💡Okay, so I have this problem about a parabola and some points on it. Let me try to figure it out step by step. First, part 1: I need to find the equation of the parabola C. The parabola is given by the equation (x^2 = 2py), and point A is ((- frac{1}{2}, y_0)) lying on this parabola. Also, the distance from point A to the focus of the parabola is (frac{5}{8}). Alright, so since point A is on the parabola, it must satisfy the equation (x^2 = 2py). Plugging in the coordinates of A, we get:[left(-frac{1}{2}right)^2 = 2p y_0][frac{1}{4} = 2p y_0]So, (2p y_0 = frac{1}{4}). That's one equation.Next, the distance from point A to the focus is given as (frac{5}{8}). For a parabola (x^2 = 2py), the focus is at ((0, frac{p}{2})). So, the distance between A ((- frac{1}{2}, y_0)) and the focus ((0, frac{p}{2})) is:[sqrt{left(-frac{1}{2} - 0right)^2 + left(y_0 - frac{p}{2}right)^2} = frac{5}{8}]Squaring both sides to eliminate the square root:[left(-frac{1}{2}right)^2 + left(y_0 - frac{p}{2}right)^2 = left(frac{5}{8}right)^2][frac{1}{4} + left(y_0 - frac{p}{2}right)^2 = frac{25}{64}]Subtracting (frac{1}{4}) from both sides:[left(y_0 - frac{p}{2}right)^2 = frac{25}{64} - frac{16}{64} = frac{9}{64}]Taking square roots:[y_0 - frac{p}{2} = pm frac{3}{8}]So, (y_0 = frac{p}{2} pm frac{3}{8}).But we also have from the first equation that (2p y_0 = frac{1}{4}), so (y_0 = frac{1}{8p}).Therefore, substituting (y_0) into the equation above:[frac{1}{8p} = frac{p}{2} pm frac{3}{8}]Let me write this as:[frac{1}{8p} = frac{p}{2} + frac{3}{8} quad text{or} quad frac{1}{8p} = frac{p}{2} - frac{3}{8}]Let me solve both equations.First equation:[frac{1}{8p} = frac{p}{2} + frac{3}{8}]Multiply both sides by 8p to eliminate denominators:[1 = 4p^2 + 3p]Bring all terms to one side:[4p^2 + 3p - 1 = 0]Using quadratic formula:[p = frac{-3 pm sqrt{9 + 16}}{8} = frac{-3 pm 5}{8}]So, p can be (frac{2}{8} = frac{1}{4}) or (frac{-8}{8} = -1). But p > 1/2, so p = 1/4 is less than 1/2, which doesn't satisfy the condition. So this solution is invalid.Second equation:[frac{1}{8p} = frac{p}{2} - frac{3}{8}]Multiply both sides by 8p:[1 = 4p^2 - 3p]Bring all terms to one side:[4p^2 - 3p - 1 = 0]Quadratic formula:[p = frac{3 pm sqrt{9 + 16}}{8} = frac{3 pm 5}{8}]So, p can be (frac{8}{8} = 1) or (frac{-2}{8} = -1/4). Since p > 1/2, p = 1 is the valid solution.So, p = 1. Therefore, the equation of the parabola is (x^2 = 2y).Alright, that was part 1. I think I did that correctly. Let me just double-check.Given p = 1, the parabola is (x^2 = 2y). Point A is ((-1/2, y_0)). Plugging into the equation:[left(-frac{1}{2}right)^2 = 2(1)y_0 implies frac{1}{4} = 2y_0 implies y_0 = frac{1}{8}]So, point A is ((-1/2, 1/8)). The focus is at (0, p/2) = (0, 1/2). The distance from A to the focus is:[sqrt{left(-frac{1}{2} - 0right)^2 + left(frac{1}{8} - frac{1}{2}right)^2} = sqrt{frac{1}{4} + left(-frac{3}{8}right)^2} = sqrt{frac{1}{4} + frac{9}{64}} = sqrt{frac{16}{64} + frac{9}{64}} = sqrt{frac{25}{64}} = frac{5}{8}]Yes, that checks out. So part 1 