💡Okay, so I have this problem about an arithmetic sequence. Let me try to figure it out step by step. The problem says:In the arithmetic sequence {a_n}, a_2 = 4, and a_4 + a_7 = 15.There are two parts:1. Find the general formula for the sequence {a_n}.2. Let b_n = 2^{a_n - 2} + n, calculate the sum of the first 10 terms of the sequence {b_n}.Alright, starting with part (1). I remember that in an arithmetic sequence, each term is obtained by adding a common difference, d, to the previous term. So, the general formula for an arithmetic sequence is a_n = a_1 + (n - 1)d, where a_1 is the first term.Given that a_2 = 4, let me write that out using the formula:a_2 = a_1 + (2 - 1)d = a_1 + d = 4.So, equation one is: a_1 + d = 4.Next, it says a_4 + a_7 = 15. Let me write expressions for a_4 and a_7 using the general formula.a_4 = a_1 + (4 - 1)d = a_1 + 3d.a_7 = a_1 + (7 - 1)d = a_1 + 6d.So, adding them together: a_4 + a_7 = (a_1 + 3d) + (a_1 + 6d) = 2a_1 + 9d = 15.So, equation two is: 2a_1 + 9d = 15.Now, I have two equations:1. a_1 + d = 42. 2a_1 + 9d = 15I need to solve this system of equations to find a_1 and d.Let me solve equation one for a_1: a_1 = 4 - d.Then, substitute this into equation two:2*(4 - d) + 9d = 15Let me compute that:2*4 = 8, 2*(-d) = -2d, so 8 - 2d + 9d = 15Combine like terms: 8 + 7d = 15Subtract 8 from both sides: 7d = 7Divide both sides by 7: d = 1.Now, substitute d = 1 back into equation one: a_1 + 1 = 4 => a_1 = 3.So, the first term is 3, and the common difference is 1. Therefore, the general formula is:a_n = a_1 + (n - 1)d = 3 + (n - 1)*1 = 3 + n - 1 = n + 2.So, a_n = n + 2. That seems straightforward.Let me double-check with the given terms:a_2 = 2 + 2 = 4, which matches.a_4 = 4 + 2 = 6, a_7 = 7 + 2 = 9, so a_4 + a_7 = 6 + 9 = 15, which also matches. Good.So, part (1) is done, and the general formula is a_n = n + 2.Moving on to part (2). We have b_n = 2^{a_n - 2} + n. We need to find the sum of the first 10 terms of {b_n}.First, let's express b_n in terms of n. Since we know a_n = n + 2, let's substitute that into the expression for b_n.b_n = 2^{(n + 2) - 2} + n = 2^{n} + n.So, b_n = 2^n + n.Therefore, the sum of the first 10 terms is:Sum = b_1 + b_2 + b_3 + ... + b_{10} = (2^1 + 1) + (2^2 + 2) + (2^3 + 3) + ... + (2^{10} + 10).I can separate this into two separate sums:Sum = (2^1 + 2^2 + 2^3 + ... + 2^{10}) + (1 + 2 + 3 + ... + 10).So, it's the sum of a geometric series plus the sum of the first 10 natural numbers.Let me compute each part separately.First, the geometric series: 2^1 + 2^2 + ... + 2^{10}.I remember that the sum of a geometric series from k=0 to n is S = a*(r^{n+1} - 1)/(r - 1), where a is the first term and r is the common ratio.But here, our series starts at k=1, so it's 2^1 + 2^2 + ... + 2^{10}.Alternatively, it's the same as the sum from k=0 to 10 of 2^k minus 2^0.So, sum from k=0 to 10 of 2^k = (2^{11} - 1)/(2 - 1) = 2^{11} - 1 = 2048 - 1 = 2047.Therefore, sum from k=1 to 10 of 2^k = 2047 - 1 = 2046.Wait, hold on. Let me verify that.Wait, the formula is S = a*(r^{n} - 1)/(r - 1) when starting from k=0 to n-1.Wait, maybe I confused the indices.Let me think again.The sum S = 2^1 + 2^2 + ... + 2^{10}.This is a geometric series with first term a = 2, common ratio r = 2, and number of terms n = 10.The formula for the sum is S = a*(r^n - 1)/(r - 1).So, plugging in, S = 2*(2^{10} - 1)/(2 - 1) = 2*(1024 - 1)/1 = 2*1023 = 2046.Yes, that's correct.Alternatively, if I think of it as sum from k=0 to 10 of 2^k is 2^{11} - 1 = 2047, so subtracting the k=0 term (which is 1) gives 2046. So, same result.Okay, so the geometric series sum is 2046.Now, the sum of the first 10 natural numbers: 1 + 2 + 3 + ... + 10.I remember that the formula for the sum of the first n natural numbers is S = n(n + 1)/2.So, for n=10, S = 10*11/2 = 55.Therefore, the total sum is 2046 + 55 = 2101.Wait, let me compute that: 2046 + 55. 2046 + 50 is 2096, plus 5 is 2101. Yep.So, the sum of the first 10 terms of {b_n} is 2101.Let me just verify the calculations again to make sure I didn't make any mistakes.First, for the geometric series:Number of terms: 10, starting from 2^1 to 2^{10}.Sum formula: S = a*(r^n - 1)/(r - 1) = 2*(2^{10} - 1)/(2 - 1) = 2*(1024 - 1)/1 = 2*1023 = 2046. Correct.Sum of natural numbers: 1 to 10. Formula: 10*11/2 = 55. Correct.Total sum: 2046 + 55 = 2101. Correct.So, I think that's the right answer.Therefore, summarizing:1. The general formula for {a_n} is a_n = n + 2.2. The sum of the first 10 terms of {b_n} is 2101.**Final Answer**(1) The general formula is boxed{a_n = n + 2}.(2) The sum of the first 10 terms is boxed{2101}.

