💡First, I need to determine Dana's hourly wage based on the total hours she worked and the total amount she earned.I'll start by calculating the total number of hours Dana worked over the weekend. She worked 9 hours on Friday, 10 hours on Saturday, and 3 hours on Sunday. Adding these together gives:9 hours + 10 hours + 3 hours = 22 hoursNext, I know that Dana earned a total of 286 over these 22 hours. To find her hourly wage, I can divide the total earnings by the total hours worked:Hourly wage = Total earnings / Total hours workedHourly wage = 286 / 22 hoursPerforming the division:286 ÷ 22 = 13So, Dana earns 13 per hour.

💡First, I need to determine the number of buildings that collapsed in the first earthquake. Let's denote this number as ( B ).According to the problem, each subsequent earthquake causes double the number of buildings to collapse compared to the previous one. This means:- After the first earthquake: ( B ) buildings collapse.- After the second earthquake: ( 2B ) buildings collapse.- After the third earthquake: ( 4B ) buildings collapse.- After the fourth earthquake: ( 8B ) buildings collapse.The total number of collapsed buildings after four earthquakes is given as 60. Therefore, I can set up the equation:[B + 2B + 4B + 8B = 60]Combining like terms:[15B = 60]To find ( B ), I divide both sides of the equation by 15:[B = frac{60}{15} = 4]Thus, the number of buildings that collapsed in the first earthquake is 4.

💡First, I need to determine the capacity of a mini-van's tank, denoted as ( V ) liters. Since a truck's tank is 120% bigger, its capacity is ( 2.2V ) liters.Next, I'll calculate the total service cost for all vehicles. There are 3 mini-vans and 2 trucks, so the service cost is:[3 times 2.20 + 2 times 2.20 = 11.00 text{ dollars}]Then, I'll calculate the total fuel cost. For the mini-vans:[3V times 0.70 = 2.1V text{ dollars}]For the trucks:[2 times 2.2V times 0.70 = 3.08V text{ dollars}]Adding these together gives the total fuel cost:[2.1V + 3.08V = 5.18V text{ dollars}]The total cost, which includes both service and fuel, is 347.7. Setting up the equation:[11.00 + 5.18V = 347.7]Solving for ( V ):[5.18V = 347.7 - 11.00][5.18V = 336.7][V = frac{336.7}{5.18} approx 65 text{ liters}]Therefore, the capacity of a mini-van's tank is approximately 65 liters.

💡Alright, so I have this problem here about complex numbers. Let me read it again to make sure I understand what's being asked.It says: Given that ( z ) is a complex number, ( (1 + 3i) cdot z ) is a real number, ( w = frac{z}{2 + i} ), and ( |w| = 5sqrt{2} ), find ( w ).Okay, so I need to find the complex number ( w ) given these conditions. Let me break this down step by step.First, since ( z ) is a complex number, I can write it as ( z = x + yi ), where ( x ) and ( y ) are real numbers. That seems like a good starting point.Next, the problem states that ( (1 + 3i) cdot z ) is a real number. So, if I multiply ( (1 + 3i) ) by ( z ), the result should have no imaginary part. Let me compute that multiplication.Multiplying ( (1 + 3i) ) by ( z = x + yi ):[(1 + 3i)(x + yi) = x(1) + x(3i) + yi(1) + yi(3i)]Simplifying each term:[= x + 3xi + yi + 3yi^2]Since ( i^2 = -1 ), the last term becomes:[3yi^2 = 3y(-1) = -3y]So, combining like terms:[= (x - 3y) + (3x + y)i]Now, for this product to be a real number, the imaginary part must be zero. That means:[3x + y = 0]So, ( y = -3x ). Got it. That gives me a relationship between ( y ) and ( x ).Now, moving on to the next part. ( w ) is defined as ( frac{z}{2 + i} ). Let me express ( w ) in terms of ( x ) and ( y ).Given ( z = x + yi ), then:[w = frac{x + yi}{2 + i}]To simplify this expression, I can multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of ( 2 + i ) is ( 2 - i ). So:[w = frac{(x + yi)(2 - i)}{(2 + i)(2 - i)}]Let me compute the denominator first:[(2 + i)(2 - i) = 2 cdot 2 + 2 cdot (-i) + i cdot 2 + i cdot (-i) = 4 - 2i + 2i - i^2]Simplifying:[= 4 - i^2 = 4 - (-1) = 5]So, the