💡Okay, so I have this problem about a rectangular parallelepiped, which is like a 3D shape with six faces, each of which is a rectangle. The vertices are labeled A, B, C, D, A1, B1, C1, D1. The edges have lengths AB = 4, AD = 2, and AA1 = 6. I need to find the angle between two lines: KM and A1N. Let me break this down step by step.First, let's visualize the shape. A rectangular parallelepiped is basically a box, right? So, A is connected to B, C, and A1. Then, each of those points is connected to their respective counterparts. For example, B is connected to C, D, and B1, and so on.Now, the points mentioned are N, M, and K. - Point N is the midpoint of edge CD. Since CD is opposite to AB, and AB is 4 units long, CD should also be 4 units. So, the midpoint N would be exactly halfway along CD, which is 2 units from both C and D.- Point M is on edge CC1, and the ratio of C1M to CM is 1:2. That means if I consider the entire length of CC1, which is equal to AA1, so 6 units, then C1M is one part and CM is two parts. So, C1M = 2 units and CM = 4 units? Wait, no, wait. If the ratio is C1M:CM = 1:2, then the total parts are 1 + 2 = 3 parts. So, each part is 6/3 = 2 units. Therefore, C1M is 2 units, and CM is 4 units. So, M is 4 units away from C and 2 units away from C1.- Point K is the intersection of the diagonals of face AA1D1D. Face AA1D1D is a rectangle, right? So, the diagonals of this face would be from A to D1 and from A1 to D. The intersection point K would be the midpoint of both diagonals. Since AA1 is 6 units and AD is 2 units, the diagonals of this face can be calculated using the Pythagorean theorem. The length of each diagonal is sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40) = 2*sqrt(10). So, the midpoint K would be at half of that distance from each corner.Wait, maybe I don't need the length of the diagonals. Maybe I just need the coordinates of point K. Let me think about assigning coordinates to all the points to make this easier.Let's assign a coordinate system where point A is at the origin (0, 0, 0). Then, since AB = 4, point B would be at (4, 0, 0). AD = 2, so point D would be at (0, 2, 0). AA1 = 6, so point A1 would be at (0, 0, 6).Now, let's find the coordinates of all the other points:- Point C is opposite to A, so it would be at (4, 2, 0).- Point C1 is directly above C, so it's at (4, 2, 6).- Point D1 is directly above D, so it's at (0, 2, 6).- Point B1 is directly above B, so it's at (4, 0, 6).Now, let's find the coordinates of points N, M, and K.- Point N is the midpoint of CD. Since C is at (4, 2, 0) and D is at (0, 2, 0), the midpoint N would be at ((4+0)/2, (2+2)/2, (0+0)/2) = (2, 2, 0).Wait, hold on. Is that correct? CD goes from C (4, 2, 0) to D (0, 2, 0). So, yes, the midpoint is indeed at (2, 2, 0).- Point M is on edge CC1, with the ratio C1M:CM = 1:2. So, CC1 goes from C (4, 2, 0) to C1 (4, 2, 6). The total length is 6 units. Since the ratio is 1:2, M divides CC1 into two parts: from C to M is 4 units, and from M to C1 is 2 units. So, starting from C (4, 2, 0), moving 4 units towards C1 (which is along the z-axis), we reach M at (4, 2, 4).Wait, hold on. If the ratio is C1M:CM = 1:2, that means C1M is one part and CM is two parts. So, the total parts are 3, each part is 2 units (since CC1 is 6 units). Therefore, C1M is 2 units, so M is 2 units below C1. So, starting from C1 (4, 2, 6), moving down 2 units along CC1, we reach M at (4, 2, 4). So, yes, M is at (4, 2, 4).