💡Okay, so I have this trigonometry problem here: Given that:[ cos (alpha + beta) = cos alpha + cos beta ]I need to find the maximum value of ( cos alpha ).Hmm, alright. Let me start by recalling some trigonometric identities. I know that the cosine of a sum can be expanded using the angle addition formula:[ cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta ]So, substituting this into the given equation:[ cos alpha cos beta - sin alpha sin beta = cos alpha + cos beta ]Let me rearrange this equation to group similar terms together. Subtract ( cos alpha ) and ( cos beta ) from both sides:[ cos alpha cos beta - sin alpha sin beta - cos alpha - cos beta = 0 ]Hmm, maybe I can factor some terms here. Let's see:[ (cos alpha cos beta - cos alpha) - (sin alpha sin beta + cos beta) = 0 ][ cos alpha (cos beta - 1) - sin alpha sin beta - cos beta = 0 ]This doesn't seem immediately helpful. Maybe I should consider another approach. Let me think about expressing everything in terms of ( cos alpha ) and ( cos beta ), or perhaps using some substitution.Wait, another idea: since ( cos(alpha + beta) = cos alpha + cos beta ), maybe I can think geometrically. If I consider points on the unit circle, ( cos alpha ) and ( cos beta ) represent the x-coordinates of points at angles ( alpha ) and ( beta ) respectively.But I'm not sure if that's the right path. Let me try another algebraic manipulation.Starting again from:[ cos(alpha + beta) = cos alpha + cos beta ]Using the angle addition formula:[ cos alpha cos beta - sin alpha sin beta = cos alpha + cos beta ]Let me move all terms to one side:[ cos alpha cos beta - sin alpha sin beta - cos alpha - cos beta = 0 ]Factor ( cos alpha ) and ( cos beta ):[ cos alpha (cos beta - 1) - cos beta - sin alpha sin beta = 0 ]Hmm, still not straightforward. Maybe I can isolate terms involving ( cos beta ) and ( sin beta ). Let's try:[ (cos alpha - 1) cos beta - sin alpha sin beta = cos alpha ]Wait, that looks like the equation of a line in terms of ( cos beta ) and ( sin beta ). If I think of ( cos beta ) and ( sin beta ) as coordinates on the unit circle, this equation represents a line. So, the point ( (cos beta, sin beta) ) lies on both the unit circle and this line.Therefore, the distance from the center of the circle (which is the origin) to this line must be less than or equal to the radius of the circle, which is 1.The distance ( d ) from the origin to the line ( ax + by + c = 0 ) is given by:[ d = frac{|c|}{sqrt{a^2 + b^2}} ]In our case, the line is:[ (cos alpha - 1) cos beta - sin alpha sin beta - cos alpha = 0 ]So, comparing to ( ax + by + c = 0 ), we have:- ( a = cos alpha - 1 )- ( b = -sin alpha )- ( c = -cos alpha )Thus, the distance is:[ d = frac{|cos alpha|}{sqrt{(cos alpha - 1)^2 + (sin alpha)^2}} ]Since the point lies on the unit circle, this distance must be less than or equal to 1:[ frac{|cos alpha|}{sqrt{(cos alpha - 1)^2 + (sin alpha)^2}} leq 1 ]Let me simplify the denominator:[ (cos alpha - 1)^2 + (sin alpha)^2 = cos^2 alpha - 2 cos alpha + 1 + sin^2 alpha ][ = (cos^2 alpha + sin^2 alpha) - 2 cos alpha + 1 ][ = 1 - 2 cos alpha + 1 ][ = 2 - 2 cos alpha ]So, the inequality becomes:[ frac{|cos alpha|}{sqrt{2 - 2 cos alpha}} leq 1 ]Let me square both sides to eliminate the square root:[ frac{cos^2 alpha}{2 - 2 cos alpha} leq 1 ]Multiply both sides by ( 2 - 2 cos alpha ) (assuming ( 2 - 2 cos alpha > 0 ), which it is since ( cos alpha leq 1 )):[ cos^2 alpha leq 2 - 2 cos alpha ]Bring all terms to one side:[ cos^2 alpha + 2 cos alpha - 2 leq 0 ]This is a quadratic inequality in terms of ( cos alpha ). Let me set ( x = cos alpha ):[ x^2 + 2x - 2 leq 0 ]Solving the quadratic equation ( x^2 + 2x - 2 = 0 ) using the quadratic formula:[ x = frac{-2 pm sqrt{(2)^2 - 4(1)(-2)}}{2(1)} ][ x = frac{-2 pm sqrt{4 + 8}}{2} ][ x = frac{-2 pm sqrt{12}}{2} ][ x = frac{-2 pm 2sqrt{3}}{2} ][ x = -1 pm sqrt{3} ]So, the roots are ( x = -1 + sqrt{3} ) and ( x = -1 - sqrt{3} ). Since ( cos alpha ) must be between -1 and 1, we discard ( x = -1 - sqrt{3} ) because it is less than -1.Thus, the inequality ( x^2 + 2x - 2 leq 0 ) holds for:[ -1 - sqrt{3} leq x leq -1 + sqrt{3} ]But since ( x = cos alpha ) must be between -1 and 1, the valid interval is:[ -1 leq cos alpha leq -1 + sqrt{3} ]Calculating ( -1 + sqrt{3} ):[ sqrt{3} approx 1.732 ][ -1 + 1.732 approx 0.732 ]So, the maximum value of ( cos alpha ) is approximately 0.732, which is ( sqrt{3} - 1 ).Wait, let me verify this. If ( cos alpha = sqrt{3} - 1 ), then ( alpha ) would be an angle whose cosine is approximately 0.732. That seems reasonable.Let me check if this value satisfies the original equation. Suppose ( cos alpha = sqrt{3} - 1 ). Then, I need to find ( beta ) such that:[ cos(alpha + beta) = cos alpha + cos beta ]But I'm not sure if I need to find ( beta ) explicitly. Since the problem only asks for the maximum value of ( cos alpha ), and through the inequality, we've determined that the maximum possible value is ( sqrt{3} - 1 ), I think that's the answer.So, after going through the steps, the maximum value of ( cos alpha ) is ( sqrt{3} - 1 ).**Final Answer**The maximum value of ( cos alpha ) is (boxed{sqrt{3} - 1}).

💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking and figure out how to solve it step by step. First, the problem says that {a_n} is an arithmetic sequence. That means each term increases by a constant difference, right? So, if I denote the first term as a_1 and the common difference as d, then the nth term can be written as a_n = a_1 + (n-1)d. Got that part.Next, it mentions that a_1, a_2, a_5 form a geometric sequence. Hmm, okay, a geometric sequence has each term multiplied by a common ratio. So, if a_1, a_2, a_5 are in geometric sequence, then the ratio between a_2 and a_1 should be the same as the ratio between a_5 and a_2. Mathematically, that means (a_2 / a_1) = (a_5 / a_2). Let me write that out:(a_2 / a_1) = (a_5 / a_2)Since it's an arithmetic sequence, I can express a_2 and a_5 in terms of a_1 and d. a_2 = a_1 + da_5 = a_1 + 4dSo substituting these into the equation:(a_1 + d) / a_1 = (a_1 + 4d) / (a_1 + d)Cross-multiplying to eliminate the fractions:(a_1 + d)^2 = a_1(a_1 + 4d)Let me expand both sides:Left side: (a_1 + d)^2 = a_1^2 + 2a_1d + d^2Right side: a_1(a_1 + 4d) = a_1^2 + 4a_1dSo, setting them equal:a_1^2 + 2a_1d + d^2 = a_1^2 + 4a_1dSubtracting a_1^2 from both sides:2a_1d + d^2 = 4a_1dBringing all terms to one side:2a_1d + d^2 - 4a_1d = 0Simplify:-2a_1d + d^2 = 0Factor out a d:d(-2a_1 + d) = 0So, this gives two possibilities:1. d = 02. -2a_1 + d = 0 => d = 2a_1Okay, so either the common difference d is zero, or d is twice the first term a_1.Now, the problem also states that a_3 + a_4 = 12. Let's use this to find a_1 and d.First, express a_3 and a_4 in terms of a_1 and d:a_3 = a_1 + 2da_4 = a_1 + 3dSo, a_3 + a_4 = (a_1 + 2d) + (a_1 + 3d) = 2a_1 + 5d = 12So, 2a_1 + 5d = 12.Now, let's consider the two cases from earlier.Case 1: d = 0If d = 0, then all terms are equal to a_1. So, a_3 = a_1 and a_4 = a_1. Therefore, a_3 + a_4 = 2a_1 = 12 => a_1 = 6.So, in this case, all terms are 6. Therefore, the sum a_1 + a_2 + a_3 + a_4 + a_5 = 5 * 6 = 30.Case 2: d = 2a_1Substitute d = 2a_1 into the equation 2a_1 + 5d = 12:2a_1 + 5*(2a_1) = 122a_1 + 10a_1 = 1212a_1 = 12a_1 = 1Then, d = 2a_1 = 2*1 = 2.So, the terms are:a_1 = 1a_2 = 1 + 2 = 3a_3 = 1 + 4 = 5a_4 = 1 + 6 = 7a_5 = 1 + 8 = 9So, the sum a_1 + a_2 + a_3 + a_4 + a_5 = 1 + 3 + 5 + 7 + 9 = 25.Therefore, for part (1), the sum is either 25 or 30, depending on whether d is 2 or 0.Wait, but the problem doesn't specify whether d is zero or not. So, both cases are possible? Or is there a reason to discard one case?Looking back, when d = 0, the geometric sequence a_1, a_2, a_5 would be 6, 6, 6, which is a geometric sequence with common ratio 1. So, that's valid.Similarly, when d = 2, the geometric sequence is 1, 3, 9, which is a geometric sequence with common ratio 3. So, both cases are valid.Therefore, both sums 25 and 30 are possible.So, for part (1), the answer is 25 or 30.Moving on to part (2). Let me read it again.Let b_n = 10 - a_n. The sum of the first n terms of the sequence {b_n} is S_n. If b_1 ≠ b_2, what is the value of n when S_n is maximum? What is the maximum value of S_n?Alright, so b_n is defined as 10 - a_n. Since a_n is an arithmetic sequence, b_n will also be an arithmetic sequence because it's a linear transformation of a_n.But wait, let's check: If a_n is arithmetic, then b_n = 10 - a_n is also arithmetic because the difference between consecutive terms will be constant.But the problem says "if b_1 ≠ b_2". So, that implies that the common difference of b_n is not zero. Since b_n = 10 - a_n, the common difference of b_n is -d, where d is the common difference of a_n.So, if b_1 ≠ b_2, then the common difference of b_n is not zero, which means the common difference of a_n is not zero. Therefore, in this case, d ≠ 0.So, from part (1), we have two cases: d = 0 and d = 2. But since b_1 ≠ b_2, we must have d ≠ 0, so we are in the case where d = 2, a_1 = 1.Therefore, a_n = 1 + (n-1)*2 = 2n - 1.So, b_n = 10 - a_n = 10 - (2n - 1) = 11 - 2n.So, b_n is an arithmetic sequence with first term b_1 = 11 - 2*1 = 9, and common difference -2.So, the sequence {b_n} is 9, 7, 5, 3, 1, -1, -3, ...We need to find the value of n where the sum S_n is maximum.Since the terms of {b_n} are decreasing by 2 each time, starting from 9, the terms will eventually become negative. The sum S_n will be maximum when the terms are still positive or when adding the next term would make the sum start decreasing.So, we need to find the largest n such that b_n is non-negative.Let me find when b_n ≥ 0:11 - 2n ≥ 011 ≥ 2nn ≤ 11/2 = 5.5Since n must be an integer, the largest n where b_n is non-negative is n = 5.Therefore, the sum S_n is maximum when n = 5.Now, let's compute S_5.Since {b_n} is an arithmetic sequence, the sum of the first n terms is given by:S_n = (n/2)(2b_1 + (n - 1)d_b)Where d_b is the common difference of {b_n}, which is -2.So, plugging in n = 5, b_1 = 9, d_b = -2:S_5 = (5/2)(2*9 + (5 - 1)*(-2)) = (5/2)(18 + (-8)) = (5/2)(10) = 5*5 = 25.Alternatively, since the terms are 9, 7, 5, 3, 1, adding them up: 9 + 7 = 16, 16 + 5 = 21, 21 + 3 = 24, 24 + 1 = 25.So, the maximum sum is 25 when n = 5.Wait, just to make sure, let's check S_6:b_6 = 11 - 2*6 = 11 - 12 = -1So, S_6 = S_5 + (-1) = 25 - 1 = 24, which is less than 25. So, indeed, S_5 is the maximum.Therefore, the value of n is 5, and the maximum value of S_n is 25.So, summarizing:(1) The sum a_1 + a_2 + a_3 + a_4 + a_5 is either 25 or 30.(2) The maximum sum S_n occurs at n = 5, and the maximum value is 25.**Final Answer**(1) The sum is boxed{25} or boxed{30}.(2) The value of ( n ) when ( S_n ) is maximum is boxed{5}, and the maximum value of ( S_n ) is boxed{25}.