denominator is 5.Now, the numerator:[(x + yi)(2 - i) = x cdot 2 + x cdot (-i) + yi cdot 2 + yi cdot (-i)]Simplifying each term:[= 2x - xi + 2yi - y i^2]Again, since ( i^2 = -1 ), the last term becomes:[- y i^2 = - y (-1) = y]So, combining like terms:[= (2x + y) + (-x + 2y)i]Therefore, ( w ) is:[w = frac{(2x + y) + (-x + 2y)i}{5} = frac{2x + y}{5} + frac{-x + 2y}{5}i]But earlier, I found that ( y = -3x ). Let me substitute ( y ) with ( -3x ) in the expression for ( w ):[w = frac{2x + (-3x)}{5} + frac{-x + 2(-3x)}{5}i = frac{-x}{5} + frac{-7x}{5}i]So, ( w = frac{-x}{5} - frac{7x}{5}i ). I can factor out ( frac{-x}{5} ) to write:[w = frac{-x}{5}(1 + 7i)]Alternatively, ( w = frac{x}{5}(-1 - 7i) ).Now, the problem states that ( |w| = 5sqrt{2} ). The magnitude of a complex number ( a + bi ) is ( sqrt{a^2 + b^2} ). So, let's compute ( |w| ).From the expression of ( w ):[w = frac{-x}{5} - frac{7x}{5}i]So, ( |w| = sqrt{left(frac{-x}{5}right)^2 + left(frac{-7x}{5}right)^2} )[= sqrt{frac{x^2}{25} + frac{49x^2}{25}} = sqrt{frac{50x^2}{25}} = sqrt{2x^2} = |x|sqrt{2}]Given that ( |w| = 5sqrt{2} ), we have:[|x|sqrt{2} = 5sqrt{2}]Dividing both sides by ( sqrt{2} ):[|x| = 5]So, ( x = 5 ) or ( x = -5 ).Now, let's find ( w ) for each case.Case 1: ( x = 5 )[w = frac{-5}{5} - frac{7 cdot 5}{5}i = -1 - 7i]Case 2: ( x = -5 )[w = frac{-(-5)}{5} - frac{7 cdot (-5)}{5}i = 1 + 7i]So, the two possible values for ( w ) are ( -1 - 7i ) and ( 1 + 7i ).Let me just double-check my steps to make sure I didn't make any mistakes.1. Expressed ( z ) as ( x + yi ).2. Multiplied ( (1 + 3i) ) by ( z ) and set the imaginary part to zero, leading to ( y = -3x ).3. Expressed ( w ) as ( frac{z}{2 + i} ) and simplified using the conjugate.4. Substituted ( y = -3x ) into the expression for ( w ), resulting in ( w = frac{-x}{5} - frac{7x}{5}i ).5. Calculated the magnitude of ( w ) and set it equal to ( 5sqrt{2} ), leading to ( |x| = 5 ).6. Substituted ( x = 5 ) and ( x = -5 ) into the expression for ( w ) to find the two possible values.Everything seems to check out. So, the possible values for ( w ) are ( 1 + 7i ) and ( -1 - 7i ).**Final Answer**The possible values for ( w ) are boxed{1 + 7i} and boxed{-1 - 7i}.

💡Okay, so I have this trapezoid (ABCD) where (AD) is parallel to (BC), with lengths (AD = a) and (BC = b). The diagonals (AC) and (BD) intersect at point (P_1). Then, from (P_1), I draw a line (P_1Q_1) parallel to (AD), which intersects (CD) at (Q_1). Connecting (AQ_1) intersects (BD) at (P_2). Repeating this process, I get points (P_2Q_2), (P_3Q_3), and so on. I need to find the length (x_n = P_nQ_n) in terms of (a), (b), and (n), and then prove that (x_n leq frac{a + nb}{4n}).First, let me try to visualize the trapezoid. (AD) and (BC) are the two bases, with (AD) longer than (BC) if (a > b). The diagonals intersect at (P_1), which divides each diagonal into segments. Since (AD parallel BC), the triangles formed by the diagonals should be similar.I remember that in a trapezoid, the ratio of the lengths of the two bases is equal to the ratio of the segments created by the intersection of the diagonals. So, if (AD = a) and (BC = b), then the ratio ( frac{AP_1}{P_1C} = frac{a}{b} ). Similarly, ( frac{BP_1}{P_1D} = frac{a}{b} ).Now, drawing (P_1Q_1 parallel AD) means that (P_1Q_1) is also parallel to (BC). Since (P_1Q_1) is parallel to the bases, the segment (P_1Q_1) should be a line that creates a smaller trapezoid within (ABCD). The length of (P_1Q_1) can be found using the properties of similar triangles.Wait, actually, since (P_1Q_1) is drawn from the intersection of the diagonals, it's not just any line; it's specifically positioned such that it's parallel to the bases. So, the length of (P_1Q_1) should be the harmonic mean of (a) and (b). I think the formula is ( frac{2ab}{a + b} ). But wait, in the problem statement, they mention (x_1 = P_1Q_1), so maybe it's just ( frac{ab}{a + b} ). Hmm, I need to be careful here.Let me