- Point K is the intersection of the diagonals of face AA1D1D. Face AA1D1D has vertices at A (0, 0, 0), A1 (0, 0, 6), D1 (0, 2, 6), and D (0, 2, 0). The diagonals of this face are AD1 and A1D. Let's find their equations.Diagonal AD1 goes from A (0, 0, 0) to D1 (0, 2, 6). The parametric equations for AD1 can be written as:x = 0y = 2tz = 6twhere t ranges from 0 to 1.Diagonal A1D goes from A1 (0, 0, 6) to D (0, 2, 0). The parametric equations for A1D can be written as:x = 0y = 2sz = 6 - 6swhere s ranges from 0 to 1.To find the intersection point K, we need to solve for t and s such that the coordinates are equal.From the y-coordinate: 2t = 2s => t = s.From the z-coordinate: 6t = 6 - 6s. Since t = s, substitute:6t = 6 - 6t6t + 6t = 612t = 6t = 0.5So, t = 0.5, which means s = 0.5.Therefore, point K is at:x = 0y = 2*(0.5) = 1z = 6*(0.5) = 3So, K is at (0, 1, 3).Wait, hold on. Is that correct? Let me double-check.For diagonal AD1: at t = 0.5, we have (0, 1, 3).For diagonal A1D: at s = 0.5, we have (0, 1, 3). Yes, that's correct. So, K is indeed at (0, 1, 3).Now, I need to find the angle between lines KM and A1N.First, let's find the coordinates of points involved in these lines.- Line KM goes from K (0, 1, 3) to M (4, 2, 4).- Line A1N goes from A1 (0, 0, 6) to N (2, 2, 0).To find the angle between two lines in 3D space, we can use the dot product formula. The angle θ between two vectors u and v is given by:cosθ = (u · v) / (|u| |v|)So, first, I need to find the vectors representing KM and A1N.Vector KM is M - K = (4 - 0, 2 - 1, 4 - 3) = (4, 1, 1)Vector A1N is N - A1 = (2 - 0, 2 - 0, 0 - 6) = (2, 2, -6)Wait, hold on. Let me double-check these vectors.For KM: from K (0,1,3) to M (4,2,4). So, the vector is (4-0, 2-1, 4-3) = (4,1,1). Correct.For A1N: from A1 (0,0,6) to N (2,2,0). So, the vector is (2-0, 2-0, 0-6) = (2,2,-6). Correct.Now, compute the dot product of vectors KM and A1N.Dot product = (4)(2) + (1)(2) + (1)(-6) = 8 + 2 - 6 = 4Next, find the magnitudes of vectors KM and A1N.|KM| = sqrt(4^2 + 1^2 + 1^2) = sqrt(16 + 1 + 1) = sqrt(18) = 3*sqrt(2)|A1N| = sqrt(2^2 + 2^2 + (-6)^2) = sqrt(4 + 4 + 36) = sqrt(44) = 2*sqrt(11)Now, compute cosθ:cosθ = 4 / (3*sqrt(2) * 2*sqrt(11)) = 4 / (6*sqrt(22)) = (2) / (3*sqrt(22))Simplify this:Multiply numerator and denominator by sqrt(22):cosθ = (2*sqrt(22)) / (3*22) = (2*sqrt(22)) / 66 = sqrt(22)/33So, cosθ = sqrt(22)/33Therefore, the angle θ is arccos(sqrt(22)/33)Wait, let me double-check the calculations.Dot product: 4*2 + 1*2 + 1*(-6) = 8 + 2 - 6 = 4. Correct.|KM|: sqrt(16 + 1 + 1) = sqrt(18). Correct.|A1N|: sqrt(4 + 4 + 36) = sqrt(44). Correct.So, cosθ = 4 / (sqrt(18)*sqrt(44)) = 4 / (sqrt(792)) = 4 / (6*sqrt(22)) = 2 / (3*sqrt(22)) = sqrt(22)/33 after rationalizing.Yes, that seems correct.So, the angle between KM and A1N is arccos(sqrt(22)/33)I think that's the answer. Let me just recap to make sure I didn't make any mistakes.Assigned coordinates correctly? Yes, A at (0,0,0), B at (4,0,0), D at (0,2,0), A1 at (0,0,6). Then C at (4,2,0), C1 at (4,2,6), D1 at (0,2,6), B1 at (4,0,6). Midpoint N at (2,2,0). Point M at (4,2,4). Point K at (0,1,3). Vectors KM (4,1,1) and A1N (2,2,-6). Dot product 4, magnitudes sqrt(18) and sqrt(44). Cosθ = 4/(sqrt(18)*sqrt(44)) = sqrt(22)/33. So, θ = arccos(sqrt(22)/33). Seems correct.**Final Answer**The angle between lines ( KM ) and ( A_1N ) is (boxed{arccos left( frac{sqrt{22}}{33} right)}).