💡Okay, so I have this problem here: T is equal to (1 + i) raised to the 19th power minus (1 - i) raised to the 19th power, and I need to find the absolute value of T. Hmm, okay. Let me try to figure this out step by step.First, I remember that when dealing with complex numbers, especially when they're raised to powers, it's often helpful to convert them into polar form. Polar form makes exponentiation easier because of De Moivre's theorem, which allows us to raise complex numbers to powers more straightforwardly.So, let me recall how to convert a complex number from rectangular form (a + bi) to polar form. The polar form is given by r*(cosθ + i*sinθ), where r is the modulus (or absolute value) of the complex number, and θ is the argument (or angle). The modulus r is calculated as the square root of (a² + b²), and the argument θ is the arctangent of (b/a).Alright, let's apply this to both (1 + i) and (1 - i).Starting with (1 + i):- The modulus r is sqrt(1² + 1²) = sqrt(2).- The argument θ is arctan(1/1) = arctan(1) = π/4 radians.So, (1 + i) in polar form is sqrt(2)*(cos(π/4) + i*sin(π/4)).Similarly, for (1 - i):- The modulus r is also sqrt(1² + (-1)²) = sqrt(2).- The argument θ is arctan(-1/1) = -π/4 radians, or equivalently, 7π/4 radians if we want a positive angle.So, (1 - i) in polar form is sqrt(2)*(cos(-π/4) + i*sin(-π/4)).Now, I need to raise both of these to the 19th power. That's where De Moivre's theorem comes in handy. De Moivre's theorem states that [r*(cosθ + i*sinθ)]^n = r^n*(cos(nθ) + i*sin(nθ)).Applying this to both (1 + i)^19 and (1 - i)^19:For (1 + i)^19:- r = sqrt(2), so r^19 = (sqrt(2))^19.- θ = π/4, so nθ = 19*(π/4) = (19π)/4.Similarly, for (1 - i)^19:- r = sqrt(2), so r^19 = (sqrt(2))^19.- θ = -π/4, so nθ = 19*(-π/4) = -19π/4.Hmm, okay. Now, I need to simplify these angles because they are more than 2π, which is a full circle. So, let's find the equivalent angles between 0 and 2π.Starting with (19π)/4:19 divided by 4 is 4.75, so that's 4π + 3π/4. Since 4π is two full circles, we can subtract 4π to get the equivalent angle: 3π/4.Similarly, for -19π/4:-19 divided by 4 is -4.75, so that's -4π - 3π/4. Adding 4π to get a positive angle gives us -3π/4. But angles can also be represented as positive by adding 2π, so -3π/4 + 2π = 5π/4. Wait, is that right? Let me double-check.Actually, -3π/4 is the same as 5π/4 when measured in the positive direction. But wait, no, 5π/4 is in the third quadrant, while -3π/4 is in the fourth quadrant. Hmm, maybe I should think differently.Alternatively, since angles are periodic with period 2π, we can add 2π until we get an angle between 0 and 2π.So, for -19π/4:Let's add 2π repeatedly until we get within 0 to 2π.First, -19π/4 + 2π = -19π/4 + 8π/4 = (-19 + 8)π/4 = -11π/4.Still negative, so add another 2π: -11π/4 + 8π/4 = (-11 + 8)π/4 = -3π/4.Still negative, add another 2π: -3π/4 + 8π/4 = (5π)/4.Okay, so -19π/4 is equivalent to 5π/4 in the range [0, 2π). So, that's the angle for (1 - i)^19.Wait, but hold on. Let me verify that. Because when you have a negative angle, adding 2π multiple times until it's positive is the way to go. So, starting from -19π/4:-19π/4 + 2π = -19π/4 + 8π/4 = -11π/4.Still negative, add another 2π: -11π/4 + 8π/4 = -3π/4.Still negative, add another 2π: -3π/4 + 8π/4 = 5π/4.Yes, so -19π/4 is equivalent to 5π/4.Wait, but in the case of (1 - i)^19, the angle was -π/4, so 19 times that is -19π/4, which is equivalent to 5π/4.But hold on, when we have a negative angle, does that affect the sine and cosine values? Let me recall that cos(-θ) = cosθ and sin(-θ) = -sinθ. So, perhaps I don't need to adjust the angle to be positive if I can handle the negative sine.But in this case, since we're dealing with exponents, it's probably better to express the angles in their positive equivalents to make it easier when applying De Moivre's theorem.So, (1 + i)^19 is (sqrt(2))^19*(cos(3π/4) + i*sin(3π/4)).Similarly, (1 - i)^19 is (sqrt(2))^19*(cos(5π/4) + i*sin(5π/4)).Wait, hold on, no. Because when we have (1 - i)^19, the angle was -π/4, so 19 times that is -19π/4, which we converted to 5π/4.But let me think again: is 5π/4 the correct equivalent angle for -19π/4?Wait, 19π/4 is 4π + 3π/4, so subtracting 4π gives 3π/4. Similarly, -19π/4 is -4π - 3π/4, so adding 4π gives -3π/4, which is equivalent to 5π/4 when measured in the positive direction.Yes, that's correct.So, now, let's write both expressions:(1 + i)^19 = (sqrt(2))^19 * [cos(3π/4) + i*sin(3π/4)](1 - i)^19 = (sqrt(2))^19 * [cos(5π/4) + i*sin(5π/4)]Now, let's compute T = (1 + i)^19 - (1 - i)^19.So, T = (sqrt(2))^19 * [cos(3π/4) + i*sin(3π/4)] - (sqrt(2))^19 * [cos(5π/4) + i*sin(5π/4)]Factor out (sqrt(2))^19:T = (sqrt(2))^19 * [cos(3π/4) + i*sin(3π/4) - cos(5π/4) - i*sin(5π/4)]Now, let's compute the terms inside the brackets.First, cos(3π/4) is equal to -sqrt(2)/2, and sin(3π/4) is sqrt(2)/2.Similarly, cos(5π/4) is -sqrt(2)/2, and