💡Alright, so I have this problem about a geometric sequence and then another sequence related to it. Let me try to figure it out step by step.First, part (1) says that {a_n} is a geometric sequence with the first term a₁ = 1 and common ratio q > 0. The sum of the first n terms is Sₙ. Also, it's given that S₁ + a₁, S₃ + a₃, and S₂ + a₂ form an arithmetic sequence. I need to find the general formula for {a_n}.Okay, so let's recall what a geometric sequence is. A geometric sequence has each term as the previous term multiplied by a common ratio q. So, a₁ = 1, a₂ = q, a₃ = q², and so on. The sum of the first n terms, Sₙ, for a geometric sequence is given by Sₙ = a₁(1 - qⁿ)/(1 - q) if q ≠ 1.Now, the problem mentions that S₁ + a₁, S₃ + a₃, S₂ + a₂ form an arithmetic sequence. An arithmetic sequence has a common difference between consecutive terms. So, the difference between the second and first term should be equal to the difference between the third and second term.Let me write down what S₁, S₂, and S₃ are.S₁ is just the sum of the first term, so S₁ = a₁ = 1.S₂ is the sum of the first two terms: S₂ = a₁ + a₂ = 1 + q.S₃ is the sum of the first three terms: S₃ = a₁ + a₂ + a₃ = 1 + q + q².Now, let's compute S₁ + a₁, S₃ + a₃, and S₂ + a₂.S₁ + a₁ = 1 + 1 = 2.S₃ + a₃ = (1 + q + q²) + q² = 1 + q + 2q².S₂ + a₂ = (1 + q) + q = 1 + 2q.So, the three terms are 2, 1 + q + 2q², and 1 + 2q. These form an arithmetic sequence. Therefore, the difference between the second and first term should be equal to the difference between the third and second term.Let me write that equation:(S₃ + a₃) - (S₁ + a₁) = (S₂ + a₂) - (S₃ + a₃)Plugging in the values:(1 + q + 2q²) - 2 = (1 + 2q) - (1 + q + 2q²)Simplify both sides:Left side: 1 + q + 2q² - 2 = (1 - 2) + q + 2q² = -1 + q + 2q²Right side: 1 + 2q - 1 - q - 2q² = (1 - 1) + (2q - q) - 2q² = q - 2q²So, we have:-1 + q + 2q² = q - 2q²Let me bring all terms to one side:-1 + q + 2q² - q + 2q² = 0Simplify:-1 + (q - q) + (2q² + 2q²) = -1 + 4q² = 0So, 4q² - 1 = 0Solving for q:4q² = 1q² = 1/4Since q > 0, q = 1/2.So, the common ratio is 1/2. Therefore, the general term a_n is a₁ * q^{n-1} = 1 * (1/2)^{n-1} = (1/2)^{n-1}.Okay, that seems straightforward. Let me just double-check.Given q = 1/2, let's compute S₁ + a₁, S₃ + a₃, S₂ + a₂.S₁ = 1, so S₁ + a₁ = 2.S₃ = 1 + 1/2 + 1/4 = 1.75, so S₃ + a₃ = 1.75 + 1/4 = 2.25.S₂ = 1 + 1/2 = 1.5, so S₂ + a₂ = 1.5 + 1/2 = 2.Wait, so the three terms are 2, 2.25, 2. But that doesn't seem like an arithmetic sequence because the differences are 0.25 and -0.25, which aren't equal. Hmm, did I make a mistake?Wait, no. Let me recalculate S₃ + a₃.Wait, S₃ is 1 + 1/2 + 1/4 = 1.75. Then a₃ is 1/4, so S₃ + a₃ = 1.75 + 0.25 = 2.0. Wait, that's 2.0, not 2.25. So, actually, S₃ + a₃ is 2.0.Similarly, S₂ + a₂ = 1.5 + 0.5 = 2.0.So, the three terms are 2, 2, 2. That is indeed an arithmetic sequence with common difference 0. So, that works.Okay, so my initial calculation was correct, but my intermediate step had a miscalculation. Good to verify.So, part (1) is solved: a_n = (1/2)^{n-1}.Now, moving on to part (2). It says that the sequence {b_n} satisfies a_{n+1} = (1/2)^{a_n b_n}, and T_n is the sum of the first n terms of {b_n}. It also says that T_n ≥ m always holds, and we need to find the maximum value of m.Alright, so first, let's write down what a_{n+1} is. From part (1), a_n = (1/2)^{n-1}, so a_{n+1} = (1/2)^n.Given that a_{n+1} = (1/2)^{a_n b_n}, so:(1/2)^n = (1/2)^{a_n b_n}Since the bases are the same and greater than 0, we can equate the exponents:n = a_n b_nTherefore, b_n = n / a_nBut a_n = (1/2)^{n-1}, so:b_n = n / (1/2)^{n-1} = n * 2^{n-1}So, b_n = n * 2^{n-1}Therefore, T_n is the sum from k=1 to n of b_k, which is sum_{k=1}^n k * 2^{k-1}We need to find T_n and then find the maximum m such that T_n ≥ m for all n. So, we need the minimum value of T_n across all n, and that minimum will be the maximum m.So, first, let's compute T_n.I remember that the sum sum_{k=1}^n k * r^{k} has a formula, but here we have sum_{k=1}^n k * 2^{k-1}. Let me adjust the formula accordingly.Let me recall the formula for sum_{k=1}^n k r^{k} = r(1 - (n+1) r^n + n r^{n+1}) ) / (1 - r)^2But in our case, the exponent is k-1, so let me write it as:sum_{k=1}^n k * 2^{k-1} = (1/2) sum_{k=1}^n k * 2^{k}So, let me compute sum_{k=1}^n k * 2^{k} first.Using the formula:sum_{k=1}^n k r^{k} = r(1 - (n+1) r^n + n r^{n+1}) ) / (1 - r)^2Here, r = 2.So, sum_{k=1}^n k * 2^{k} = 2(1 - (n+1)2^n + n 2^{n+1}) / (1 - 2)^2Simplify denominator: (1 - 2)^2 = 1So, numerator: 2[1 - (n+1)2^n + n 2^{n+1}]Simplify inside the brackets:1 - (n+1)2^n + n 2^{n+1} = 1 - (n+1)2^n + 2n 2^n = 1 + (2n - n - 1)2^n = 1 + (n - 1)2^nMultiply by 2:2[1 + (n - 1)2^n] = 2 + 2(n - 1)2^n = 2 + (n - 1)2^{n+1}Wait, but let me double-check:Wait, 2[1 + (n - 1)2^n] = 2*1 + 2*(n - 1)2^n = 2 + (n - 1)2^{n+1}Wait, that seems correct.But wait, let me compute step by step:sum_{k=1}^n k * 2^{k} = 2(1 - (n+1)2^n + n 2^{n+1}) / (1 - 2)^2Denominator is 1, so it's 2[1 - (n+1)2^n + n 2^{n+1}]Compute inside:1 - (n+1)2^n + n 2^{n+1} = 1 - (n+1)2^n + 2n 2^n = 1 + (2n - n - 1)2^n = 1 + (n - 1)2^nMultiply by 2:2 + 2(n - 1)2^n = 2 + (n - 1)2^{n+1}Wait, but 2(n - 1)2^n = (n - 1)2^{n+1}, yes.So, sum_{k=1}^n k * 2^{k} = 2 + (n - 1)2^{n+1}Therefore, sum_{k=1}^n k * 2^{k - 1} = (1/2) sum_{k=1}^n k * 2^{k} = (1/2)[2 + (n - 1)2^{n+1}] = 1 + (n - 1)2^{n}So, T_n = 1 + (n - 1)2^{n}Wait, let me verify this formula with small n.For n=1: T₁ = b₁ = 1*2^{0}=1. According to the formula, 1 + (1 - 1)2^{1}=1+0=1. Correct.For n=2: T₂ = b₁ + b₂ =1 + 2*2^{1}=1 +4=5. Formula:1 + (2 -1)2²=1 +1*4=5. Correct.For n=3: T₃=1 +4 + 3*2²=1+4+12=17. Formula:1 + (3 -1)2³=1 +2*8=17. Correct.Good, so the formula seems correct.So, T_n =1 + (n -1)2^{n}Now, we need to find the maximum m such that T_n ≥ m for all n. That is, m is the minimum value of T_n across all n, and the maximum m that satisfies T_n ≥ m always is this minimum.So, we need to find the minimum of T_n. Let's see how T_n behaves as n increases.Compute T_n for n=1: 1n=2:5n=3:17n=4:1 +3*16=1+48=49n=5:1 +4*32=1+128=129It's increasing rapidly. So, the minimum occurs at n=1, which is 1.But wait, let me check n=0? Wait, n starts at 1, since it's the sum of the first n terms. So, n=1 is the first term.Therefore, the minimum T_n is 1, so m can be at most 1.But let me check if T_n is always increasing. Let's compute T_{n+1} - T_n.T_{n+1} =1 +n*2^{n+1}T_n =1 + (n -1)2^{n}So, T_{n+1} - T_n = [1 +n*2^{n+1}] - [1 + (n -1)2^{n}] =n*2^{n+1} - (n -1)2^{n} =2^{n}(2n - (n -1))=2^{n}(n +1)Since 2^{n}(n +1) >0 for all n ≥1, T_n is strictly increasing.Therefore, the minimum value of T_n is at n=1, which is 1. So, the maximum m such that T_n ≥m always is 1.Hence, the maximum value of m is 1.**Final Answer**(1) The general formula for {a_n} is boxed{a_n = left(dfrac{1}{2}right)^{n-1}}.(2) The maximum value of m is boxed{1}.

