 + 84) = sqrt(148) = 2*sqrt(37)Wait, that doesn't seem to be a clear pattern. Maybe I'm approaching this the wrong way.Alternatively, perhaps instead of trying to find d_n, I can express a_n and b_n in terms of c_n and d_n.We have:a_n = (c_n + d_n)/2b_n = (c_n - d_n)/2We know c_n = 2^{n+1}So, if I can find d_n, I can express a_n and b_n.From earlier, we have:d_{n+1} = sqrt(c_n² + 3d_n²)But c_n = 2^{n+1}, so c_n² = 4^{n+1}Thus,d_{n+1} = sqrt(4^{n+1} + 3d_n²)This is a recursive relation for d_n. Let's see if we can solve it.Let me define e_n = d_n². Then,e_{n+1} = 4^{n+1} + 3e_nThis is a linear recurrence relation for e_n.Given that, we can solve it using standard techniques.The recurrence is:e_{n+1} = 3e_n + 4^{n+1}This is a nonhomogeneous linear recurrence. Let's find the homogeneous solution and a particular solution.First, solve the homogeneous equation:e_{n+1} = 3e_nThe solution is e_n^{(h)} = C * 3^nNow, find a particular solution. The nonhomogeneous term is 4^{n+1} = 4*4^n. Let's assume a particular solution of the form e_n^{(p)} = A*4^n.Substitute into the recurrence:A*4^{n+1} = 3*A*4^n + 4^{n+1}Divide both sides by 4^n:4A = 3A + 4So, 4A - 3A = 4 => A = 4Thus, the general solution is:e_n = e_n^{(h)} + e_n^{(p)} = C*3^n + 4*4^nNow, apply the initial condition. We have d₁ = 2, so e₁ = d₁² = 4.When n=1:e₁ = C*3^1 + 4*4^1 = 3C + 16 = 4So, 3C = 4 - 16 = -12 => C = -4Thus, the solution is:e_n = -4*3^n + 4*4^n = 4*4^n - 4*3^n = 4(4^n - 3^n)Therefore, d_n = sqrt(e_n) = sqrt(4(4^n - 3^n)) = 2*sqrt(4^n - 3^n)So, d_n = 2*sqrt(4^n - 3^n)Therefore, a_n and b_n can be written as:a_n = (c_n + d_n)/2 = (2^{n+1} + 2*sqrt(4^n - 3^n))/2 = 2^n + sqrt(4^n - 3^n)Similarly,b_n = (c_n - d_n)/2 = (2^{n+1} - 2*sqrt(4^n - 3^n))/2 = 2^n - sqrt(4^n - 3^n)Wait, let's check if this makes sense with the initial terms.For n=1:a₁ = 2^1 + sqrt(4^1 - 3^1) = 2 + sqrt(4 - 3) = 2 + 1 = 3 ✔️b₁ = 2^1 - sqrt(4^1 - 3^1) = 2 - 1 = 1 ✔️For n=2:a₂ = 2^2 + sqrt(4^2 - 3^2) = 4 + sqrt(16 - 9) = 4 + sqrt(7) ≈ 6.6458 ✔️b₂ = 4 - sqrt(7) ≈ 1.3542 ✔️Okay, so that seems correct.So, the general terms are:a_n = 2^n + sqrt(4^n - 3^n)b_n = 2^n - sqrt(4^n - 3^n)Alternatively, since 4^n = (2^2)^n = 2^{2n}, we can write sqrt(4^n - 3^n) as sqrt(2^{2n} - 3^n). But I don't think that simplifies further.Alternatively, we can factor out 2^n:sqrt(4^n - 3^n) = sqrt( (2^n)^2 - 3^n ) = 2^n * sqrt(1 - (3/4)^n )But I'm not sure if that helps.Alternatively, perhaps we can write sqrt(4^n - 3^n) as something else, but maybe it's fine as it is.So, for part (1), the general terms are:a_n = 2^n + sqrt(4^n - 3^n)b_n = 2^n - sqrt(4^n - 3^n)Now, moving on to part (2):We need to define S_n = sum_{i=1}^n [a_i] and T_n = sum_{i=1}^n [b_i], where [x] denotes the floor function, the greatest integer less than