sin(5π/4) is -sqrt(2)/2.So, substituting these values:cos(3π/4) = -sqrt(2)/2sin(3π/4) = sqrt(2)/2cos(5π/4) = -sqrt(2)/2sin(5π/4) = -sqrt(2)/2So, plugging these into the expression:[ (-sqrt(2)/2) + i*(sqrt(2)/2) - (-sqrt(2)/2) - i*(-sqrt(2)/2) ]Simplify term by term:First term: (-sqrt(2)/2)Second term: + i*(sqrt(2)/2)Third term: - (-sqrt(2)/2) = + sqrt(2)/2Fourth term: - i*(-sqrt(2)/2) = + i*(sqrt(2)/2)So, combining like terms:Real parts: (-sqrt(2)/2) + (sqrt(2)/2) = 0Imaginary parts: (sqrt(2)/2) + (sqrt(2)/2) = sqrt(2)So, the expression inside the brackets simplifies to 0 + i*sqrt(2) = i*sqrt(2)Therefore, T = (sqrt(2))^19 * (i*sqrt(2))Now, let's compute (sqrt(2))^19.First, note that sqrt(2) is 2^(1/2). So, (2^(1/2))^19 = 2^(19/2)Similarly, sqrt(2) is 2^(1/2), so multiplying by sqrt(2) gives 2^(1/2).Therefore, T = 2^(19/2) * 2^(1/2) * iWhen multiplying exponents with the same base, we add the exponents:2^(19/2 + 1/2) = 2^(20/2) = 2^10 = 1024So, T = 1024 * iBut wait, hold on. Let me double-check that exponent calculation.Wait, (sqrt(2))^19 is (2^(1/2))^19 = 2^(19/2). Then, multiplying by sqrt(2) is 2^(1/2), so total exponent is 19/2 + 1/2 = 20/2 = 10. So, yes, 2^10 is 1024.Therefore, T = 1024iBut the question asks for |T|, the absolute value of T.The absolute value of a complex number a + bi is sqrt(a² + b²). In this case, T is 0 + 1024i, so |T| = sqrt(0² + 1024²) = 1024.Wait, but hold on, that seems too straightforward. Let me go back through my steps to make sure I didn't make a mistake.Starting from T = (1 + i)^19 - (1 - i)^19.Converted both to polar form: sqrt(2) cis π/4 and sqrt(2) cis (-π/4).Raised to 19th power: (sqrt(2))^19 cis (19π/4) and (sqrt(2))^19 cis (-19π/4).Simplified angles: 19π/4 is 4π + 3π/4, so equivalent to 3π/4. Similarly, -19π/4 is equivalent to 5π/4.Wait, hold on, earlier I thought -19π/4 is equivalent to 5π/4, but when I computed the expression, I used cos(5π/4) and sin(5π/4). But in reality, when we have cis(-19π/4), which is equivalent to cis(5π/4), right?But when I expanded the expression, I had:[cos(3π/4) + i sin(3π/4)] - [cos(5π/4) + i sin(5π/4)]Which led to 0 + i sqrt(2). But wait, let me re-examine that step.Wait, cos(3π/4) is -sqrt(2)/2, sin(3π/4) is sqrt(2)/2.cos(5π/4) is -sqrt(2)/2, sin(5π/4) is -sqrt(2)/2.So, substituting:[ (-sqrt(2)/2) + i*(sqrt(2)/2) ] - [ (-sqrt(2)/2) + i*(-sqrt(2)/2) ]Which is:(-sqrt(2)/2 + i sqrt(2)/2) - (-sqrt(2)/2 - i sqrt(2)/2)Distribute the negative sign:(-sqrt(2)/2 + i sqrt(2)/2) + sqrt(2)/2 + i sqrt(2)/2Now, combine like terms:(-sqrt(2)/2 + sqrt(2)/2) + (i sqrt(2)/2 + i sqrt(2)/2)The real parts cancel out: 0The imaginary parts add up: i sqrt(2)/2 + i sqrt(2)/2 = i sqrt(2)So, yes, that part is correct. So, T = (sqrt(2))^19 * i sqrt(2)Which is (sqrt(2))^19 * sqrt(2) * i = (sqrt(2))^(20) * iWait, hold on, sqrt(2) is 2^(1/2), so (2^(1/2))^20 = 2^(10) = 1024Therefore, T = 1024iSo, |T| is the modulus of 1024i, which is 1024.But wait, in the initial problem statement, it's (1 + i)^19 - (1 - i)^19. So, is that correct?Wait, let me think again. Because when I subtracted (1 - i)^19 from (1 + i)^19, I ended up with an imaginary number, 1024i, whose modulus is 1024.But in the previous assistant's answer, it was 512 sqrt(2). So, which one is correct?Wait, maybe I made a mistake in the exponent calculation.Let me recast the problem.Alternatively, perhaps I can compute (1 + i)^n - (1 - i)^n for some n, and see if there's a pattern.But let me try another approach. Let me compute (1 + i)^n - (1 - i)^n.Note that (1 + i) and (1 - i) are complex conjugates. So, when we raise them to the same power, their difference will be twice the imaginary part of (1 + i)^n.Because (1 + i)^n - (1 - i)^n = 2i * Im[(1 + i)^n]So, perhaps that's a more straightforward way to compute T.Let me try that.So, T = (1 + i)^19 - (1 - i)^19 = 2i * Im[(1 + i)^19]So, if I can compute Im[(1 + i)^19], then multiply by 2i, and then find the modulus.But let's see.First, compute (1 + i)^19.As before, (1 + i) is sqrt(2) cis π/4.So, (1 + i)^19 = (sqrt(2))^19 cis (19π/4)Which is 2^(19/2) cis (3π/4), since 19π/4 = 4π + 3π/4, and cis is periodic with period 2π.So, cis(3π/4) is cos(3π/4) + i sin(3π/4) = -sqrt(2)/2 + i sqrt(2)/2.Therefore, (1 + i)^19 = 2^(19/2) * (-sqrt(2)/2 + i sqrt(2)/2)Simplify this:2^(19/2) * (-sqrt(2)/2) = 2^(19/2) * (-2^(1/2)/2) = -2^(19/2 + 1/2 - 1) = -2^(20/2 - 1) = -2^(10 - 1) = -2^9 = -512Similarly, 2^(19/2) * (sqrt(2)/2) = 2^(19/2) * (2^(1/2)/2) = 2^(19/2 + 1/2 - 1) = 2^(20/2 - 1) = 2^(10 - 1) = 2^9 = 512So, (1 + i)^19 = -512 + 512iTherefore, Im[(1 + i)^19] = 512So, T = 2i * 512 = 1024iHence, |T| = |1024i| = 1024Wait, so according to this method, the modulus is 1024.But in the initial assistant's answer, it was 512 sqrt(2). So, which one is correct?Wait, let me check the calculations again.First, (1 + i)^19:(1 + i) is sqrt(2) cis π/4So, (1 + i)^19 = (sqrt(2))^19 cis (19π/4)Which is 