think step by step. The diagonals intersect at (P_1), and since (AD parallel BC), triangles (AP_1D) and (CP_1B) are similar. The ratio of similarity is ( frac{a}{b} ). Therefore, the length of (P_1Q_1) can be found by considering the similar triangles.Alternatively, maybe using coordinate geometry would help. Let me assign coordinates to the trapezoid. Let’s place (A) at ((0, 0)), (D) at ((a, 0)), (B) at ((c, h)), and (C) at ((d, h)), where (h) is the height of the trapezoid. Since (AD parallel BC), the y-coordinates of (B) and (C) are the same.The diagonals (AC) and (BD) intersect at (P_1). To find the coordinates of (P_1), I can solve the equations of lines (AC) and (BD). The coordinates of (A) are ((0, 0)), and (C) is ((d, h)). The equation of line (AC) is (y = frac{h}{d}x).Similarly, the coordinates of (B) are ((c, h)), and (D) is ((a, 0)). The equation of line (BD) is (y = frac{h}{c - a}(x - a)).Setting the two equations equal to find (P_1):[frac{h}{d}x = frac{h}{c - a}(x - a)]Simplifying:[frac{x}{d} = frac{x - a}{c - a}]Cross-multiplying:[x(c - a) = d(x - a)]Expanding:[xc - ax = dx - ad]Bringing like terms together:[xc - dx = -ad + ax]Factor:[x(c - d) = a(x - d)]Wait, this seems complicated. Maybe I made a mistake in setting up the coordinates. Alternatively, perhaps I should use vectors or another method.Wait, another approach: in a trapezoid, the intersection point of the diagonals divides them proportionally. Specifically, the ratio is ( frac{AP_1}{P_1C} = frac{AD}{BC} = frac{a}{b} ). So, (AP_1 = frac{a}{a + b} AC) and (P_1C = frac{b}{a + b} AC).Similarly, (BP_1 = frac{a}{a + b} BD) and (P_1D = frac{b}{a + b} BD).Now, drawing (P_1Q_1 parallel AD). Since (AD) is the base, (P_1Q_1) is a line segment parallel to the base, so it should create a smaller similar trapezoid (AP_1Q_1D).The length of (P_1Q_1) can be found using the ratio of similarity. Since (P_1) divides the diagonals in the ratio (a:b), the length (P_1Q_1) should be the weighted average of (a) and (b). Specifically, (P_1Q_1 = frac{ab}{a + b}).Wait, that seems familiar. So, (x_1 = frac{ab}{a + b}).Now, moving on to (x_2). After drawing (P_1Q_1), we connect (AQ_1), which intersects (BD) at (P_2). Then, drawing (P_2Q_2 parallel AD), intersecting (CD) at (Q_2). So, this process is iterative.I need to find a pattern or a recursive relation for (x_n). Let me consider the first few terms.We have (x_1 = frac{ab}{a + b}).Now, to find (x_2), we need to find the length of (P_2Q_2). Since (P_2) is the intersection of (AQ_1) and (BD), and (Q_2) is the intersection of (P_2Q_2) with (CD), where (P_2Q_2 parallel AD).Perhaps I can use similar triangles again. The line (AQ_1) intersects (BD) at (P_2). The ratio in which (P_2) divides (BD) can be found using the properties of similar triangles or coordinate geometry.Alternatively, maybe using the concept of harmonic division or projective geometry. But perhaps a better approach is to use coordinate geometry consistently.Let me try to assign coordinates again, but more carefully.Let’s place (A) at ((0, 0)), (D) at ((a, 0)), (B) at ((0, h)), and (C) at ((b, h)). Wait, but in a trapezoid, the sides (AB) and (CD) are the non-parallel sides. So, actually, if (AD) is parallel to (BC), then (AD) is the lower base and (BC) is the upper base.So, let me adjust the coordinates: Let’s place (A) at ((0, 0)), (D) at ((a, 0)), (B) at ((c, h)), and (C) at ((c + b, h)). This way, (AD) is from ((0, 0)) to ((a, 0)), and (BC) is from ((c, h)) to ((c + b, h)), so they are parallel.Now, the diagonals are (AC) from ((0, 0)) to ((c + b, h)), and (BD) from ((c, h)) to ((a, 0)).Let me find the coordinates of (P_1), the intersection of (AC) and (BD).The parametric equation of (AC) can be written as:[x = t(c + b), quad y = t h, quad t in [0, 1]]The parametric equation of (BD) can be written as:[x = c + s(a - c), quad y = h - s h, quad s in [0, 1]]Setting them equal:[t(c + b) = c + s(a - c)][t h = h - s h]From the second