💡Alright, so I have this problem about an ellipse with foci at (0, 3) and (4, 0). It says that the ellipse has two x-intercepts, and one of them is the origin, which is (0, 0). I need to find the other x-intercept and present it as an ordered pair.Hmm, okay. I remember that an ellipse is defined as the set of all points where the sum of the distances from the two foci is constant. So, for any point on the ellipse, if I add the distance from that point to each focus, it should equal the same constant value.Since one of the x-intercepts is the origin, (0, 0), I can use this point to find that constant sum. Let me calculate the distance from (0, 0) to each focus.First, the distance from (0, 0) to (0, 3). That's straightforward because they share the same x-coordinate. The distance is just the difference in y-coordinates, which is |3 - 0| = 3.Next, the distance from (0, 0) to (4, 0). Again, they share the same y-coordinate, so the distance is the difference in x-coordinates, which is |4 - 0| = 4.So, the sum of these distances is 3 + 4 = 7. That means for any other point on the ellipse, the sum of its distances to (0, 3) and (4, 0) should also be 7.Now, I need to find the other x-intercept. An x-intercept is a point where y = 0, so the other intercept will be at some point (x, 0). Let's denote this point as (x, 0).Using the definition of an ellipse, the sum of the distances from (x, 0) to each focus should be 7. Let's write that out mathematically.The distance from (x, 0) to (0, 3) can be found using the distance formula: √[(x - 0)² + (0 - 3)²] = √(x² + 9).Similarly, the distance from (x, 0) to (4, 0) is √[(x - 4)² + (0 - 0)²] = |x - 4|. Since distance can't be negative, this simplifies to |x - 4|.So, according to the ellipse definition, we have:√(x² + 9) + |x - 4| = 7.Now, I need to solve this equation for x. Let's think about the absolute value term |x - 4|. The absolute value function can be tricky because it depends on whether x is greater than or less than 4.Since we already know that (0, 0) is an x-intercept, and we're looking for another one, it's reasonable to assume that the other intercept is on the other side of the foci. The foci are at (0, 3) and (4, 0), so the ellipse is likely stretched out along the line connecting these two points. Therefore, the other x-intercept is probably to the right of (4, 0), meaning x > 4. If that's the case, then |x - 4| simplifies to x - 4 because x - 4 is positive.So, substituting that into our equation, we get:√(x² + 9) + (x - 4) = 7.Let me rewrite that:√(x² + 9) = 7 - (x - 4).Simplify the right side:7 - x + 4 = 11 - x.So now, the equation is:√(x² + 9) = 11 - x.To solve for x, I can square both sides to eliminate the square root. Let's do that:(√(x² + 9))² = (11 - x)².Simplifying both sides:x² + 9 = 121 - 22x + x².Wait, that seems interesting. Let me write it out step by step.Left side: x² + 9.Right side: (11 - x)² = 11² - 2*11*x + x² = 121 - 22x + x².So, putting it together:x² + 9 = 121 - 22x + x².Hmm, I notice that x² appears on both sides. Maybe I can subtract x² from both sides to simplify.Subtracting x² from both sides:9 = 121 - 22x.Now, let's solve for x.First, subtract 121 from both sides:9 - 121 = -22x.Calculating 9 - 121: that's -112.So, -112 = -22x.Now, divide both sides by -22 to solve for x:x = (-112)/(-22) = 112/22.Simplify that fraction. Both numerator and denominator are divisible by 2:112 ÷ 2 = 56.22 ÷ 2 = 11.So, x = 56/11.Hmm, 56 divided by 11 is approximately 5.09, which is greater than 4, so that makes sense with our earlier assumption that x > 4.Therefore, the other x-intercept is at (56/11, 0).But wait, let me double-check my steps to make sure I didn't make a mistake.Starting from the equation:√(x² + 9) + |x - 4| = 7.Assuming x > 4, so |x - 4| = x - 4.Then:√(x² + 9) + x - 4 = 7.Subtract x - 4 from both sides:√(x² + 9) = 11 - x.Square both sides:x² + 9 = (11 - x)².Expand the right side:x² + 9 = 121 - 22x + x².Subtract x² from both sides:9 = 121 - 22x.Subtract 121:-112 = -22x.Divide by -22:x = 56/11.Yes, that seems correct.Just to be thorough, let me plug x = 56/11 back into the original equation to verify.First, compute |x - 4|:x = 56/11 ≈ 5.09.So, |5.09 - 4| = 1.09, which is 56/11 - 44/11 = 12/11.Now, compute √(x² + 9):x² = (56/11)² = 3136/121.So, x² + 9 = 3136/121 + 1089/121 = (3136 + 1089)/121 = 4225/121.√(4225/121) = √4225 / √121 = 65/11 ≈ 5.91.Now, add |x - 4|:65/11 + 12/11 = 77/11 = 7.Which matches the left side of the equation. So, yes, it checks out.Therefore, the other x-intercept is indeed at (56/11, 0).I think that's solid. I don't see any mistakes in my reasoning or calculations. So, I'm confident that (56/11, 0) is the correct answer.**Final Answer**The other x-intercept is boxed{left(dfrac{56}{11}, 0right)}.