2^(19/2) cis (3π/4)cis(3π/4) is cos(3π/4) + i sin(3π/4) = -sqrt(2)/2 + i sqrt(2)/2Therefore, (1 + i)^19 = 2^(19/2) * (-sqrt(2)/2 + i sqrt(2)/2)Compute 2^(19/2) * (-sqrt(2)/2):2^(19/2) * (-2^(1/2)/2) = -2^(19/2 + 1/2 - 1) = -2^(20/2 - 1) = -2^(10 - 1) = -2^9 = -512Similarly, 2^(19/2) * (sqrt(2)/2) = 2^(19/2 + 1/2 - 1) = 2^(10 - 1) = 2^9 = 512So, (1 + i)^19 = -512 + 512iSimilarly, (1 - i)^19:(1 - i) is sqrt(2) cis (-π/4)So, (1 - i)^19 = (sqrt(2))^19 cis (-19π/4)Which is 2^(19/2) cis (-19π/4 + 4π) = 2^(19/2) cis (3π/4)Wait, no. Wait, -19π/4 + 4π = -19π/4 + 16π/4 = (-19 + 16)π/4 = -3π/4So, cis(-3π/4) is cos(-3π/4) + i sin(-3π/4) = cos(3π/4) - i sin(3π/4) = -sqrt(2)/2 - i sqrt(2)/2Therefore, (1 - i)^19 = 2^(19/2) * (-sqrt(2)/2 - i sqrt(2)/2)Compute this:2^(19/2) * (-sqrt(2)/2) = -5122^(19/2) * (-sqrt(2)/2) = -512Wait, no:Wait, 2^(19/2) * (-sqrt(2)/2) = -2^(19/2 + 1/2 - 1) = -2^(20/2 - 1) = -2^(10 - 1) = -2^9 = -512Similarly, 2^(19/2) * (-sqrt(2)/2) = -512Wait, but the imaginary part is -sqrt(2)/2, so:2^(19/2) * (-sqrt(2)/2) = -512And 2^(19/2) * (-sqrt(2)/2) = -512Wait, that can't be right. Wait, no, let me clarify.(1 - i)^19 = 2^(19/2) * [cos(-3π/4) + i sin(-3π/4)] = 2^(19/2) * (-sqrt(2)/2 - i sqrt(2)/2)So, real part: 2^(19/2) * (-sqrt(2)/2) = -512Imaginary part: 2^(19/2) * (-sqrt(2)/2) = -512Therefore, (1 - i)^19 = -512 - 512iTherefore, T = (1 + i)^19 - (1 - i)^19 = (-512 + 512i) - (-512 - 512i) = (-512 + 512i) + 512 + 512i = (0) + (1024i) = 1024iSo, |T| = |1024i| = 1024But earlier, the assistant's answer was 512 sqrt(2). So, which one is correct?Wait, perhaps I made a mistake in the initial step when I thought that T = 2i * Im[(1 + i)^19]. Let me verify that.Yes, because (1 + i)^n - (1 - i)^n = 2i * Im[(1 + i)^n]So, if Im[(1 + i)^19] = 512, then T = 2i * 512 = 1024i, which has modulus 1024.But in the assistant's answer, they had T = 2^9 i sqrt(2), which is 512 * sqrt(2) i, so |T| = 512 sqrt(2). But according to my calculation, it's 1024.Wait, perhaps the assistant made a mistake in the exponent calculation.Wait, let's see:In the assistant's answer, they had:(√2 cis π/4)^19 = 2^(19/2) cis (19π/4) = 2^(19/2) cis (3π/4)Similarly, (√2 cis (-π/4))^19 = 2^(19/2) cis (-19π/4) = 2^(19/2) cis (-3π/4)Then, they subtracted these two:2^(19/2) [cis(3π/4) - cis(-3π/4)] = 2^(19/2) [ (cos(3π/4) + i sin(3π/4)) - (cos(-3π/4) + i sin(-3π/4)) ]Which simplifies to:2^(19/2) [ (-sqrt(2)/2 + i sqrt(2)/2) - (-sqrt(2)/2 - i sqrt(2)/2) ]Which is:2^(19/2) [ (-sqrt(2)/2 + i sqrt(2)/2 + sqrt(2)/2 + i sqrt(2)/2) ] = 2^(19/2) [ (0) + i sqrt(2) ]So, T = 2^(19/2) * i sqrt(2) = 2^(19/2 + 1/2) * i = 2^(20/2) * i = 2^10 * i = 1024iTherefore, |T| = 1024But in the assistant's answer, they had:2^(19/2) * 2i * sqrt(2)/2 = 2^9 i sqrt(2)Wait, that seems incorrect. Because 2^(19/2) * 2 * sqrt(2)/2 = 2^(19/2) * sqrt(2) = 2^(19/2 + 1/2) = 2^10 = 1024So, the assistant incorrectly wrote it as 2^9 i sqrt(2), which is 512 sqrt(2) i, but it should be 1024i.Therefore, the correct modulus is 1024.Wait, but in the initial problem, the assistant's answer was 512 sqrt(2). So, perhaps they made a mistake in the exponent calculation.Alternatively, perhaps I made a mistake in my calculation.Wait, let me check the modulus of T.If T = 1024i, then |T| = 1024.But if T = 512 sqrt(2) i, then |T| = 512 sqrt(2).So, which one is correct?Wait, let's compute (1 + i)^19 - (1 - i)^19 numerically.Compute (1 + i)^19:We can compute (1 + i)^n using binomial expansion, but that's tedious. Alternatively, note that (1 + i)^2 = 2i, so (1 + i)^4 = (2i)^2 = -4, (1 + i)^8 = (-4)^2 = 16, (1 + i)^16 = 16^2 = 256.Then, (1 + i)^19 = (1 + i)^16 * (1 + i)^3 = 256 * (1 + i)^3Compute (1 + i)^3:(1 + i)^3 = (1 + i)*(1 + i)^2 = (1 + i)*(2i) = 2i + 2i^2 = 2i - 2 = -2 + 2iTherefore, (1 + i)^19 = 256 * (-2 + 2i) = -512 + 512iSimilarly, (1 - i)^19:(1 - i)^2 = -2i, (1 - i)^4 = (-2i)^2 = -4, (1 - i)^8 = (-4)^2 = 16, (1 - i)^16 = 16^2 = 256Then, (1 - i)^19 = (1 - i)^16 * (1 - i)^3 = 256 * (1 - i)^3Compute (1 - i)^3:(1 - i)^3 = (1 - i)*(1 - i)^2 = (1 - i)*(-2i) = -2i + 2i^2 = -2i - 2 = -2 - 2iTherefore, (1 - i)^19 = 256 * (-2 - 2i) = -512 - 512iTherefore, T = (1 + i)^19 - (1 - i)^19 = (-512 + 512i) - (-512 - 512i) = (-512 + 512i) + 512 + 512i = 0 + 1024i = 1024iSo, |T| = |1024i| = 1024Therefore, the correct answer is 1024.But in the initial assistant's answer, they had |T| = 512 sqrt(2). So, that must be incorrect.Wait, perhaps the assistant made a mistake in the exponent calculation when they wrote:"= 2^{19/2} cdot 2i cdot frac{sqrt{2}}{2} = 2^9 i sqrt{2}"Wait, let's parse that.They had:2^{19/2} * 2i * sqrt(2)/2Which is 2^{19/2} * 2 * sqrt(2)/2 * iSimplify:2^{19/2} * (2 * sqrt(2)/2) = 2^{19/2} * sqrt(2) = 2^{19/2 + 1/2} = 2^{20/2} = 2^{10} = 1024So, the correct expression is 1024i, not 2^9 i sqrt(2). So, the assistant made a mistake in the calculation there.Therefore, the correct modulus is 1024.So, in conclusion, |T| = 1024.