equation:[t = 1 - s]Substituting into the first equation:[(1 - s)(c + b) = c + s(a - c)]Expanding:[c + b - s c - s b = c + s a - s c]Simplify:[c + b - s c - s b = c + s a - s c]Subtract (c) from both sides:[b - s c - s b = s a - s c]Add (s c) to both sides:[b - s b = s a]Factor:[b(1 - s) = s a]Solving for (s):[b - b s = a s][b = s(a + b)][s = frac{b}{a + b}]Therefore, (t = 1 - s = frac{a}{a + b}).So, the coordinates of (P_1) are:From (AC):[x = t(c + b) = frac{a}{a + b}(c + b)][y = t h = frac{a}{a + b} h]Alternatively, from (BD):[x = c + s(a - c) = c + frac{b}{a + b}(a - c)][y = h - s h = h - frac{b}{a + b} h = frac{a}{a + b} h]So, both give consistent coordinates.Now, drawing (P_1Q_1 parallel AD). Since (AD) is horizontal (from ((0, 0)) to ((a, 0))), (P_1Q_1) must also be horizontal. Therefore, (Q_1) is the point on (CD) with the same y-coordinate as (P_1), which is ( frac{a}{a + b} h ).The line (CD) goes from (D(a, 0)) to (C(c + b, h)). Let me find the equation of line (CD).The slope of (CD) is:[m = frac{h - 0}{(c + b) - a} = frac{h}{c + b - a}]The equation of (CD) is:[y - 0 = m(x - a)][y = frac{h}{c + b - a}(x - a)]We need to find the point (Q_1) on (CD) with (y = frac{a}{a + b} h).Setting (y = frac{a}{a + b} h):[frac{a}{a + b} h = frac{h}{c + b - a}(x - a)]Divide both sides by (h):[frac{a}{a + b} = frac{1}{c + b - a}(x - a)]Multiply both sides by (c + b - a):[frac{a(c + b - a)}{a + b} = x - a]Solve for (x):[x = a + frac{a(c + b - a)}{a + b} = frac{a(a + b) + a(c + b - a)}{a + b}][= frac{a^2 + a b + a c + a b - a^2}{a + b}][= frac{2 a b + a c}{a + b}][= frac{a(2 b + c)}{a + b}]Therefore, the coordinates of (Q_1) are:[left( frac{a(2 b + c)}{a + b}, frac{a}{a + b} h right)]Now, the length (P_1Q_1) is the horizontal distance between (P_1) and (Q_1), since they lie on a horizontal line.Coordinates of (P_1):[left( frac{a(c + b)}{a + b}, frac{a}{a + b} h right)]Coordinates of (Q_1):[left( frac{a(2 b + c)}{a + b}, frac{a}{a + b} h right)]So, the x-coordinate difference is:[frac{a(2 b + c)}{a + b} - frac{a(c + b)}{a + b} = frac{a(2 b + c - c - b)}{a + b} = frac{a b}{a + b}]Therefore, (x_1 = P_1Q_1 = frac{a b}{a + b}).Okay, so that confirms the first part. Now, moving on to (x_2).We need to connect (AQ_1), which is the line from (A(0, 0)) to (Q_1left( frac{a(2 b + c)}{a + b}, frac{a}{a + b} h right)). Let me find the equation of line (AQ_1).The slope (m_{AQ_1}) is:[m_{AQ_1} = frac{frac{a}{a + b} h - 0}{frac{a(2 b + c)}{a + b} - 0} = frac{frac{a h}{a + b}}{frac{a(2 b + c)}{a + b}} = frac{h}{2 b + c}]So, the equation is:[y = frac{h}{2 b + c} x]This line intersects (BD) at (P_2). The equation of (BD) is from (B(c, h)) to (D(a, 0)). The parametric equations can be written as:[x = c + t(a - c), quad y = h - t h, quad t in [0, 1]]We need to find the intersection point (P_2) between (AQ_1) and (BD).From (AQ_1): (y = frac{h}{2 b + c} x)From (BD): (y = h - frac{h}{a - c}(x - c))Set them equal:[frac{h}{2 b + c} x = h - frac{h}{a - c}(x - c)]Divide both sides by (h):[frac{x}{2 b + c} = 1 - frac{1}{a - c}(x - c)]Multiply both sides by ((2 b + c)(a - c)) to eliminate denominators:[x(a - c) = (2 b + c)(a - c) - (2 b + c)(x - c)]Expand the right side:[x(a - c) = (2 b + c)(a - c) - (2 b + c)x + (2 b + c)c]Bring all terms to the left:[x(a - c) + (2 b + c)x = (2 b + c)(a - c) + (2 b + c)c]Factor (x):[x(a - c + 2 b + c) = (2 b + c)(a - c + c)][x(a + 2 b) = (2 b + c)a][x = frac{a(2 b + c)}{a + 2 b}]Now, substitute back into (y = frac{h}{2 b + c} x):[y = frac{h}{2 b + c} cdot frac{a(2 b + c)}{a + 2 b} = frac{a h}{a + 2 b}]So, the coordinates of (P_2) are:[left( frac{a(2 b + c)}{a + 2 b}, frac{a h}{a + 2 b} right)]Now, drawing (P_2Q_2 parallel AD), which is horizontal. So, (Q_2) is the point on (CD) with the same y-coordinate as (P_2), which is ( frac{a h}{a + 2 b} ).Using the equation of (CD):[y = frac{h}{c + b - a}(x - a)]Set (y = frac{a h}{a + 2 b}):[frac{a h}{a + 2 b} = frac{h}{c + b - a}(x - a)]Divide both sides by (h):[frac{a}{a + 2 b} = frac{1}{c + b - a}(x - a)]Multiply both sides by (c + b - a):[frac{a(c + b - a)}{a + 2 b} = x - a]Solve for (x):[x = a + frac{a(c + b - a)}{a + 2 b} = frac{a(a + 2 b) + a(c + b - a)}{a + 2 b}][= frac{a^2 + 2 a b + a c + a b - a^2}{a + 2 b}][= frac{3 a b + a c}{a + 2 b}][= frac{a(3 b + c)}{a + 2 b}]Therefore, the coordinates of (Q_2) are:[left( frac{a(3 b + c)}{a + 2 b}, frac{a h}{a + 2 b} right)]Now, the length (P_2Q_2) is the horizontal distance between (P_2) and (Q_2):Coordinates of (P_2):[left( frac{a(2 b + c)}{a + 2 b}, frac{a h}{a + 2 b} right)]Coordinates of (Q_2):[left( frac{a(3 b + c)}{a + 2 b}, frac{a h}{a + 2 b} right)]Difference in x-coordinates:[frac{a(3 b + c)}{a + 2 b} - frac{a(2 b + c)}{a + 2 b} = frac{a(3 b + c - 2 b - c)}{a + 2 b} = frac{a b}{a + 2 b}]So, (x_2 = P_2Q_2 = frac{a b}{a + 2 b}).Hmm, interesting. So, (x_1 = frac{a b}{a + b}), (x_2 = frac{a b}{a + 2 b}). It seems like a pattern is emerging where (x_n = frac{a b}{a + n b}).Let me test this hypothesis for (x_3). Following the same process, (P_3) would be the intersection of (AQ_2) and (BD), and then (P_3Q_3 parallel AD).Assuming (x_n = frac{a b}{a + n b}), then for (n = 3), (x_3 = frac{a b}{a + 3 b}).To confirm, let's go through the steps quickly.Coordinates of (Q_2): (left( frac{a(3 b + c)}{a + 2 b}, frac{a h}{a + 2 b} right))Equation of (AQ_2): from (A(0, 0)) to (Q_2). Slope:[m_{AQ_2} = frac{frac{a h}{a + 2 b}}{frac{a(3 b + c)}{a + 2 b}} = frac{h}{3 b + c}]Equation: (y = frac{h}{3 b + c} x)Intersection with (BD): parametric equations of (BD) are (x = c + t(a - c)), (y = h - t h). Set equal:[frac{h}{3 b + c} x = h - frac{h}{a - c}(x - c)]Divide by (h):[frac{x}{3 b + c} = 1 - frac{1}{a - c}(x - c)]Multiply both sides by ((3 b + c)(a - c)):[x(a - c) = (3 b + c)(a - c) - (3 b + c)(x - c)]Expand:[x(a - c) = (3 b + c)(a - c) - (3 b + c)x + (3 b + c)c]Bring all terms to left:[x(a - c) + (3 b + c)x = (3 b + c)(a - c) + (3 b + c)c]Factor (x):[x(a - c + 3 b + c) = (3 b + c)(a - c + c)][x(a + 3 b) = (3 b + c)a][x = frac{a(3 b + c)}{a + 3 b}]Then, (y = frac{h}{3 b + c} x = frac{h}{3 b + c} cdot frac{a(3 b + c)}{a + 3 b} = frac{a h}{a + 3 b})Coordinates of (P_3): (left( frac{a(3 b + c)}{a + 3 b}, frac{a h}{a + 3 b} right))Drawing (P_3Q_3 parallel AD), so (Q_3) is on (CD) with (y = frac{a h}{a + 3 b}).Using equation of (CD):[y = frac{h}{c + b - a}(x - a)]Set (y = frac{a h}{a + 3 b}):[frac{a h}{a + 3 b} = frac{h}{c + b - a}(x - a)]Divide by (h):[frac{a}{a + 3 b} = frac{1}{c + b - a}(x - a)]Multiply both sides by (c + b - a):[frac{a(c + b - a)}{a + 3 b} = x - a]Solve for (x):[x = a + frac{a(c + b - a)}{a + 3 b} = frac{a(a + 3 b) + a(c + b - a)}{a + 3 b}][= frac{a^2 + 3 a b + a c + a b - a^2}{a + 3 b}][= frac{4 a b + a c}{a + 3 b}][= frac{a(4 b + c)}{a + 3 b}]Coordinates of (Q_3): (left( frac{a(4 b + c)}{a + 3 b}, frac{a h}{a + 3 b} right))Length (P_3Q_3):Difference in x-coordinates:[frac{a(4 b + c)}{a + 3 b} - frac{a(3 b + c)}{a + 3 b} = frac{a(4 b + c - 3 b - c)}{a + 3 b} = frac{a b}{a + 3 b}]Thus, (x_3 = frac{a b}{a + 3 b}).This confirms the pattern. Therefore, it seems that (x_n = frac{a b}{a + n b}).So, for part (1), the expression for (x_n) is:[x_n = frac{a b}{a + n b}]Now, moving on to part (2): Prove that (x_n leq frac{a + n b}{4 n}).Given (x_n = frac{a b}{a + n b}), we need to show:[frac{a b}{a + n b} leq frac{a + n b}{4 n}]Let me cross-multiply to eliminate denominators (since all terms are positive):[4 n a b leq (a + n b)^2]Expand the right side:[4 n a b leq a^2 + 2 a n b + n^2 b^2]Subtract (4 n a b) from both sides:[0 leq a^2 - 2 a n b + n^2 b^2]Notice that the right side is a perfect square:[0 leq (a - n b)^2]Which is always true since squares are non-negative. Therefore, the inequality holds.Hence, (x_n leq frac{a + n b}{4 n}).**Final Answer**(1) The length (x_n) is (boxed{dfrac{ab}{a + nb}}).(2) It is proved that (x_n leq dfrac{a + nb}{4n}), so the final result is (boxed{x_n leq dfrac{a + nb}{4n}}).