is done.Now, part 2. This seems more complicated. Let me parse the problem again.We have point P moving on parabola C, not on line l: (y = 2x + 9y_0). Since y_0 is 1/8, line l is (y = 2x + 9*(1/8) = 2x + 9/8).Line l intersects parabola C at points E and F. So, we can find E and F by solving the system:[x^2 = 2y][y = 2x + frac{9}{8}]Substitute y from the second equation into the first:[x^2 = 2*(2x + 9/8) = 4x + frac{9}{4}]Bring all terms to one side:[x^2 - 4x - frac{9}{4} = 0]Multiply both sides by 4 to eliminate fractions:[4x^2 - 16x - 9 = 0]Quadratic equation:[x = frac{16 pm sqrt{256 + 144}}{8} = frac{16 pm sqrt{400}}{8} = frac{16 pm 20}{8}]So, x = (16 + 20)/8 = 36/8 = 4.5 = 9/2, or x = (16 - 20)/8 = (-4)/8 = -1/2.Therefore, the points E and F are at x = -1/2 and x = 9/2.Wait, but x = -1/2 is point A, right? Because point A is (-1/2, 1/8). So, is E or F equal to A? Let me check.When x = -1/2, y = 2*(-1/2) + 9/8 = -1 + 9/8 = 1/8. So yes, point A is one of the intersection points, specifically E or F.But the problem says "line l intersects parabola C at points E and F". So, one of them is A, and the other is (9/2, y). Let's compute y for x = 9/2.y = 2*(9/2) + 9/8 = 9 + 9/8 = 81/8.So, points E and F are (-1/2, 1/8) and (9/2, 81/8). So, E is A, and F is (9/2, 81/8).Now, the problem continues: A vertical line through P intersects l at point M. The perpendicular from P to l meets it at N. We need to prove that |AM|² / |AN| = |EF|.Hmm. Let me break this down.First, let me consider point P on the parabola C, which is (x^2 = 2y). So, any point P can be represented as (m, m²/2), where m is a parameter.Since P is not on line l, which is y = 2x + 9/8, so m²/2 ≠ 2m + 9/8.A vertical line through P is a line with constant x-coordinate m. So, the equation is x = m. This intersects line l at point M. So, to find M, we plug x = m into l's equation:y = 2m + 9/8. So, point M is (m, 2m + 9/8).Next, the perpendicular from P to l meets l at N. So, we need to find the foot of the perpendicular from P to l.First, let's find the equation of the line l: y = 2x + 9/8. Its slope is 2, so the slope of the perpendicular is -1/2.So, the line perpendicular to l through P(m, m²/2) is:y - m²/2 = (-1/2)(x - m)We can find the intersection point N between this perpendicular and line l.So, set up the equations:1. y = 2x + 9/82. y = (-1/2)x + (m²/2 + m/2)Set them equal:2x + 9/8 = (-1/2)x + (m²/2 + m/2)Multiply both sides by 8 to eliminate denominators:16x + 9 = (-4x) + 4m² + 4mBring all terms to left:16x + 9 + 4x - 4m² - 4m = 020x + 9 - 4m² - 4m = 0Solve for x:20x = 4m² + 4m - 9x = (4m² + 4m - 9)/20Simplify:x = (m² + m - 9/4)/5So, x-coordinate of N is (m² + m - 9/4)/5.Then, y-coordinate is from line l:y = 2x + 9/8 = 2*(m² + m - 9/4)/5 + 9/8Compute:= (2m² + 2m - 9/2)/5 + 9/8= (2m² + 2m - 9/2)/5 + 9/8To combine, find common denominator, which is 40:= (16m² + 16m - 180)/40 + (45)/40= (16m² + 16m - 180 + 45)/40= (16m² + 16m - 135)/40So, point N is ((m² + m - 9/4)/5, (16m² + 16m - 135)/40)Now, we need to compute |AM|² / |AN| and show it's equal to |EF|.First, let's compute |EF|.Points E and F are (-1/2, 1/8) and (9/2, 81/8). So, the distance between E and F is:√[(9/2 - (-1/2))² + (81/8 - 1/8)²]Compute differences:x: 9/2 - (-1/2) = 10/2 = 5y: 81/8 - 1/8 = 80/8 = 10So, |EF| = √(5² + 10²) = √(25 + 100) = √125 = 5√5.So, |EF| = 5√5.Now, let's compute |AM|² and |AN|.Point A is (-1/2, 1/8). Point M is (m, 2m + 9/8).So, vector AM is (m - (-1/2), 2m + 9/8 - 1/8) = (m + 1/2, 2m + 1).So, |AM|² is (m + 1/2)² + (2m + 1)².Compute:= (m² + m + 1/4) + (4m² + 4m + 1)= 5m² + 5m + 5/4So, |AM|² = 5m² + 5m + 5/4.Now, compute |AN|.Point N is ((m² + m - 9/4)/5, (16m² + 16m - 135)/40)Point A is (-1/2, 1/8). So, vector AN is:x: (m² + m - 9/4)/5 - (-1/2) = (m² + m - 9/4)/5 + 1/2y: (16m² + 16m - 135)/40 - 1/8 = (16m² + 16m - 135)/40 - 5/40 = (16m² + 16m - 140)/40Simplify x-coordinate:= (m² + m - 9/4)/5 + 1/2= (2(m² + m - 9/4) + 5)/10= (2m² + 2m - 9/2 + 5)/10= (2m² + 2m - 9/2 + 10/2)/10= (2m² + 2m + 1/2)/10= (4m² + 4m + 1)/20Similarly, y-coordinate:= (16m² + 16m - 140)/40= (4m² + 4m - 35)/10So, vector AN is ((4m² + 4m + 1)/20, (4m² + 4m - 35)/10)Compute |AN|:= √[ ((4m² + 4m + 1)/20)^2 + ((4m² + 4m - 35)/10)^2 ]Factor out 1/20:= √[ ( (4m² + 4m + 1)^2 + 4*(4m² + 4m - 35)^2 ) / 400 ]This seems complicated. Maybe there's a smarter way.Wait, perhaps instead of computing |AN| directly, notice that |AN| is the distance from A to N, which is the length of the projection or something? Maybe not.Alternatively, perhaps we can relate |AM|² / |AN| to |EF| by expressing them in terms of m and see if they are equal.Given that |EF| = 5√5, and |AM|² = 5m² + 5m + 5/4, and |AN| is something.Wait, let me compute |AM|² / |AN|.We have |AM|² = 5m² + 5m + 5/4.Compute |AN|:From earlier, vector AN is ((4m² + 4m + 1)/20, (4m² + 4m - 35)/10)Compute the squared distance:= [ (4m² + 4m + 1)/20 ]² + [ (4m² + 4m - 35)/10 ]²= (16m^4 + 32m^3 + 24m² + 8m + 1)/400 + (16m^4 + 32m^3 + 24m² - 280m + 1225)/100Wait, that seems messy. Maybe factor out terms.Wait, notice that 4m² + 4m + 1 = (2m + 1)^2, and 4m² + 4m - 35 is another quadratic.Let me denote t = 4m² + 4m.Then, 4m² + 4m + 1 = t + 1, and 4m² + 4m - 35 = t - 35.So, vector AN is ( (t + 1)/20, (t - 35)/10 )So, squared distance:= [ (t + 1)/20 ]² + [ (t - 35)/10 ]²= (t² + 2t + 1)/400 + (t² - 70t + 1225)/100= (t² + 2t + 1)/400 + 4(t² - 70t + 1225)/400= [ t² + 2t + 1 + 4t² - 280t + 4900 ] / 400= (5t² - 278t + 4901)/400Hmm, not sure if that helps.Alternatively, perhaps compute |AN| in terms of |AM|².Wait, let's see:We have |AM|² = 5m² + 5m + 5/4.If I factor 5/4:= 5/4 (4m² + 4m + 1)= 5/4 (2m + 1)^2So, |AM|² = (5/4)(2m + 1)^2So, |AM| = (sqrt(5)/2)(2m + 1)But then, |AM|² / |AN| = (5/4)(2m + 1)^2 / |AN|We need this to be equal to |EF| = 5√5.So,(5/4)(2m + 1)^2 / |AN| = 5√5Simplify:(1/4)(2m + 1)^2 / |AN| = √5So,(2m + 1)^2 / |AN| = 4√5Hmm, not sure.Wait, maybe express |AN| in terms of (2m + 1).From earlier, vector AN is ((4m² + 4m + 1)/20, (4m² + 4m - 35)/10 )Note that 4m² + 4m + 1 = (2m + 1)^2Similarly, 4m² + 4m - 35 = (2m + 1)^2 - 36So, vector AN is ( (2m + 1)^2 / 20, ( (2m + 1)^2 - 36 ) / 10 )So, squared distance |AN|²:= [ (2m + 1)^4 / 400 ] + [ ( (2m + 1)^2 - 36 )² / 100 ]Let me compute this:= ( (2m + 1)^4 ) / 400 + ( (2m + 1)^4 - 72(2m + 1)^2 + 1296 ) / 100= [ (2m + 1)^4 / 400 ] + [ 4(2m + 1)^4 - 288(2m + 1)^2 + 5184 ) / 400 ]= [ (2m + 1)^4 + 4(2m + 1)^4 - 288(2m + 1)^2 + 5184 ] / 400= [5(2m + 1)^4 - 288(2m + 1)^2 + 5184 ] / 400This is getting too complicated. Maybe there's a different approach.Wait, perhaps instead of computing |AN| directly, notice that |AN| is related to |AM|².From earlier, |AM|² = 5m² + 5m + 5/4 = 5(m² + m + 1/4) = 5(m + 1/2)^2.So, |AM|² = 5(m + 1/2)^2.So, |AM|² / |AN| = 5(m + 1/2)^2 / |AN|We need this to equal |EF| = 5√5.So,5(m + 1/2)^2 / |AN| = 5√5Simplify:(m + 1/2)^2 / |AN| = √5So,|AN| = (m + 1/2)^2 / √5But from earlier, |AN| is the distance from A to N, which is a point on l. Maybe we can express |AN| in terms of projections or something.Alternatively, perhaps use coordinates.Point A is (-1/2, 1/8). Point N is ((m² + m - 9/4)/5, (16m² + 16m - 135)/40 )So, the vector AN is:x: (m² + m - 9/4)/5 - (-1/2) = (m² + m - 9/4)/5 + 1/2 = (2m² + 2m - 9/2 + 5)/10 = (2m² + 2m + 1/2)/10 = (4m² + 4m + 1)/20y: (16m² + 16m - 135)/40 - 1/8 = (16m² + 16m - 135)/40 - 5/40 = (16m² + 16m - 140)/40 = (4m² + 4m - 35)/10So, vector AN is ( (4m² + 4m + 1)/20, (4m² + 4m - 35)/10 )Let me denote t = 4m² + 4m.Then, vector AN is ( (t + 1)/20, (t - 35)/10 )So, |AN|² = ( (t + 1)/20 )² + ( (t - 35)/10 )²= (t² + 2t + 1)/400 + (t² - 70t + 1225)/100= (t² + 2t + 1)/400 + 4(t² - 70t + 1225)/400= (t² + 2t + 1 + 4t² - 280t + 4900)/400= (5t² - 278t + 4901)/400Hmm, not sure.But from earlier, |AM|² = 5(m + 1/2)^2 = 5(m² + m + 1/4) = 5m² + 5m + 5/4.Let me compute |AM|² / |AN|:= (5m² + 5m + 5/4) / sqrt( (5t² - 278t + 4901)/400 )But t = 4m² + 4m, so t = 4(m² + m).Let me express 5m² + 5m + 5/4 in terms of t:5m² + 5m + 5/4 = 5(m² + m) + 5/4 = (5/4)t + 5/4 = (5t + 5)/4So, |AM|² = (5t + 5)/4Therefore, |AM|² / |AN| = (5t + 5)/4 / sqrt( (5t² - 278t + 4901)/400 )= (5(t + 1)/4) / ( sqrt(5t² - 278t + 4901)/20 )= (5(t + 1)/4) * (20 / sqrt(5t² - 278t + 4901))= (25(t + 1)) / sqrt(5t² - 278t + 4901)We need this to equal |EF| = 5√5.So,25(t + 1) / sqrt(5t² - 278t + 4901) = 5√5Divide both sides by 5:5(t + 1) / sqrt(5t² - 278t + 4901) = √5Multiply both sides by sqrt(5t² - 278t + 4901):5(t + 1) = √5 * sqrt(5t² - 278t + 4901)Square both sides:25(t + 1)^2 = 5(5t² - 278t + 4901)Simplify:25(t² + 2t + 1) = 25t² - 1390t + 24505Expand left side:25t² + 50t + 25 = 25t² - 1390t + 24505Subtract 25t² from both sides:50t + 25 = -1390t + 24505Bring all terms to left:50t + 25 + 1390t - 24505 = 01440t - 24480 = 01440t = 24480t = 24480 / 1440 = 17So, t = 17.But t = 4m² + 4m = 17So,4m² + 4m - 17 = 0Quadratic in m:m = [-4 ± sqrt(16 + 272)] / 8 = [-4 ± sqrt(288)] / 8 = [-4 ± 12√2]/8 = [-1 ± 3√2]/2Wait, but this suggests that the equality holds only for specific m, but the problem states that P is any point on the parabola (except on line l). So, this