💡First, I need to determine how much profit Sarah makes per burger. She earns 15 for each burger sold and spends 4 on ingredients for each burger. Next, I'll calculate the profit per burger by subtracting the cost from the earnings:[15 - 4 = 11 text{ dollars per burger}]Sarah's initial investment is 7000. To find out how many burgers she needs to sell to recover this investment, I'll set up the inequality:[11b geq 7000]Solving for ( b ), I'll divide both sides by 11:[b geq frac{7000}{11} approx 636.36]Since Sarah can't sell a fraction of a burger, I'll round up to the nearest whole number. Therefore, Sarah must sell at least 637 burgers to recover her initial investment.

💡First, I need to evaluate ( f(-2) ). Since (-2 < 1), I use the first part of the piecewise function:[f(-2) = 1 + log_{2}(2 - (-2)) = 1 + log_{2}(4) = 1 + 2 = 3]Next, I need to evaluate ( f(3) ). Since (3 > 1), I use the second part of the piecewise function:[f(3) = 2^{3 - 1} = 2^{2} = 4]Therefore, ( f(f(-2)) = f(3) = 4 ).

💡Okay, so I have this problem where there are seven line segments with integer lengths in centimeters: a₁, a₂, a₃, a₄, a₅, a₆, a₇. They are ordered such that a₁ < a₂ < a₃ < a₄ < a₅ < a₆ < a₇. The key condition here is that any three of these segments cannot form a triangle. I know that for three lengths to form a triangle, the sum of any two sides must be greater than the third side. So, in this case, for any three segments, the sum of the two shorter ones must be less than or equal to the longest one.Given that a₁ is 1 cm and a₇ is 21 cm, I need to find the value of a₆. First, let me recall that if no three segments can form a triangle, then for any three segments a_i, a_j, a_k where i < j < k, we must have a_i + a_j ≤ a_k. This is the opposite of the triangle inequality condition. So, each segment must be at least the sum of all the previous segments. Wait, is that correct? Let me think.Actually, for the condition that no three can form a triangle, it's sufficient that for each segment a_k, it is at least the sum of the two previous segments. Because if a_k is at least the sum of a_{k-1} and a_{k-2}, then certainly a_k is greater than the sum of any two segments before it, right? So, this seems similar to the Fibonacci sequence, where each term is the sum of the two preceding ones.Given that, let's try to construct such a sequence starting from a₁ = 1. So, a₁ = 1. Then, a₂ must be at least a₁ + 1, but since they have to be integers and in increasing order, the smallest possible a₂ is 2. Then, a₃ must be at least a₁ + a₂, which is 1 + 2 = 3. So, a₃ = 3. Next, a₄ must be at least a₂ + a₃ = 2 + 3 = 5. So, a₄ = 5. Then, a₅ must be at least a₃ + a₄ = 3 + 5 = 8. So, a₅ = 8. Following that, a₆ must be at least a₄ + a₅ = 5 + 8 = 13. So, a₆ = 13. Finally, a₇ must be at least a₅ + a₆ = 8 + 13 = 21. And indeed, a₇ is given as 21, which fits perfectly.So, the sequence is 1, 2, 3, 5, 8, 13, 21. Each term is the sum of the two previous terms, ensuring that no three terms can form a triangle because each term is too large compared to the previous ones. Therefore, a₆ is 13.Wait, let me double-check if this sequence actually satisfies the condition that any three segments cannot form a triangle. Let's pick any three segments and see if the sum of the two smaller ones is less than or equal to the largest one.Take a₁=1, a₂=2, a₃=3: 1 + 2 = 3, which is equal to a₃, so they can't form a triangle.Take a₂=2, a₃=3, a₄=5: 2 + 3 = 5, equal to a₄, so no triangle.a₃=3, a₄=5, a₅=8: 3 + 5 = 8, equal to a₅, so no triangle.a₄=5, a₅=8, a₆=13: 5 + 8 = 13, equal to a₆, so no triangle.a₅=8, a₆=13, a₇=21: 8 + 13 = 21, equal to a₇, so no triangle.What about non-consecutive terms? For example, a₁=1, a₂=2, a₄=5: 1 + 2 = 3 < 5, so no triangle.a₁=1, a₃=3, a₅=8: 1 + 3 = 4 < 8, no triangle.a₂=2, a₄=5, a₆=13: 2 + 5 = 7 < 13, no triangle.a₃=3, a₅=8, a₇=21: 3 + 8 = 11 < 21, no triangle.It seems like all combinations satisfy the condition that the sum of the two smaller segments is less than or equal to the largest one, thus preventing the formation of a triangle. So, the sequence is correctly constructed.Therefore, the value of a₆ is 13.