💡Okay, so I have this geometry problem here: In a given circle, two equal parallel chords are drawn, and the distance between them is equal to the radius of the circle. I need to find the acute angle between the lines connecting the ends of the chords. Hmm, let me try to visualize this.First, I imagine a circle with center O and radius R. There are two equal parallel chords, let's call them AB and A₁B₁. Since they are equal and parallel, they must be equidistant from the center but on opposite sides. The distance between these two chords is given as R, which is the radius of the circle.I remember that the perpendicular distance from the center of the circle to a chord can be found using the formula: [ d = sqrt{R^2 - left(frac{c}{2}right)^2} ]where d is the distance from the center to the chord, and c is the length of the chord. Since both chords are equal, their distances from the center must be the same. Let's denote this distance as d. Given that the distance between the two chords is R, and since they are on opposite sides of the center, the total distance from one chord to the other would be 2d. So, 2d = R, which means d = R/2.Now, plugging this back into the formula to find the length of the chord:[ frac{R}{2} = sqrt{R^2 - left(frac{c}{2}right)^2} ]Squaring both sides:[ left(frac{R}{2}right)^2 = R^2 - left(frac{c}{2}right)^2 ][ frac{R^2}{4} = R^2 - frac{c^2}{4} ]Subtracting (frac{R^2}{4}) from both sides:[ 0 = frac{3R^2}{4} - frac{c^2}{4} ][ frac{c^2}{4} = frac{3R^2}{4} ][ c^2 = 3R^2 ][ c = Rsqrt{3} ]So, each chord has a length of ( Rsqrt{3} ).Now, I need to find the acute angle between the lines connecting the ends of the chords. Let's consider the lines AA₁ and BB₁. These lines connect the endpoints of the two chords. Since AB and A₁B₁ are parallel and equal, the figure formed by connecting these endpoints is a parallelogram. In fact, since the chords are equal and the distances are equal, this parallelogram is a rhombus.But wait, actually, since the chords are parallel and equal, and the distance between them is R, which is the radius, the figure might be a rectangle or something else. Let me think.If I connect A to A₁ and B to B₁, these lines will intersect at some point. But since the chords are parallel, the lines AA₁ and BB₁ might form an angle. I need to find that angle.Alternatively, maybe it's better to consider the triangle formed by the center O and the endpoints of the chords. Let's consider triangle OAB and OA₁B₁. Since AB and A₁B₁ are equal chords, triangles OAB and OA₁B₁ are congruent.Now, the distance between the chords is R, which is the distance between the two lines AB and A₁B₁. Since each chord is at a distance of R/2 from the center, the total distance between them is R.Let me try to find the coordinates of these points to make it easier. Let's place the circle with center at the origin (0,0). Let’s assume chord AB is horizontal for simplicity. Then, chord A₁B₁ is also horizontal and parallel to AB.The distance from the center to chord AB is R/2, so the y-coordinate of chord AB is R/2. Similarly, the y-coordinate of chord A₁B₁ is -R/2 because it's on the opposite side.Since the length of chord AB is ( Rsqrt{3} ), the x-coordinates of points A and B can be found using the circle equation:[ x^2 + y^2 = R^2 ]For chord AB at y = R/2:[ x^2 + left(frac{R}{2}right)^2 = R^2 ][ x^2 + frac{R^2}{4} = R^2 ][ x^2 = frac{3R^2}{4} ][ x = pm frac{Rsqrt{3}}{2} ]So, the coordinates of A and B are ( left(frac{Rsqrt{3}}{2}, frac{R}{2}right) ) and ( left(-frac{Rsqrt{3}}{2}, frac{R}{2}right) ).Similarly, for chord A₁B₁ at y = -R/2, the coordinates are ( left(frac{Rsqrt{3}}{2}, -frac{R}{2}right) ) and ( left(-frac{Rsqrt{3}}{2}, -frac{R}{2}right) ).Now, let's label the points:- A: ( left(frac{Rsqrt{3}}{2}, frac{R}{2}right) )- B: ( left(-frac{Rsqrt{3}}{2}, frac{R}{2}right) )- A₁: ( left(frac{Rsqrt{3}}{2}, -frac{R}{2}right) )- B₁: ( left(-frac{Rsqrt{3}}{2}, -frac{R}{2}right) )Now, I need to find the acute angle between lines AA₁ and BB₁.First, let's find the coordinates of these lines.Line AA₁ connects point A ( left(frac{Rsqrt{3}}{2}, frac{R}{2}right) ) to point A₁ ( left(frac{Rsqrt{3}}{2}, -frac{R}{2}right) ).Similarly, line BB₁ connects point B ( left(-frac{Rsqrt{3}}{2}, frac{R}{2}right) ) to point B₁ ( left(-frac{Rsqrt{3}}{2}, -frac{R}{2}right) ).Wait a minute, both AA₁ and BB₁ are vertical lines because their x-coordinates are constant. So, line AA₁ is a vertical line at x = ( frac{Rsqrt{3}}{2} ), and line BB₁ is a vertical line at x = ( -frac{Rsqrt{3}}{2} ). But vertical lines are parallel, so the angle between them is zero, which can't be right because the problem asks for an acute angle, and zero is not acute. Hmm, maybe I made a mistake.Wait, no, actually, the lines connecting the ends of the chords are not AA₁ and BB₁, but rather lines like AB₁ and A₁B. Let me clarify.The problem says "the lines connecting the ends of the chords." So, the ends are A, B, A₁, B₁. So, the lines connecting these ends would be AA₁, AB₁, BA₁, and BB₁. But the angle between which two lines? Probably the angle between AA₁ and AB₁ or something like that.Wait, maybe I need to consider the angle between the two lines that connect the ends, meaning the angle between AA₁ and BB₁, but since they are vertical, that's zero. Alternatively, maybe the angle between AA₁ and AB₁.Let me think again. The problem says "the acute angle between the lines connecting the ends of the chords." So, the ends are A, B, A₁, B₁. So, the lines connecting the ends would be AA₁, AB₁, BA₁, and BB₁. So, the angle between AA₁ and AB₁.Let me calculate the vectors for these lines.First, vector AA₁ goes from A ( left(frac{Rsqrt{3}}{2}, frac{R}{2}right) ) to A₁ ( left(frac{Rsqrt{3}}{2}, -frac{R}{2}right) ). So, the vector is (0, -R).Vector AB₁ goes from A ( left(frac{Rsqrt{3}}{2}, frac{R}{2}right) ) to B₁ ( left(-frac{Rsqrt{3}}{2}, -frac{R}{2}right) ). So, the vector is ( left(-Rsqrt{3}, -Rright) ).Now, to find the angle between vectors AA₁ (0, -R) and AB₁ (-R√3, -R). The angle θ between two vectors can be found using the dot product formula:[ costheta = frac{vec{u} cdot vec{v}}{|vec{u}| |vec{v}|} ]So, let's compute the dot product:[ vec{AA₁} cdot vec{AB₁} = (0)(-Rsqrt{3}) + (-R)(-R) = 0 + R^2 = R^2 ]The magnitude of vector AA₁ is:[ |vec{AA₁}| = sqrt{0^2 + (-R)^2} = R ]The magnitude of vector AB₁ is:[ |vec{AB₁}| = sqrt{(-Rsqrt{3})^2 + (-R)^2} = sqrt{3R^2 + R^2} = sqrt{4R^2} = 2R ]So,[ costheta = frac{R^2}{R times 2R} = frac{R^2}{2R^2} = frac{1}{2} ]Therefore,[ theta = arccosleft(frac{1}{2}right) = 60^circ ]Since 60 degrees is acute, that's our answer.Wait, but let me double-check. If I consider the angle between