approach might not be the right way.Perhaps I made a mistake in the algebra. Let me check.Wait, when I squared both sides, I might have introduced extraneous solutions. Maybe the equation holds for all m, but my approach is complicating things.Alternatively, perhaps instead of computing |AN| directly, notice that |AN| is related to the projection of AM onto AN or something.Wait, another idea: Since N is the foot of the perpendicular from P to l, and M is the projection of P onto l via vertical line, maybe there's a relationship between AM and AN.Alternatively, perhaps use coordinates to express |AM|² / |AN|.Wait, from earlier, |AM|² = 5(m + 1/2)^2.And |AN| is the distance from A to N, which is a point on l. Since l is a straight line, maybe |AN| can be expressed in terms of the distance from A to l, but A is on l, so the distance is zero? Wait, no, A is on l, so the distance from A to l is zero. But N is the foot of the perpendicular from P to l, so AN is the distance from A to N along l.Wait, but A is on l, so AN is just the distance along l from A to N.But l is a straight line, so the distance from A to N is just the length along l from A to N.But since N is the foot of the perpendicular from P to l, and M is the vertical projection, maybe there's a relation between AM and AN.Alternatively, perhaps use similar triangles or parametric equations.Wait, let me parametrize line l.Line l: y = 2x + 9/8.Parametric equations can be written as:x = ty = 2t + 9/8So, any point on l can be written as (t, 2t + 9/8).Point A is (-1/2, 1/8), which corresponds to t = -1/2.Point M is (m, 2m + 9/8), which corresponds to t = m.Point N is ((m² + m - 9/4)/5, (16m² + 16m - 135)/40 ), which also lies on l, so it corresponds to some t.Let me find t for N:From parametric equations, x = t, so t = (m² + m - 9/4)/5.So, t_N = (m² + m - 9/4)/5.Similarly, t_A = -1/2.So, the distance from A to N along l is |t_N - t_A| * sqrt(1 + 2²) = |t_N - t_A| * sqrt(5).Because the distance along the line with slope 2 is scaled by sqrt(1 + (2)^2) = sqrt(5).So, |AN| = |t_N - t_A| * sqrt(5)Compute t_N - t_A:= (m² + m - 9/4)/5 - (-1/2)= (m² + m - 9/4)/5 + 1/2= (2m² + 2m - 9/2 + 5)/10= (2m² + 2m + 1/2)/10= (4m² + 4m + 1)/20So, |AN| = |(4m² + 4m + 1)/20| * sqrt(5)Since 4m² + 4m + 1 is always positive (discriminant 16 - 16 = 0, so it's a perfect square), we can drop the absolute value:|AN| = (4m² + 4m + 1)/20 * sqrt(5)But 4m² + 4m + 1 = (2m + 1)^2So, |AN| = (2m + 1)^2 / 20 * sqrt(5)Therefore, |AN| = sqrt(5) * (2m + 1)^2 / 20Now, recall that |AM|² = 5(m + 1/2)^2 = 5*(2m + 1)^2 / 4So, |AM|² = (5/4)(2m + 1)^2Therefore, |AM|² / |AN| = (5/4)(2m + 1)^2 / [ sqrt(5)*(2m + 1)^2 / 20 ]Simplify:= (5/4) / [ sqrt(5)/20 ]= (5/4) * (20 / sqrt(5))= (5 * 20) / (4 * sqrt(5))= 100 / (4 * sqrt(5))= 25 / sqrt(5)= 5√5Which is equal to |EF|.Therefore, |AM|² / |AN| = |EF|.So, the proof is complete.**Final Answer**The equation of the parabola ( C ) is (boxed{x^2 = 2y}), and the required relationship is proven as (boxed{frac{|AM|^2}{|AN|} = |EF|}).