💡Alright, let's tackle this problem step by step. So, we have a sequence ({a_n}) that satisfies the recurrence relation (a_{n+1} + 1 = a_n) for all (n in mathbb{N}^*). We need to determine what type of sequence this is from the given options.First, let's understand the recurrence relation. The equation (a_{n+1} + 1 = a_n) can be rearranged to find the relationship between consecutive terms. If I subtract (a_{n+1}) from both sides, I get:[a_n - a_{n+1} = 1]But actually, it's more straightforward to express (a_{n+1}) in terms of (a_n). Let's rearrange the original equation:[a_{n+1} = a_n - 1]So, each term is obtained by subtracting 1 from the previous term. This suggests that the sequence is decreasing by a constant difference each time. Now, recalling the definitions of arithmetic and geometric sequences:- An **arithmetic sequence** is one where each term after the first is obtained by adding a constant difference to the preceding term. The general form is (a_n = a_1 + (n-1)d), where (d) is the common difference. - A **geometric sequence** is one where each term after the first is obtained by multiplying the preceding term by a constant ratio. The general form is (a_n = a_1 cdot r^{n-1}), where (r) is the common ratio.Looking back at our recurrence relation (a_{n+1} = a_n - 1), we can see that each term is obtained by subtracting 1 from the previous term. This is equivalent to adding a common difference of (-1). Therefore, this sequence is an arithmetic sequence with a common difference of (-1).Let's verify this by considering the first few terms. Suppose the first term is (a_1). Then:- (a_2 = a_1 - 1)- (a_3 = a_2 - 1 = (a_1 - 1) - 1 = a_1 - 2)- (a_4 = a_3 - 1 = (a_1 - 2) - 1 = a_1 - 3)And so on. So, the (n)-th term can be expressed as:[a_n = a_1 + (n - 1)(-1) = a_1 - (n - 1)]This confirms that the sequence is arithmetic with a common difference of (-1).Now, let's look at the options provided:A: An arithmetic sequence with a common difference of 1 B: A geometric sequence with a common ratio of 1 C: An arithmetic sequence with a common difference of -1 D: A geometric sequence with a common ratio of -1From our analysis, the correct answer is option C, as the sequence decreases by 1 each time, which is an arithmetic sequence with a common difference of (-1).Just to be thorough, let's eliminate the other options:- **Option A**: A common difference of 1 would mean each term is 1 more than the previous term, which contradicts our recurrence relation where each term is 1 less. - **Option B**: A geometric sequence with a common ratio of 1 would mean every term is the same, which isn't the case here since each term decreases by 1. - **Option D**: A geometric sequence with a common ratio of -1 would alternate the sign of each term, which doesn't match our sequence where each term is consistently decreasing by 1.Therefore, after careful consideration and elimination, the correct answer is indeed option C.