or equal to x.Then, we need to find the smallest n such that sum_{k=1}^n (S_k + T_k) > 2017.First, let's understand what [a_i] and [b_i] are.Given that a_i = 2^i + sqrt(4^i - 3^i)Similarly, b_i = 2^i - sqrt(4^i - 3^i)Let's see if we can find expressions for [a_i] and [b_i].First, note that sqrt(4^i - 3^i) is less than 2^i, because 4^i - 3^i < 4^i, so sqrt(4^i - 3^i) < 2^i.Therefore, a_i = 2^i + sqrt(4^i - 3^i) < 2^i + 2^i = 2^{i+1}Similarly, b_i = 2^i - sqrt(4^i - 3^i) > 2^i - 2^i = 0But more precisely, since sqrt(4^i - 3^i) is less than 2^i, b_i is positive.But let's see if we can find [a_i] and [b_i].Let me compute a few terms to see the pattern.For i=1:a₁ = 3, so [a₁] = 3b₁ = 1, so [b₁] = 1For i=2:a₂ = 4 + sqrt(16 - 9) = 4 + sqrt(7) ≈ 4 + 2.6458 ≈ 6.6458, so [a₂] = 6b₂ = 4 - sqrt(7) ≈ 1.3542, so [b₂] = 1For i=3:a₃ = 8 + sqrt(64 - 27) = 8 + sqrt(37) ≈ 8 + 6.08276 ≈ 14.08276, so [a₃] = 14b₃ = 8 - sqrt(37) ≈ 1.91724, so [b₃] = 1For i=4:a₄ = 16 + sqrt(256 - 81) = 16 + sqrt(175) ≈ 16 + 13.2288 ≈ 29.2288, so [a₄] = 29b₄ = 16 - sqrt(175) ≈ 2.7712, so [b₄] = 2Wait, b₄ is approximately 2.7712, so [b₄] = 2Similarly, for i=5:a₅ = 32 + sqrt(1024 - 243) = 32 + sqrt(781) ≈ 32 + 27.95 ≈ 59.95, so [a₅] = 59b₅ = 32 - sqrt(781) ≈ 4.05, so [b₅] = 4Wait, that's interesting. Let's see:For i=1: [a₁]=3, [b₁]=1i=2: [a₂]=6, [b₂]=1i=3: [a₃]=14, [b₃]=1i=4: [a₄]=29, [b₄]=2i=5: [a₅]=59, [b₅]=4Hmm, seems like [b_i] is roughly 2^{i-2} for i >=2Wait, let's check:For i=2: [b₂]=1=2^{0}i=3: [b₃]=1=2^{0}i=4: [b₄]=2=2^{1}i=5: [b₅]=4=2^{2}i=6: Let's compute a₆ and b₆.a₆ = 64 + sqrt(4096 - 729) = 64 + sqrt(3367) ≈ 64 + 58.03 ≈ 122.03, so [a₆]=122b₆ = 64 - sqrt(3367) ≈ 5.97, so [b₆]=5Wait, 5 is not 2^{3}=8. Hmm, so maybe my initial thought was wrong.Wait, let's compute more precisely.sqrt(3367) is approximately 58.03, so b₆ = 64 - 58.03 ≈ 5.97, which is less than 6, so [b₆]=5Similarly, for i=6, [b₆]=5Wait, so perhaps [b_i] is roughly 2^{i-2} -1 or something else.Alternatively, maybe [b_i] = floor(2^{i-1} - sqrt(4^{i-1} - 3^{i-1}))But that seems recursive.Alternatively, perhaps we can find a pattern for [a_i] and [b_i].Looking at the values:i | a_i | [a_i] | b_i | [b_i]---|-----|-------|-----|-------1 | 3 | 3 | 1 | 12 | ~6.6458 | 6 | ~1.3542 | 13 | ~14.0828 |14 | ~1.9172 |14 | ~29.2288 |29 | ~2.7712 |25 | ~59.95 |59 | ~4.05 |46 | ~122.03 |122 | ~5.97 |57 | ~244.06 |244 | ~8.94 |88 | ~488.12 |488 | ~14.06 |149 | ~976.24 |976 | ~22.76 |2210| ~1952.48 |1952 | ~37.52 |37Wait, looking at [b_i], it seems to follow roughly the pattern of [b_i] = [a_{i-2}]Wait, for i=2, [b₂]=1, which is [a₀], but a₀ is not defined.Wait, maybe not. Alternatively, [b_i] seems to be roughly half of [a_i], but not exactly.Wait, let's see:For i=2: [a₂]=6, [b₂]=1 (1/6)i=3: [a₃]=14, [b₃]=1 (1/14)i=4: [a₄]=29, [b₄]=2 (2/29)i=5: [a₅]=59, [b₅]=4 (4/59)i=6: [a₆]=122, [b₆]=5 (5/122)i=7: [a₇]=244, [b₇]=8 (8/244)i=8: [a₈]=488, [b₈]=14 (14/488)i=9: [a₉]=976, [b₉]=22 (22/976)i=10: [a_{10}]=1952, [b_{10}]=37 (37/1952)Hmm, not a clear ratio. Alternatively, perhaps [b_i] is roughly equal to [a_{i-1}]/2 or something.Wait, for i=2: [b₂]=1, [a₁]=3, 3/2=1.5, floor is 1.i=3: [b₃]=1, [a₂]=6, 6/2=3, floor is 3, but [b₃]=1, so no.i=4: [b₄]=2, [a₃]=14, 14/2=7, floor is 7, but [b₄]=2.Hmm, not matching.Alternatively, perhaps [b_i] = [sqrt(4^{i} - 3^{i})]Wait, for i=2: sqrt(16 - 9)=sqrt(7)≈2.6458, floor is 2, but [b₂]=1No, that doesn't match.Wait, [b_i] = [2^{i} - sqrt(4^{i} - 3^{i})]Which is exactly b_i, so [b_i] is the floor of b_i.But b_i = 2^{i} - sqrt(4^{i} - 3^{i})Wait, let's see:sqrt(4^i - 3^i) = sqrt((2^i)^2 - 3^i) = 2^i * sqrt(1 - (3/4)^i)So, b_i = 2^i - 2^i * sqrt(1 - (3/4)^i) = 2^i (1 - sqrt(1 - (3/4)^i))Hmm, not sure if that helps.Alternatively, perhaps we can approximate sqrt(4^i - 3^i) for large i.For large i, 3^i is much smaller than 4^i, so sqrt(4^i - 3^i) ≈ 2^i - (3^i)/(2*2^i) = 2^i - (3/2)^i / 2Wait, let's see:sqrt(4^i - 3^i) = sqrt(4^i (1 - (3/4)^i)) = 2^i sqrt(1 - (3/4)^i)Using the approximation sqrt(1 - x) ≈ 1 - x/2 for small x.So, sqrt(1 - (3/4)^i) ≈ 1 - (3/4)^i / 2Thus,sqrt(4^i - 3^i) ≈ 2^i (1 - (3/4)^i / 2) = 2^i - 2^{i} * (3/4)^i / 2 = 2^i - (3/2)^i / 2Therefore,b_i = 2^i - sqrt(4^i - 3^i) ≈ 2^i - [2^i - (3/2)^i / 2] = (3/2)^i / 2So, for large i, [b_i] ≈ floor( (3/2)^i / 2 )But let's check for i=5:(3/2)^5 / 2 = (243/32)/2 ≈ 7.59375 / 2 ≈ 3.796875, floor is 3, but [b₅]=4Hmm, not exact.For i=6:(3/2)^6 / 2 = (729/64)/2 ≈ 11.390625 / 2 ≈ 5.6953125, floor is 5, which matches [b₆]=5i=7:(3/2)^7 / 2 = (2187/128)/2 ≈ 17.0859375 / 2 ≈ 8.54296875, floor is 8, which matches [b₇]=8i=8:(3/2)^8 / 2 = (6561/256)/2 ≈ 25.62890625 / 2 ≈ 12.814453125, floor is 12, but [b₈]=14Hmm, discrepancy here.Wait, maybe the approximation is better for larger i.Alternatively, perhaps [b_i] = floor( (3/2)^i )For i=5: (3/2)^5 = 7.59375, floor is 7, but [b₅]=4No, that doesn't match.Alternatively, perhaps [b_i] = floor( (3/2)^{i-1} )For i=5: (3/2)^4 = 5.0625, floor is 5, which matches [b₅]=4? No, 5≠4.Wait, not matching.Alternatively, perhaps [b_i] = floor( (3/2)^{i} / 2 )For i=5: (3/2)^5 / 2 ≈ 7.59375 / 2 ≈ 3.796875, floor is 3, but [b₅]=4No.Alternatively, perhaps [b_i] = floor( (3/2)^{i-1} )For i=5: (3/2)^4 ≈ 5.0625, floor is 5, but [b₅]=4No.Hmm, maybe it's better to accept that [b_i] doesn't follow a simple pattern and instead try to find a general expression or recurrence for S_n and T_n.Given that S_n = sum_{i=1}^n [a_i] and T_n = sum_{i=1}^n [b_i]We need to compute sum_{k=1}^n (S_k + T_k) > 2017But this seems complicated because S_k and T_k are cumulative