💡First, I need to find all the positive integer divisors of 450, excluding 1. To do this, I'll start by factorizing 450 into its prime components.450 can be factorized as (2^1 times 3^2 times 5^2). Using this prime factorization, I can list all the divisors by considering the different combinations of these prime factors.The divisors of 450, excluding 1, are: 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, and 450.Next, I need to arrange these divisors around a circle such that every pair of adjacent numbers has a common factor greater than 1. My focus is on the number 15, which has prime factors 3 and 5.To determine the numbers adjacent to 15, I'll look for divisors that share a common factor with 15. The potential candidates are 30 (which is 15 multiplied by 2) and 75 (which is 15 multiplied by 5). Both 30 and 75 share the common factor 15 with 15, satisfying the condition.Finally, I'll calculate the sum of these two adjacent numbers: 30 and 75. Adding them together gives (30 + 75 = 105).

💡First, I need to determine the distance traveled by ferry P. Since ferry P travels for 2 hours at a speed of 8 kilometers per hour, the distance can be calculated by multiplying the time by the speed.Next, ferry Q takes a route that is three times longer than ferry P's route. Therefore, I will multiply the distance traveled by ferry P by 3 to find the distance for ferry Q.I know that the journey of ferry Q is 2 hours longer than that of ferry P. Since ferry P takes 2 hours, ferry Q takes 4 hours in total.With the distance and time for ferry Q, I can calculate its speed by dividing the distance by the time.Finally, to find out how many kilometers per hour slower ferry P is compared to ferry Q, I will subtract the speed of ferry P from the speed of ferry Q.

💡First, I need to determine the fraction of the Earth's surface that is land. According to the problem, one third of the Earth's surface is land.Next, I'll calculate the fraction of this land that is inhabitable. The problem states that only one fourth of the exposed land area is inhabitable.Then, I need to find out what portion of this inhabitable land is available for residential purposes. The problem mentions that 60% of the inhabitable land is available for residential use.Finally, I'll multiply these fractions together to find the overall fraction of the Earth's surface that is available for human living.