💡Okay, so I have this problem where a student found that five different trigonometric expressions all equal the same constant. I need to do two things: first, choose one of these expressions and calculate the constant, and second, generalize this into a trigonometric identity and prove it.Let me start by looking at the expressions given:1. sin^2 13^circ + cos^2 17^circ - sin 13^circ cos 17^circ2. sin^2 15^circ + cos^2 15^circ - sin 15^circ cos 15^circ3. sin^2 18^circ + cos^2 12^circ - sin 18^circ cos 12^circ4. sin^2 (-18^circ) + cos^2 48^circ - sin (-18^circ) cos 48^circ5. sin^2 (-25^circ) + cos^2 55^circ - sin (-25^circ) cos 55^circHmm, okay. I notice that in each expression, there are two angles involved, and sometimes one is negative. Maybe there's a relationship between these angles? Let me check:- For expression 1: 13° and 17°, which add up to 30°.- Expression 2: 15° and 15°, which add up to 30°.- Expression 3: 18° and 12°, which also add up to 30°.- Expression 4: -18° and 48°, which add up to 30° (since -18 + 48 = 30).- Expression 5: -25° and 55°, which add up to 30° (since -25 + 55 = 30).Oh, interesting! So in each case, the two angles add up to 30°. That seems like a key observation. So maybe the identity has something to do with angles that sum to 30°. Let me choose one of these expressions to calculate the constant. The second one seems straightforward because both angles are the same: 15°. So let's pick expression ②.Expression ② is: sin^2 15^circ + cos^2 15^circ - sin 15^circ cos 15^circI know that sin^2 theta + cos^2 theta = 1, so that simplifies the first two terms:1 - sin 15^circ cos 15^circNow, I need to compute sin 15^circ cos 15^circ. I remember that sin 2theta = 2sin theta cos theta, so maybe I can use that identity.Let me set theta = 15^circ, then:sin 30^circ = 2sin 15^circ cos 15^circSo, sin 15^circ cos 15^circ = frac{1}{2} sin 30^circI know that sin 30^circ = frac{1}{2}, so substituting back:sin 15^circ cos 15^circ = frac{1}{2} times frac{1}{2} = frac{1}{4}Therefore, expression ② becomes:1 - frac{1}{4} = frac{3}{4}So, the constant is frac{3}{4}.Now, for part (2), I need to generalize this into a trigonometric identity. From the first part, I saw that in each expression, the two angles add up to 30°. So, perhaps the identity is:sin^2 alpha + cos^2 (30^circ - alpha) - sin alpha cos (30^circ - alpha) = frac{3}{4}Let me test this with another expression to see if it holds. Let's take expression ①: sin^2 13^circ + cos^2 17^circ - sin 13^circ cos 17^circHere, 13° + 17° = 30°, so it fits the pattern. Let me compute it step by step.First, compute sin^2 13^circ + cos^2 17^circ. I know that sin^2 theta + cos^2 theta = 1, but here the angles are different. So, I can't directly use that identity. Hmm, maybe I need to express cos 17^circ in terms of sin or something else.Wait, 17° is 30° - 13°, so maybe I can write cos 17^circ = cos (30° - 13°). Using the cosine of a difference identity:cos (A - B) = cos A cos B + sin A sin BSo, cos (30° - 13°) = cos 30° cos 13° + sin 30° sin 13°Similarly, sin 13° cos 17° = sin 13° (cos 30° cos 13° + sin 30° sin 13°)Let me compute each part step by step.First, compute sin^2 13° + cos^2 17°:sin^2 13° + cos^2 17° = sin^2 13° + (cos 30° cos 13° + sin 30° sin 13°)^2Let me expand the square:= sin^2 13° + cos^2 30° cos^2 13° + 2cos 30° cos 13° sin 30° sin 13° + sin^2 30° sin^2 13°Now, substitute the known values:cos 30° = frac{sqrt{3}}{2}, sin 30° = frac{1}{2}So,= sin^2 13° + left(frac{sqrt{3}}{2}right)^2 cos^2 13° + 2 times frac{sqrt{3}}{2} times cos 13° times frac{1}{2} times sin 13° + left(frac{1}{2}right)^2 sin^2 13°Simplify each term:= sin^2 13° + frac{3}{4} cos^2 13° + frac{sqrt{3}}{2} cos 13° sin 13° + frac{1}{4} sin^2 13°Combine like terms:sin^2 13° + frac{1}{4} sin^2 13° = frac{5}{4} sin^2 13°So,= frac{5}{4} sin^2 13° + frac{3}{4} cos^2 13° + frac{sqrt{3}}{2} cos 13° sin 13°Now, let's compute the entire expression ①:sin^2 13° + cos^2 17° - sin 13° cos 17°We already have sin^2 13° + cos^2 17° as above, and we need to subtract sin 13° cos 17°.Earlier, we expressed cos 17° as cos 30° cos 13° + sin 30° sin 13°, so:sin 13° cos 17° = sin 13° (cos 30° cos 13° + sin 30° sin 13°) = frac{sqrt{3}}{2} sin 13° cos 13° + frac{1}{2} sin^2 13°So, subtracting this from the earlier expression:frac{5}{4} sin^2 13° + frac{3}{4} cos^2 13° + frac{sqrt{3}}{2} cos 13° sin 13° - left( frac{sqrt{3}}{2} sin 13° cos 13° + frac{1}{2} sin^2 13° right )Simplify term by term:- The frac{sqrt{3}}{2} cos 13° sin 13° and -frac{sqrt{3}}{2} sin 13° cos 13° cancel each other out.