💡Okay, so I have this problem about an arithmetic sequence. Let me try to figure it out step by step. First, part (Ⅰ) says that Sₙ is the sum of the first n terms of the arithmetic sequence {aₙ}, and we're given that a₅ = 11 and S₅ = 35. I need to find the general formula for {aₙ}. Alright, let's recall some formulas about arithmetic sequences. The nth term of an arithmetic sequence is given by aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference. The sum of the first n terms, Sₙ, is given by Sₙ = n/2 [2a₁ + (n-1)d] or Sₙ = n(a₁ + aₙ)/2.Given that a₅ = 11, so using the formula for the nth term, that would be a₁ + 4d = 11. Let me write that down as equation (1): a₁ + 4d = 11.Also, we're given that S₅ = 35. Using the sum formula, S₅ = 5/2 [2a₁ + 4d] = 35. Let me compute that: 5/2 [2a₁ + 4d] = 35. Multiplying both sides by 2/5 gives [2a₁ + 4d] = 14. Simplifying that, divide both sides by 2: a₁ + 2d = 7. Let me write that as equation (2): a₁ + 2d = 7.Now, I have two equations:1. a₁ + 4d = 112. a₁ + 2d = 7I can subtract equation (2) from equation (1) to eliminate a₁. So, (a₁ + 4d) - (a₁ + 2d) = 11 - 7. That simplifies to 2d = 4, so d = 2.Now that I have d, I can plug it back into equation (2) to find a₁. So, a₁ + 2*2 = 7, which is a₁ + 4 = 7, so a₁ = 3.Therefore, the first term is 3 and the common difference is 2. So the general formula for aₙ is a₁ + (n-1)d, which is 3 + (n-1)*2. Let me simplify that: 3 + 2n - 2 = 2n + 1. So, aₙ = 2n + 1.Okay, that seems straightforward. Let me check if that makes sense. For n=5, a₅ should be 11. Plugging in, 2*5 +1 = 11. Yes, that works. And the sum S₅ should be 35. Let me compute S₅ using the sum formula: S₅ = 5/2 [2*3 + 4*2] = 5/2 [6 + 8] = 5/2 *14 = 5*7=35. Perfect, that matches.So part (Ⅰ) is done. Now, moving on to part (Ⅱ). It says let bₙ = a^{aₙ}, where a is a real constant and a > 0. We need to find the sum of the first n terms of {bₙ}, denoted as Tₙ.Hmm, so bₙ is defined as a raised to the power of aₙ. Since aₙ is 2n +1, then bₙ = a^{2n +1}. Let me write that down: bₙ = a^{2n +1}. So, each term is a^{2n +1}. Let me see if this forms a geometric sequence.To check if {bₙ} is a geometric sequence, I can compute the ratio of consecutive terms: b_{n+1}/b_n.So, b_{n+1} = a^{2(n+1) +1} = a^{2n + 3}, and bₙ = a^{2n +1}. Therefore, the ratio is a^{2n +3}/a^{2n +1} = a^{2}. Since the ratio is constant (a²) for all n, this is indeed a geometric sequence with common ratio q = a².Now, the first term of the geometric sequence {bₙ} is b₁. Let's compute that: b₁ = a^{2*1 +1} = a³.So, the first term is a³ and the common ratio is a². Now, we need to find the sum of the first n terms, Tₙ.The formula for the sum of the first n terms of a geometric sequence is Tₙ = b₁ (1 - qⁿ)/(1 - q) when q ≠ 1. If q = 1, then Tₙ = b₁ * n.In this case, q = a². So, if a² ≠ 1, which is equivalent to a ≠ 1 (since a > 0), then we can use the formula. If a = 1, then q = 1, and the sum is simply n times the first term.Let me write that