💡Okay, so I have this problem here about arithmetic and geometric sequences. Let me try to understand what it's asking step by step.First, there's an arithmetic sequence {a_n} where each term is non-zero. It satisfies the equation a_3 - (a_7)^2 / 2 + a_11 = 0. Then, there's a geometric sequence {b_n} where b_7 = a_7. I need to find the value of b_1 multiplied by b_13. The options are 25, 16, 8, and 4.Alright, let's start with the arithmetic sequence. I remember that in an arithmetic sequence, each term is obtained by adding a common difference to the previous term. So, the general formula for the nth term is a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.Given that, let's write expressions for a_3, a_7, and a_11.a_3 = a_1 + 2da_7 = a_1 + 6da_11 = a_1 + 10dNow, the equation given is a_3 - (a_7)^2 / 2 + a_11 = 0. Let's substitute the expressions we have into this equation.So, substituting:(a_1 + 2d) - [(a_1 + 6d)^2 / 2] + (a_1 + 10d) = 0Let me simplify this step by step.First, combine the terms without the squared part:(a_1 + 2d) + (a_1 + 10d) = 2a_1 + 12dSo, the equation becomes:2a_1 + 12d - [(a_1 + 6d)^2 / 2] = 0Hmm, that looks a bit complicated. Maybe I can factor out some terms or find a relationship between a_1 and d.Wait, I remember that in an arithmetic sequence, the terms are equally spaced, so the average of terms equidistant from the center is equal. For example, a_3 and a_11 are both 4 terms away from a_7. So, a_3 + a_11 = 2a_7.Is that correct? Let me check:a_3 = a_1 + 2da_11 = a_1 + 10dAdding them: (a_1 + 2d) + (a_1 + 10d) = 2a_1 + 12dAnd 2a_7 is 2*(a_1 + 6d) = 2a_1 + 12d. Yes, that's correct. So, a_3 + a_11 = 2a_7.So, going back to the original equation:a_3 - (a_7)^2 / 2 + a_11 = 0We can rewrite this as:(a_3 + a_11) - (a_7)^2 / 2 = 0But since a_3 + a_11 = 2a_7, substitute that in:2a_7 - (a_7)^2 / 2 = 0So, now the equation is:2a_7 - (a_7)^2 / 2 = 0Let me write that as:(2a_7) - (a_7^2)/2 = 0To make it easier, multiply both sides by 2 to eliminate the denominator:4a_7 - a_7^2 = 0So, we have:4a_7 - a_7^2 = 0Factor out a_7:a_7(4 - a_7) = 0So, either a_7 = 0 or 4 - a_7 = 0, which gives a_7 = 4.But the problem states that each term is non-zero, so a_7 can't be zero. Therefore, a_7 = 4.Alright, so a_7 is 4. Now, moving on to the geometric sequence {b_n} where b_7 = a_7 = 4.I need to find b_1 * b_13.In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio r. The general formula is b_n = b_1 * r^(n-1).So, let's write expressions for b_7 and b_13.b_7 = b_1 * r^(6)b_13 = b_1 * r^(12)We know that b_7 = 4, so:b_1 * r^6 = 4We need to find b_1 * b_13.Let's compute b_1 * b_13:b_1 * b_13 = b_1 * (b_1 * r^12) = (b_1)^2 * r^12But from b_7, we have b_1 * r^6 = 4. Let's square both sides of this equation:(b_1 * r^6)^2 = 4^2(b_1)^2 * r^12 = 16But wait, (b_1)^2 * r^12 is exactly what we have for b_1 * b_13.So, b_1 * b_13 = 16.Therefore, the answer should be 16, which is option B.Let me just double-check my steps to make sure I didn't make a mistake.1. I used the property of arithmetic sequences that a_3 + a_11 = 2a_7. That seems correct because they are equidistant from a_7.2. Substituted into the given equation and simplified to find a_7 = 4. That seems solid.3. For the geometric sequence, I used the general term formula and found that b_1 * b_13 is equal to (b_1)^2 * r^12, which is the square of b_7. Since b_7 is 4, squaring it gives 16. That makes sense.So, I think my answer is correct.**Final Answer**boxed{B}

💡First, I need to determine the total number of possible two-digit numbers that can be formed by randomly drawing two cards from the five available cards numbered 1, 2, 3, 4, and 5. Since the order matters in forming a two-digit number, this is a permutation problem. The number of permutations of 5 cards taken 2 at a time is calculated as 5 × 4 = 20. So, there are 20 possible two-digit numbers.Next, I need to identify which of these two-digit numbers are even. A number is even if its last digit is even. From the cards, the even numbers are 2 and 4. Therefore, the possible two-digit even numbers are those where the second digit is either 2 or 4. Listing these out, the even numbers are: 12, 14, 24, 32, 34, 42, 52, and 54. This gives a total of 8 even numbers.Finally, to find the probability of forming an even two-digit number, I divide the number of favorable outcomes (even numbers) by the total number of possible outcomes. So, the probability is 8 divided by 20, which simplifies to 2/5.