AA₁ and AB₁, it's 60 degrees. Alternatively, if I consider the angle between AB and A₁B₁, but since they are parallel, the angle is zero. So, the acute angle between the lines connecting the ends is 60 degrees.Alternatively, maybe I should consider the angle between the diagonals of the quadrilateral formed by the chords. The quadrilateral is a rectangle because the chords are equal and parallel, and the distances are equal. In a rectangle, the diagonals are equal and bisect each other. The angle between the diagonals can be found using the dot product as well.But in this case, the diagonals are AA₁ and BB₁, which are vertical lines, so the angle between them is zero. Hmm, that doesn't make sense. Maybe I'm confusing the lines.Wait, perhaps the lines connecting the ends are the diagonals of the quadrilateral formed by the four points. So, the diagonals would be AA₁ and BB₁, but as I saw, they are vertical lines, so the angle between them is zero. Alternatively, maybe the lines are AB₁ and A₁B, which are the other diagonals.Let me check that. The vector AB₁ is from A to B₁, which we already calculated as (-R√3, -R). The vector A₁B is from A₁ to B, which is from ( left(frac{Rsqrt{3}}{2}, -frac{R}{2}right) ) to ( left(-frac{Rsqrt{3}}{2}, frac{R}{2}right) ). So, the vector is ( left(-Rsqrt{3}, Rright) ).So, vector AB₁ is (-R√3, -R) and vector A₁B is (-R√3, R). Let's find the angle between these two vectors.Dot product:[ (-Rsqrt{3})(-Rsqrt{3}) + (-R)(R) = 3R^2 - R^2 = 2R^2 ]Magnitudes:Both vectors have magnitude:[ sqrt{(-Rsqrt{3})^2 + (-R)^2} = sqrt{3R^2 + R^2} = 2R ][ sqrt{(-Rsqrt{3})^2 + (R)^2} = sqrt{3R^2 + R^2} = 2R ]So,[ costheta = frac{2R^2}{2R times 2R} = frac{2R^2}{4R^2} = frac{1}{2} ]Thus,[ theta = arccosleft(frac{1}{2}right) = 60^circ ]Again, we get 60 degrees. So, whether we consider the angle between AA₁ and AB₁ or between AB₁ and A₁B, we get 60 degrees. Since the problem asks for the acute angle, 60 degrees is the answer.Alternatively, another approach is to consider the triangle formed by the center and the endpoints. Let's consider triangle OAB. Since OA and OB are radii, they are equal to R. The chord AB has length ( Rsqrt{3} ). So, triangle OAB has sides R, R, and ( Rsqrt{3} ). Using the Law of Cosines:[ c^2 = a^2 + b^2 - 2abcostheta ][ (Rsqrt{3})^2 = R^2 + R^2 - 2R^2costheta ][ 3R^2 = 2R^2 - 2R^2costheta ][ 3R^2 - 2R^2 = -2R^2costheta ][ R^2 = -2R^2costheta ][ costheta = -frac{1}{2} ][ theta = 120^circ ]So, the angle at the center O is 120 degrees. But we need the acute angle between the lines connecting the ends of the chords. Since the angle at the center is 120 degrees, the angle between the lines connecting the ends (which are the chords) would be supplementary to this, but wait, no. Actually, the lines connecting the ends are not the chords themselves but the lines like AA₁ and AB₁.Wait, perhaps the angle between the lines connecting the ends is the same as the angle between the radii. But in this case, the angle between OA and OA₁ is 120 degrees, so the acute angle would be 60 degrees.Yes, that makes sense. So, the acute angle between the lines connecting the ends of the chords is 60 degrees.I think I've confirmed it through multiple approaches: coordinate geometry, vectors, and triangle properties. All lead to the conclusion that the acute angle is 60 degrees.