down:Case 1: a = 1. Then, bₙ = 1^{2n +1} = 1 for all n. So, Tₙ = 1 + 1 + ... +1 (n times) = n.Case 2: a ≠ 1. Then, Tₙ = b₁ (1 - qⁿ)/(1 - q) = a³ (1 - (a²)ⁿ)/(1 - a²) = a³ (1 - a^{2n})/(1 - a²).So, putting it all together, Tₙ is equal to n when a =1, and a³ (1 - a^{2n})/(1 - a²) when a ≠1.Let me check if this makes sense. If a =1, then each term is 1, so the sum is n, which is correct. If a ≠1, then it's a geometric series with ratio a², so the formula applies. Let me test with a specific value. Suppose a =2, n=2. Then, b₁ =2³=8, b₂=2^{5}=32. So, T₂=8+32=40. Using the formula: a³(1 - a^{4})/(1 - a²)=8(1 -16)/(1 -4)=8*(-15)/(-3)=8*5=40. Correct.Another test: a=3, n=3. b₁=27, b₂=81, b₃=243. Sum is 27+81+243=351. Formula: 27(1 - 3^6)/(1 -9)=27(1 -729)/(-8)=27*(-728)/(-8)=27*91=2457? Wait, that doesn't match. Wait, 27*(1 -729)=27*(-728)= -19656. Divided by -8: -19656/-8=2457. But the actual sum is 27+81+243=351. Hmm, that's not matching. Did I make a mistake?Wait, hold on. Let me compute the formula again. For a=3, n=3:Tₙ = a³(1 - a^{2n})/(1 - a²) =27(1 - 3^6)/(1 -9)=27(1 -729)/(-8)=27*(-728)/(-8)=27*91=2457. But the actual sum is 27+81+243=351. That's a big discrepancy. So, I must have made a mistake.Wait, hold on. Let me check the formula again. The formula is Tₙ = b₁(1 - qⁿ)/(1 - q). Here, b₁ =a³, q=a², so Tₙ =a³(1 - (a²)^n)/(1 -a²)=a³(1 -a^{2n})/(1 -a²). But when I plug in a=3, n=3, I get 27*(1 -729)/(1 -9)=27*(-728)/(-8)=27*91=2457. But the actual sum is 27 + 81 + 243 = 351. So, 2457 vs 351. That's way off. So, I must have made a mistake in the formula.Wait, hold on. Let me compute Tₙ again. For a=3, n=3:b₁=3^{2*1 +1}=3^3=27b₂=3^{2*2 +1}=3^5=243b₃=3^{2*3 +1}=3^7=2187Wait, hold on, that's not correct. Wait, 2n +1 for n=1 is 3, n=2 is 5, n=3 is 7. So, b₁=3^3=27, b₂=3^5=243, b₃=3^7=2187. So, the sum is 27 + 243 + 2187= 2457. So, actually, my initial calculation was correct, but my manual addition was wrong. I thought b₂ was 81, but no, it's 243. So, 27 +243=270, plus 2187 is indeed 2457. So, the formula is correct.Wait, but when I thought a=2, n=2: b₁=8, b₂=32, sum=40. Formula: 8*(1 -16)/(1 -4)=8*(-15)/(-3)=40. Correct.Another test: a=1/2, n=2. Then, b₁=(1/2)^3=1/8, b₂=(1/2)^5=1/32. Sum=1/8 +1/32=5/32. Formula: (1/8)(1 - (1/2)^4)/(1 - (1/2)^2)= (1/8)(1 -1/16)/(1 -1/4)= (1/8)(15/16)/(3/4)= (1/8)*(15/16)*(4/3)= (1/8)*(15/12)= (1/8)*(5/4)=5/32. Correct.So, the formula works. So, my initial confusion was because I miscalculated the terms when a=3, n=3. So, the formula is correct.Therefore, summarizing:For part (Ⅰ), the general term is aₙ=2n +1.For part (Ⅱ), Tₙ is equal to n when a=1, and a³(1 -a^{2n})/(1 -a²) when a≠1.So, the final answer is Tₙ= { n, if a=1; [a³(1 -a^{2n})]/(1 -a²), if a≠1 }.**Final Answer**Ⅰ. The general formula for the arithmetic sequence is boxed{a_n = 2n + 1}.Ⅱ. The sum of the first ( n ) terms of the sequence ( {b_n} ) is boxed{T_n = begin{cases} n & text{if } a = 1, dfrac{a^3(1 - a^{2n})}{1 - a^2} & text{if } a neq 1. end{cases}}