- frac{5}{4} sin^2 13° - frac{1}{2} sin^2 13° = frac{5}{4} - frac{2}{4} = frac{3}{4} sin^2 13°- The remaining term is frac{3}{4} cos^2 13°So, putting it together:frac{3}{4} sin^2 13° + frac{3}{4} cos^2 13° = frac{3}{4} (sin^2 13° + cos^2 13°) = frac{3}{4} times 1 = frac{3}{4}So, it checks out for expression ① as well. That gives me more confidence that the identity holds.Let me try another one, maybe expression ④: sin^2 (-18^circ) + cos^2 48^circ - sin (-18^circ) cos 48^circFirst, note that sin (-18^circ) = -sin 18^circ, so:sin^2 (-18^circ) = sin^2 18^circSo, the expression becomes:sin^2 18^circ + cos^2 48^circ - (-sin 18^circ) cos 48^circ = sin^2 18^circ + cos^2 48^circ + sin 18^circ cos 48^circAgain, 18° + 48° = 66°, which is not 30°, but wait, in the previous cases, the angles added up to 30°, but here, it's 66°. Hmm, that seems inconsistent. Wait, actually, in expression ④, the angles are -18° and 48°, which add up to 30°, as -18 + 48 = 30. So, it still fits the pattern.So, let me apply the identity:sin^2 alpha + cos^2 (30° - alpha) - sin alpha cos (30° - alpha)Here, alpha = -18°, so:sin^2 (-18°) + cos^2 (30° - (-18°)) - sin (-18°) cos (30° - (-18°)) = sin^2 (-18°) + cos^2 48° - sin (-18°) cos 48°Which is exactly expression ④. So, if the identity holds, it should equal frac{3}{4}.Let me compute it step by step.First, compute sin^2 (-18°) + cos^2 48°:sin^2 (-18°) = sin^2 18°, as sine is odd.cos 48° = cos (30° + 18°), but maybe it's better to express cos 48° as cos (30° - (-18°)).Using the cosine of a difference identity:cos (A - B) = cos A cos B + sin A sin BSo, cos (30° - (-18°)) = cos 30° cos (-18°) + sin 30° sin (-18°)But cos (-18°) = cos 18°, and sin (-18°) = -sin 18°, so:cos 48° = cos 30° cos 18° - sin 30° sin 18°Similarly, sin (-18°) cos 48° = -sin 18° cos 48° = -sin 18° (cos 30° cos 18° - sin 30° sin 18°)Let me compute each part:First, sin^2 18° + cos^2 48°:sin^2 18° + (cos 30° cos 18° - sin 30° sin 18°)^2Expanding the square:= sin^2 18° + cos^2 30° cos^2 18° - 2cos 30° cos 18° sin 30° sin 18° + sin^2 30° sin^2 18°Substitute the known values:cos 30° = frac{sqrt{3}}{2}, sin 30° = frac{1}{2}So,= sin^2 18° + left(frac{sqrt{3}}{2}right)^2 cos^2 18° - 2 times frac{sqrt{3}}{2} times cos 18° times frac{1}{2} times sin 18° + left(frac{1}{2}right)^2 sin^2 18°Simplify each term:= sin^2 18° + frac{3}{4} cos^2 18° - frac{sqrt{3}}{2} cos 18° sin 18° + frac{1}{4} sin^2 18°Combine like terms:sin^2 18° + frac{1}{4} sin^2 18° = frac{5}{4} sin^2 18°So,= frac{5}{4} sin^2 18° + frac{3}{4} cos^2 18° - frac{sqrt{3}}{2} cos 18° sin 18°Now, compute the entire expression ④:sin^2 (-18°) + cos^2 48° - sin (-18°) cos 48° = sin^2 18° + cos^2 48° + sin 18° cos 48°Wait, earlier I had:sin^2 18° + cos^2 48° + sin 18° cos 48°But from the expansion, I have:frac{5}{4} sin^2 18° + frac{3}{4} cos^2 18° - frac{sqrt{3}}{2} cos 18° sin 18° + sin 18° cos 48°But I need to express sin 18° cos 48° in terms of the identity.Wait, cos 48° = cos (30° - (-18°)) = cos 30° cos (-18°) - sin 30° sin (-18°) = frac{sqrt{3}}{2} cos 18° + frac{1}{2} sin 18°So, sin 18° cos 48° = sin 18° left( frac{sqrt{3}}{2} cos 18° + frac{1}{2} sin 18° right ) = frac{sqrt{3}}{2} sin 18° cos 18° + frac{1}{2} sin^2 18°Therefore, adding this to the previous expression:frac{5}{4} sin^2 18° + frac{3}{4} cos^2 18° - frac{sqrt{3}}{2} cos 18° sin 18° + frac{sqrt{3}}{2} sin 18° cos 18° + frac{1}{2} sin^2 18°Simplify term by term:- The -frac{sqrt{3}}{2} cos 18° sin 18° and +frac{sqrt{3}}{2} sin 18° cos 18° cancel each other out.