💡Okay, so I have this problem here: Paul was packing up his old toys. He managed to squeeze some toys into a box. If Paul filled up four boxes and packed a total of 32 toys, how many toys did he squeeze into each box?Alright, let's break this down. First, Paul has a certain number of toys, which is 32 in total. He's putting these toys into boxes, and he filled up four boxes. The question is asking how many toys went into each box.So, I think this is a division problem because we're trying to find out how many toys are in each of the four boxes when we have a total of 32 toys. Division is about splitting a total into equal parts, which seems to fit here.Let me write that down: 32 toys divided by 4 boxes. So, 32 ÷ 4. Hmm, what's 32 divided by 4? I know that 4 times 8 is 32, so that means 32 divided by 4 is 8. So, each box should have 8 toys.Wait, let me double-check that to make sure I didn't make a mistake. If each box has 8 toys and there are 4 boxes, then 8 times 4 should give me back the total number of toys, which is 32. 8 times 4 is indeed 32, so that checks out.Is there another way to think about this? Maybe using multiplication instead of division? If I consider that each box has the same number of toys, let's call that number x. So, 4 boxes times x toys per box equals 32 toys. That would be 4x = 32. To find x, I divide both sides by 4, which again gives me x = 8.Okay, so both division and multiplication approaches lead me to the same answer: 8 toys per box. That makes sense because division and multiplication are inverse operations. It's like saying if I have 4 groups and I want to know how many are in each group when I have a total of 32, I divide by the number of groups.I guess another way to visualize this is by using counters or objects. If I have 32 counters and I want to distribute them equally into 4 groups, I would give one counter to each group at a time until all are distributed. After doing that, each group would end up with 8 counters.Also, thinking about it in terms of real-life scenarios, if I have 32 apples and I want to put them into 4 baskets equally, each basket would have 8 apples. It's the same principle here with toys and boxes.I wonder if there's a way to use addition to solve this problem as well. If I think about adding the number of toys in each box until I reach 32, and I know there are 4 boxes, then I'm essentially adding the same number four times. So, if I let the number of toys in each box be y, then y + y + y + y = 32. That simplifies to 4y = 32, and solving for y again gives me y = 8.So, whether I use division, multiplication, or even repeated addition, I consistently arrive at the same answer: 8 toys per box. This consistency makes me more confident that the answer is correct.I also think about the concept of fair sharing. If Paul wants to distribute his toys equally among the four boxes, he needs to ensure each box gets the same number of toys. By dividing the total number of toys by the number of boxes, he achieves this fair distribution.Another angle to consider is the relationship between the total number of toys and the number of boxes. The total is a multiple of the number of boxes, which means that the number of toys per box is a whole number. If the total wasn't a multiple, we might have to deal with fractions or decimals, but in this case, it's straightforward.I suppose it's also useful to understand why division is the appropriate operation here. Division helps us find the number of groups or the size of each group when we know the total and one of these quantities. In this problem, we know the total number of toys and the number of boxes, so division helps us find out how many toys are in each box.To summarize, the problem involves distributing 32 toys equally into 4 boxes. By using division, multiplication, or even repeated addition, we determine that each box contains 8 toys. This makes sense logically and mathematically, and the consistency across different methods reinforces the correctness of the answer.

💡Okay, so I have this problem here where I need to define a new operation, A spadesuit B, which is given by the formula A² + 2AB + 3B + 7. Then, I need to find the value of A when A spadesuit 5 equals 97. Hmm, let me try to break this down step by step.First, I need to understand what the operation A spadesuit B actually does. It's a combination of A squared, twice the product of A and B, three times B, and then adding 7. So, if I plug in B as 5, the operation becomes A² + 2A*5 + 3*5 + 7. Let me write that out:A spadesuit 5 = A² + 10A + 15 + 7.Wait, that simplifies to A² + 10A + 22. Okay, so the equation I need to solve is A² + 10A + 22 = 97. Hmm, that makes sense because the problem says A spadesuit 5 equals 97.Now, I need to solve for A. Let me subtract 97 from both sides to set the equation to zero. So, A² + 10A + 22 - 97 = 0. That simplifies to A² + 10A - 75 = 0. Alright, so now I have a quadratic equation: A² + 10A - 75 = 0.I remember that quadratic equations can be solved using the quadratic formula, which is A = [-b ± √(b² - 4ac)] / (2a). In this equation, a is 1 (the coefficient of A²), b is 10 (the coefficient of A), and c is -75 (the constant term). Let me plug these values into the quadratic formula.So, A = [-10 ± √(10² - 4*1*(-75))] / (2*1). Calculating the discriminant first: 10 squared is 100, and 4 times 1 times -75 is -300. So, the discriminant becomes 100 - (-300), which is 100 + 300, equaling 400. That's a perfect square, which is good because it means the roots will be rational.Now, plugging back into the formula: A = [-10 ± √400] / 2. The square root of 400 is 20, so this becomes A = [-10 ± 20] / 2. This gives me two possible solutions: one where I add 20 to -10 and divide by 2, and another where I subtract 20 from -10 and divide by 2.Calculating the first solution: (-10 + 20) / 2 = 10 / 2 = 5. The second solution: (-10 - 20) / 2 = (-30) / 2 = -15. So, the two possible values for A are 5 and -15.Let me double-check these solutions to make sure they work in the original equation. Starting with A = 5:A spadesuit 5 = 5² + 2*5*5 + 3*5 + 7 = 25 + 50 + 15 + 7 = 97. Yep, that adds up correctly.Now, checking A = -15:A spadesuit 5 = (-15)² + 2*(-15)*5 + 3*5 + 7 = 225 - 150 + 15 + 7. Let's compute that: 225 - 150 is 75, plus 15 is 90, plus 7 is 97. That also works out. So both solutions are valid.Therefore, the values of A that satisfy the equation A spadesuit 5 = 97 are 5 and -15.