💡Okay, so I have this problem where I need to find the maximum value of t, which is the x-coordinate of the midpoint of segment MN. Point P is moving on the graph of f(x) = ln x, where x is greater than or equal to 1. The tangent line at P intersects the x-axis at M, and a line perpendicular to this tangent at P intersects the x-axis at N. Then, t is the midpoint of MN. Hmm, that sounds a bit complicated, but let me try to break it down step by step.First, I should probably parameterize point P. Since P is on the graph of f(x) = ln x, its coordinates can be written as (m, ln m), where m is some x-value greater than or equal to 1. That makes sense because ln x is defined for x > 0, but since x >= 1, m will be at least 1.Next, I need to find the equation of the tangent line at point P. To do that, I need the derivative of f(x) at point P. The derivative of ln x is 1/x, so at point P, the slope of the tangent line is 1/m. Therefore, the equation of the tangent line l is:y - ln m = (1/m)(x - m)Let me write that out more clearly:y = (1/m)(x - m) + ln mSimplifying that, it becomes:y = (x/m) - 1 + ln mOkay, so that's the equation of the tangent line. Now, this tangent line intersects the x-axis at point M. To find M, I need to set y = 0 and solve for x.0 = (x/m) - 1 + ln mLet me solve for x:(x/m) = 1 - ln mx = m(1 - ln m)So, point M has coordinates (m(1 - ln m), 0). That seems right because if I plug x = m(1 - ln m) into the tangent line equation, y should be 0.Now, moving on to the line perpendicular to l passing through P. Since l has a slope of 1/m, the slope of the perpendicular line should be the negative reciprocal, which is -m. So, the equation of this perpendicular line is:y - ln m = -m(x - m)Let me write that out:y = -m(x - m) + ln mSimplifying:y = -mx + m² + ln mNow, to find point N where this perpendicular line intersects the x-axis, set y = 0:0 = -mx + m² + ln mSolving for x:mx = m² + ln mx = (m² + ln m)/mSimplify that:x = m + (ln m)/mSo, point N has coordinates (m + (ln m)/m, 0). Got that.Now, I need to find the midpoint of segment MN. The midpoint's x-coordinate is the average of the x-coordinates of M and N. So, let's compute that.Coordinates of M: (m(1 - ln m), 0)Coordinates of N: (m + (ln m)/m, 0)Midpoint x-coordinate, t:t = [m(1 - ln m) + m + (ln m)/m] / 2Let me simplify that expression step by step.First, expand m(1 - ln m):m(1 - ln m) = m - m ln mSo, plug that back into t:t = [ (m - m ln m) + m + (ln m)/m ] / 2Combine like terms:m - m ln m + m = 2m - m ln mSo, t = [2m - m ln m + (ln m)/m] / 2Let me write that as:t = (2m - m ln m + (ln m)/m) / 2Alternatively, I can factor out m in some terms:t = (2m + (ln m)/m - m ln m) / 2Hmm, that seems a bit messy, but maybe I can factor it differently or find a common denominator. Let me see.Alternatively, maybe I can write it as:t = m - (m ln m)/2 + (ln m)/(2m)But I don't know if that helps. Maybe it's better to just keep it as:t = (2m + (ln m)/m - m ln m) / 2Now, the goal is to find the maximum value of t as m varies over [1, ∞). So, I need to find the maximum of this function t(m). To do that, I can take the derivative of t with respect to m, set it equal to zero, and solve for m. Then, check if that critical point is a maximum.So, let's compute t'(m). First, let me write t(m) as:t(m) = (2m + (ln m)/m - m ln m) / 2Which can be rewritten as:t(m) = m + (ln m)/(2m) - (m ln m)/2Now, let's compute the derivative term by term.First term: d/dm [m] = 1Second term: d/dm [(ln m)/(2m)]Let me compute that. Let me denote u = ln m and v = 2m. Then, the derivative is (u'v - uv') / v².u' = 1/mv' = 2So, derivative is ( (1/m)(2m) - (ln m)(2) ) / (2m)^2Simplify numerator:(2 - 2 ln m) / (4m²) = (1 - ln m)/(2m²)So, the derivative of the second term is (1 - ln m)/(2m²)Third term: d/dm [ - (m ln m)/2 ]Let me compute that. Let me denote u = m ln m. Then, derivative is (u')/2.u' = ln m + m*(1/m) = ln m + 1So, derivative is (ln m + 1)/2But since it's negative, it's - (ln m + 1)/2Putting it all together, the derivative t'(m) is:1 + (1 - ln m)/(2m²) - (ln m + 1)/2Let me write that out:t'(m) = 1 + (1 - ln m)/(2m²) - (ln m + 1)/2Hmm, that seems a bit complicated. Maybe I can combine the terms.Let me first combine the constant terms and the ln m terms.Let me write:t'(m) = 1 - (ln m + 1)/2 + (1 - ln m)/(2m²)Let me compute 1 - (ln m + 1)/2:1 - (ln m + 1)/2 = (2 - ln m - 1)/2 = (1 - ln m)/2So, t'(m) = (1 - ln m)/2 + (1 - ln m)/(2m²)Factor out (1 - ln m)/2:t'(m) = (1 - ln m)/2 * (1 + 1/m²)So, t'(m) = (1 - ln m)(1 + 1/m²)/2Wait, that seems a bit different from what I had before, but let me check.Wait, no, actually, when I combined 1 - (ln m + 1)/2, I got (1 - ln m)/2. Then, adding (1 - ln m)/(2m²), so it's (1 - ln m)/2 + (1 - ln m)/(2m²) = (1 - ln m)(1/2 + 1/(2m²)) = (1 - ln m)( (m² + 1)/(2m²) )Wait, let me see:(1 - ln m)/2 + (1 - ln m)/(2m²) = (1 - ln m)(1/2 + 1/(2m²)) = (1 - ln m)( (m² + 1)/(2m²) )Yes, that's correct. So, t'(m) = (1 - ln m)(m² + 1)/(2m²)So, t'(m) = (1 - ln m)(m² + 1)/(2m²)Now, to find critical points, set t'(m) = 0.So, (1 - ln m)(m² + 1)/(2m²) = 0Since m >= 1, m² + 1 is always positive, and 2m² is positive. So, the only way for t'(m) to be zero is when (1 - ln m) = 0.So, 1 - ln m = 0 => ln m = 1 => m = eSo, the critical point is at m = e.Now, we need to check whether this critical point is a maximum or a minimum.Looking at the derivative t'(m):- When m < e, ln m < 1, so (1 - ln m) > 0, so t'(m) > 0.- When m > e, ln m > 1, so (1 - ln m) < 0, so t'(m) < 0.Therefore, t(m) is increasing on [1, e) and decreasing on (e, ∞). So, the function t(m) reaches its maximum at m = e.Therefore, the maximum value of t is t(e).So, let's compute t(e).Recall that t(m) = (2m + (ln m)/m - m ln m)/2So, plug in m = e:t(e) = (2e + (ln e)/e - e ln e)/2Compute each term:ln e = 1So,t(e) = (2e + (1)/e - e * 1)/2Simplify:2e - e = eSo,t(e) = (e + 1/e)/2Which can be written as:t(e) = (e² + 1)/(2e)Because e + 1/e = (e² + 1)/eSo, t(e) = (e² + 1)/(2e)Therefore, the maximum value of t is (e² + 1)/(2e)Let me just double-check my steps to make sure I didn't make any mistakes.1. Parameterized P as (m, ln m). Correct.2. Found the tangent line at P: y = (1/m)(x - m) + ln m. Correct.3. Found M by setting y=0: x = m(1 - ln m). Correct.4. Found the perpendicular line: slope is -m, equation y = -m(x - m) + ln m. Correct.5. Found N by setting y=0: x = m + (ln m)/m. Correct.6. Midpoint t: [m(1 - ln m) + m + (ln m)/m]/2. Simplified correctly to (2m + (ln m)/m - m ln m)/2.7. Took derivative t'(m): Correctly computed as (1 - ln m)(m² + 1)/(2m²). Critical point at m = e.8. Evaluated t(e): Correctly simplified to (e² + 1)/(2e).Yes, everything seems to check out. So, the maximum value of t is (e² + 1)/(2e).