- frac{5}{4} sin^2 18° + frac{1}{2} sin^2 18° = frac{5}{4} + frac{2}{4} = frac{7}{4} sin^2 18°- The remaining term is frac{3}{4} cos^2 18°Wait, that doesn't seem right because I was expecting the total to be frac{3}{4}. Did I make a mistake?Wait, no. Let me check again. The expression is:sin^2 18° + cos^2 48° + sin 18° cos 48°But when I expanded sin^2 18° + cos^2 48°, I got:frac{5}{4} sin^2 18° + frac{3}{4} cos^2 18° - frac{sqrt{3}}{2} cos 18° sin 18°Then, adding sin 18° cos 48° which is frac{sqrt{3}}{2} sin 18° cos 18° + frac{1}{2} sin^2 18°So, total expression:frac{5}{4} sin^2 18° + frac{3}{4} cos^2 18° - frac{sqrt{3}}{2} cos 18° sin 18° + frac{sqrt{3}}{2} sin 18° cos 18° + frac{1}{2} sin^2 18°Simplify:- The cos 18° sin 18° terms cancel.- frac{5}{4} sin^2 18° + frac{1}{2} sin^2 18° = frac{5}{4} + frac{2}{4} = frac{7}{4} sin^2 18°- Remaining term: frac{3}{4} cos^2 18°So, total is frac{7}{4} sin^2 18° + frac{3}{4} cos^2 18°Factor out frac{1}{4}:frac{1}{4} (7 sin^2 18° + 3 cos^2 18°)Hmm, that doesn't look like frac{3}{4}. Did I make a mistake in the expansion?Wait, maybe I should approach it differently. Let me try using the identity I proposed:sin^2 alpha + cos^2 (30° - alpha) - sin alpha cos (30° - alpha) = frac{3}{4}Here, alpha = -18°, so:sin^2 (-18°) + cos^2 (30° - (-18°)) - sin (-18°) cos (30° - (-18°)) = sin^2 (-18°) + cos^2 48° - sin (-18°) cos 48°Which is exactly expression ④. So, according to the identity, it should equal frac{3}{4}. But when I tried expanding, I got a different expression. Maybe I made an error in the expansion.Let me try a different approach. Let me use the identity directly.Given the identity:sin^2 alpha + cos^2 (30° - alpha) - sin alpha cos (30° - alpha) = frac{3}{4}Let me prove this identity in general, which will also confirm that expression ④ equals frac{3}{4}.So, let's let alpha be any angle. Then,sin^2 alpha + cos^2 (30° - alpha) - sin alpha cos (30° - alpha)First, express cos (30° - alpha) using the cosine of difference identity:cos (30° - alpha) = cos 30° cos alpha + sin 30° sin alpha = frac{sqrt{3}}{2} cos alpha + frac{1}{2} sin alphaSo, substitute back into the expression:sin^2 alpha + left( frac{sqrt{3}}{2} cos alpha + frac{1}{2} sin alpha right )^2 - sin alpha left( frac{sqrt{3}}{2} cos alpha + frac{1}{2} sin alpha right )Let me expand the square term:left( frac{sqrt{3}}{2} cos alpha + frac{1}{2} sin alpha right )^2 = left( frac{sqrt{3}}{2} cos alpha right )^2 + 2 times frac{sqrt{3}}{2} cos alpha times frac{1}{2} sin alpha + left( frac{1}{2} sin alpha right )^2Simplify each term:= frac{3}{4} cos^2 alpha + frac{sqrt{3}}{2} cos alpha sin alpha + frac{1}{4} sin^2 alphaNow, substitute back into the main expression:sin^2 alpha + left( frac{3}{4} cos^2 alpha + frac{sqrt{3}}{2} cos alpha sin alpha + frac{1}{4} sin^2 alpha right ) - sin alpha left( frac{sqrt{3}}{2} cos alpha + frac{1}{2} sin alpha right )Expand the subtraction term:= sin^2 alpha + frac{3}{4} cos^2 alpha + frac{sqrt{3}}{2} cos alpha sin alpha + frac{1}{4} sin^2 alpha - frac{sqrt{3}}{2} sin alpha cos alpha - frac{1}{2} sin^2 alphaNow, let's combine like terms:- sin^2 alpha + frac{1}{4} sin^2 alpha - frac{1}{2} sin^2 alpha = left(1 + frac{1}{4} - frac{1}{2}right) sin^2 alpha = frac{3}{4} sin^2 alpha- frac{3}{4} cos^2 alpha- frac{sqrt{3}}{2} cos alpha sin alpha - frac{sqrt{3}}{2} sin alpha cos alpha = 0So, the expression simplifies to:frac{3}{4} sin^2 alpha + frac{3}{4} cos^2 alpha = frac{3}{4} (sin^2 alpha + cos^2 alpha) = frac{3}{4} times 1 = frac{3}{4}Therefore, the identity holds for any angle alpha. This means that all the given expressions equal frac{3}{4} because in each case, the two angles add up to 30°, fitting the identity.So, to generalize, the trigonometric identity is:sin^2 alpha + cos^2 (30° - alpha) - sin alpha cos (30° - alpha) = frac{3}{4}And I've proven it using algebraic manipulation and trigonometric identities.