sums.Wait, perhaps we can find expressions for S_n and T_n in terms of a_n and b_n.Given that a_i = 2^i + sqrt(4^i - 3^i) and b_i = 2^i - sqrt(4^i - 3^i)But [a_i] = floor(a_i) and [b_i] = floor(b_i)But since a_i and b_i are not integers, their floors are the greatest integers less than or equal to them.But perhaps we can find that [a_i] = 2^i + floor(sqrt(4^i - 3^i))Similarly, [b_i] = 2^i - ceil(sqrt(4^i - 3^i))Wait, let's check:For i=2:a₂ = 4 + sqrt(7) ≈ 6.6458, so [a₂]=6 = 4 + floor(sqrt(7))=4+2=6 ✔️b₂ = 4 - sqrt(7) ≈1.3542, so [b₂]=1 = 4 - ceil(sqrt(7))=4-3=1 ✔️Similarly, for i=3:a₃=8 + sqrt(37)≈14.0828, [a₃]=14=8 + floor(sqrt(37))=8+6=14 ✔️b₃=8 - sqrt(37)≈1.9172, [b₃]=1=8 - ceil(sqrt(37))=8-7=1 ✔️i=4:a₄=16 + sqrt(175)≈29.2288, [a₄]=29=16 + floor(sqrt(175))=16+13=29 ✔️b₄=16 - sqrt(175)≈2.7712, [b₄]=2=16 - ceil(sqrt(175))=16-14=2 ✔️i=5:a₅=32 + sqrt(781)≈59.95, [a₅]=59=32 + floor(sqrt(781))=32+27=59 ✔️b₅=32 - sqrt(781)≈4.05, [b₅]=4=32 - ceil(sqrt(781))=32-28=4 ✔️i=6:a₆=64 + sqrt(3367)≈122.03, [a₆]=122=64 + floor(sqrt(3367))=64+58=122 ✔️b₆=64 - sqrt(3367)≈5.97, [b₆]=5=64 - ceil(sqrt(3367))=64-59=5 ✔️So, it seems that:[a_i] = 2^i + floor(sqrt(4^i - 3^i))[b_i] = 2^i - ceil(sqrt(4^i - 3^i))Therefore, S_n = sum_{i=1}^n [a_i] = sum_{i=1}^n (2^i + floor(sqrt(4^i - 3^i))) = sum_{i=1}^n 2^i + sum_{i=1}^n floor(sqrt(4^i - 3^i))Similarly, T_n = sum_{i=1}^n [b_i] = sum_{i=1}^n (2^i - ceil(sqrt(4^i - 3^i))) = sum_{i=1}^n 2^i - sum_{i=1}^n ceil(sqrt(4^i - 3^i))Therefore, S_n + T_n = [sum_{i=1}^n 2^i + sum_{i=1}^n floor(sqrt(4^i - 3^i))] + [sum_{i=1}^n 2^i - sum_{i=1}^n ceil(sqrt(4^i - 3^i))] = 2*sum_{i=1}^n 2^i + [sum_{i=1}^n floor(sqrt(4^i - 3^i)) - sum_{i=1}^n ceil(sqrt(4^i - 3^i))]But sum_{i=1}^n floor(sqrt(4^i - 3^i)) - sum_{i=1}^n ceil(sqrt(4^i - 3^i)) = -n, because for each i, floor(x) - ceil(x) = -1 if x is not integer, which it isn't here because sqrt(4^i - 3^i) is irrational for i>=2.Wait, let's check:For each i, floor(sqrt(4^i - 3^i)) - ceil(sqrt(4^i - 3^i)) = -1, because sqrt(4^i - 3^i) is not an integer, so ceil(sqrt(4^i - 3^i)) = floor(sqrt(4^i - 3^i)) +1Therefore, sum_{i=1}^n [floor(sqrt(4^i - 3^i)) - ceil(sqrt(4^i - 3^i))] = sum_{i=1}^n (-1) = -nTherefore, S_n + T_n = 2*sum_{i=1}^n 2^i - nSum_{i=1}^n 2^i is a geometric series:sum_{i=1}^n 2^i = 2^{n+1} - 2Therefore, S_n + T_n = 2*(2^{n+1} - 2) - n = 2^{n+2} - 4 - nThus, sum_{k=1}^n (S_k + T_k) = sum_{k=1}^n (2^{k+2} - 4 - k)Let's compute this sum:sum_{k=1}^n (2^{k+2} - 4 - k) = sum_{k=1}^n 2^{k+2} - sum_{k=1}^n 4 - sum_{k=1}^n kCompute each part:1. sum_{k=1}^n 2^{k+2} = 4*sum_{k=1}^n 2^k = 4*(2^{n+1} - 2) = 2^{n+3} - 82. sum_{k=1}^n 4 = 4n3. sum_{k=1}^n k = n(n+1)/2Therefore, the total sum is:(2^{n+3} - 8) - 4n - n(n+1)/2Simplify:= 2^{n+3} - 8 - 4n - (n² + n)/2= 2^{n+3} - 8 - (8n + n² + n)/2= 2^{n+3} - 8 - (n² + 9n)/2We need this to be greater than 2017:2^{n+3} - 8 - (n² + 9n)/2 > 2017Let me rewrite it:2^{n+3} - (n² + 9n)/2 - 8 > 2017Multiply both sides by 2 to eliminate the fraction:2^{n+4} - (n² + 9n) - 16 > 4034So,2^{n+4} - n² - 9n - 16 > 4034We need to find the smallest n such that this inequality holds.Let me compute 2^{n+4} and see when it exceeds 4034 + n² + 9n +16Let's try n=8:2^{12}=40964096 - 64 -72 -16=4096-152=39443944 > 4034? No, 3944<4034n=9:2^{13}=81928192 -81 -81 -16=8192-178=80148014>4034? Yes.But wait, let's compute the exact value:For n=8:sum = 2^{8+3} - (8² +9*8)/2 -8 = 2^{11} - (64 +72)/2 -8 = 2048 - 136/2 -8=2048 -68 -8=2048-76=19721972 >2017? No.n=9:sum=2^{12} - (81 +81)/2 -8=4096 -162/2 -8=4096-81-8=4096-89=40074007>2017? Yes.But wait, the sum for n=9 is 4007, which is greater than 2017.But wait, the question is to find the smallest n such that sum_{k=1}^n (S_k + T_k) > 2017.But when n=8, the sum is 1972, which is less than 2017.When n=9, the sum is 4007, which is greater than 2017.Therefore, the smallest n is 9.But wait, let me double-check the calculations.Wait, earlier I had:sum_{k=1}^n (S_k + T_k) = 2^{n+3} - 8 - (n² +9n)/2For n=8:2^{11} -8 - (64 +72)/2=2048 -8 -136/2=2048-8-68=2048-76=1972Yes.For n=9:2^{12} -8 - (81 +81)/2=4096 -8 -162/2=4096-8-81=4096-89=4007Yes.So, the sum jumps from 1972 at n=8 to 4007 at n=9, which is greater than 2017.Therefore, the smallest n is 9.But wait, let me check if n=8 is the last one below 2017 and n=9 is the first above.Yes, because 1972 <2017<4007.Therefore, the answer is n=9.

💡First, I need to simplify the given complex fraction (frac{1+i}{2-i}). To do this, I'll multiply both the numerator and the denominator by the conjugate of the denominator, which is (2+i). This will help eliminate the imaginary unit from the denominator.Multiplying out the numerator:[(1+i)(2+i) = 1 cdot 2 + 1 cdot i + i cdot 2 + i cdot i = 2 + i + 2i + i^2]Since (i^2 = -1), this simplifies to:[2 + 3i - 1 = 1 + 3i]Next, multiplying out the denominator:[(2-i)(2+i) = 2 cdot 2 + 2 cdot i - i cdot 2 - i cdot i = 4 + 2i - 2i - i^2]Again, since (i^2 = -1), this simplifies to:[4 - (-1) = 5]So, the simplified form of the fraction is:[frac{1+3i}{5} = frac{1}{5} + frac{3}{5}i]According to the problem, this is equal to (a + (b+1)i). By comparing the real and imaginary parts, I can set up the following equations:[a = frac{1}{5}][b + 1 = frac{3}{5}]Solving for (b):[b = frac{3}{5} - 1 = -frac{2}{5}]Therefore, the complex number (z) is:[z = a + bi = frac{1}{5} - frac{2}{5}i]The point corresponding to (z) in the complex plane has coordinates (left(frac{1}{5}, -frac{2}{5}right)). Since the real part is positive and the imaginary part is negative, the point lies in the fourth quadrant.