💡First, I need to understand how the code word "GREAT DISCOVERY" maps to the digits 0-11. Each letter in the code word represents a unique digit based on its position.Next, I'll list out the letters and their corresponding digits:- G = 0- R = 1- E = 2- A = 3- T = 4- D = 5- I = 6- S = 7- C = 8- O = 9- V = 10- E = 11Now, I'll decode the word "RIDE" using this mapping:- R corresponds to 1- I corresponds to 6- D corresponds to 5- E corresponds to 2Combining these digits, "RIDE" translates to 1652.

💡Alright, so I have this problem here where I need to find the modulus of a complex number z, which is given by z equals i raised to the power of 2018 divided by (i raised to the power of 2019 minus 1). Hmm, okay, let's break this down step by step.First, I remember that i is the imaginary unit, where i squared is equal to -1. The powers of i cycle every four exponents. So, i^1 is i, i^2 is -1, i^3 is -i, and i^4 is 1, then it repeats. That seems useful because 2018 and 2019 are pretty large exponents, but maybe I can simplify them using this cyclicity.Let me start by figuring out what i^2018 is. Since the cycle repeats every 4, I can divide 2018 by 4 and find the remainder. So, 2018 divided by 4 is 504 with a remainder of 2. That means i^2018 is the same as i^2, which is -1. Okay, so the numerator simplifies to -1.Now, let's tackle the denominator: i^2019 minus 1. Again, using the same method, I'll divide 2019 by 4. 2019 divided by 4 is 504 with a remainder of 3. So, i^2019 is the same as i^3, which is -i. Therefore, the denominator becomes (-i) minus 1, which is -1 - i.So now, z is equal to (-1) divided by (-1 - i). Hmm, that looks a bit messy. I think I can simplify this by rationalizing the denominator. To do that, I'll multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of (-1 - i) is (-1 + i). Let me write that out:z = (-1) / (-1 - i) * (-1 + i)/(-1 + i)Multiplying the numerators: (-1) * (-1 + i) = 1 - iMultiplying the denominators: (-1 - i) * (-1 + i). Let me expand this using the distributive property:(-1)*(-1) + (-1)*(i) + (-i)*(-1) + (-i)*(i) = 1 - i + i - i^2Simplify each term: 1 - i + i is just 1, and -i^2 is -(-1) which is 1. So, the denominator becomes 1 + 1 = 2.So now, z is equal to (1 - i)/2, which can be written as (1/2) - (i/2).Now, to find the modulus of z, which is |z|, I remember that for a complex number a + bi, the modulus is the square root of (a squared plus b squared). So, in this case, a is 1/2 and b is -1/2.Calculating |z|: sqrt[(1/2)^2 + (-1/2)^2] = sqrt[(1/4) + (1/4)] = sqrt[2/4] = sqrt[1/2] = sqrt(2)/2.So, the modulus of z is sqrt(2)/2. Looking back at the options, that's option A.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the exponents, 2018 divided by 4 gives a remainder of 2, so i^2018 is indeed -1. For 2019, the remainder is 3, so i^2019 is -i. Then, the denominator is -1 - i. Multiplying numerator and denominator by the conjugate, (-1 + i), gives me 1 - i in the numerator and 2 in the denominator. So, z is (1 - i)/2, which gives a modulus of sqrt(2)/2. Yep, that seems correct.I think I'm confident with this answer. It's always good to verify each step, especially with complex numbers, as it's easy to mix up signs or make a mistake in the exponent cycles. But everything checks out here.**Final Answer**boxed{A}