💡Okay, so I have this problem here: "The terminal side of angle α passes through the fixed point P of the function y = log_a(x - 3) + 2. Then, sin2α + cos2α = （　　）" with options A: 7/5, B: 6/5, C: 4, D: 5.Hmm, I need to figure out what sin2α + cos2α is. First, let me understand the problem step by step.The function given is y = log_a(x - 3) + 2. It says that the terminal side of angle α passes through the fixed point P of this function. So, I need to find the fixed point P first.Wait, what is a fixed point of a function? I think a fixed point is a point where the function intersects a specific line, maybe the line y = x? Or is it a point that doesn't change regardless of the parameter? Hmm, in the context of logarithmic functions, maybe it's a point that remains constant regardless of the base a. Let me think.For logarithmic functions, the general form is y = log_a(x). The graph of this function passes through (1, 0) because log_a(1) = 0 for any base a. So, maybe the fixed point is a point that is always on the graph no matter what the base a is.Given the function y = log_a(x - 3) + 2, let's see. If I set x - 3 = 1, then log_a(1) = 0, so y = 0 + 2 = 2. So, when x - 3 = 1, x = 4, and y = 2. Therefore, the fixed point P is (4, 2). That makes sense because regardless of the base a, log_a(1) is always 0, so y will always be 2 when x is 4.Okay, so point P is (4, 2). Now, the terminal side of angle α passes through this point. So, in the coordinate plane, the terminal side of angle α is a ray starting at the origin and passing through (4, 2). Therefore, we can consider this point as a point on the terminal side of angle α in standard position.To find sin α and cos α, I need to consider the coordinates of point P. In trigonometry, for any point (x, y) on the terminal side of an angle α, we can find sin α and cos α using the formulas:sin α = y / rcos α = x / rwhere r is the distance from the origin to the point (x, y). So, first, I need to calculate r.Given point P is (4, 2), so x = 4, y = 2.r = sqrt(x^2 + y^2) = sqrt(4^2 + 2^2) = sqrt(16 + 4) = sqrt(20) = 2*sqrt(5).So, r = 2*sqrt(5).Now, let's compute sin α and cos α.sin α = y / r = 2 / (2*sqrt(5)) = 1 / sqrt(5). To rationalize the denominator, multiply numerator and denominator by sqrt(5): (1*sqrt(5)) / (sqrt(5)*sqrt(5)) = sqrt(5)/5.Similarly, cos α = x / r = 4 / (2*sqrt(5)) = 2 / sqrt(5). Rationalizing: (2*sqrt(5)) / (sqrt(5)*sqrt(5)) = 2*sqrt(5)/5.So, sin α = sqrt(5)/5 and cos α = 2*sqrt(5)/5.Now, the question asks for sin2α + cos2α. I remember that there are double-angle formulas for sine and cosine.The double-angle formula for sine is sin2α = 2 sin α cos α.The double-angle formula for cosine can be written in a few ways, but one common form is cos2α = cos^2 α - sin^2 α. Alternatively, it can also be written as 2 cos^2 α - 1 or 1 - 2 sin^2 α. I think I'll use cos2α = 2 cos^2 α - 1 because I already have cos α.So, let's compute sin2α and cos2α separately and then add them together.First, sin2α:sin2α = 2 sin α cos α = 2*(sqrt(5)/5)*(2*sqrt(5)/5).Let's compute that:2*(sqrt(5)/5)*(2*sqrt(5)/5) = 2*(2*5)/25 = 2*(10)/25 = 20/25 = 4/5.Wait, hold on, let me check that step again.Wait, sqrt(5)*sqrt(5) is 5, right? So, (sqrt(5)/5)*(2*sqrt(5)/5) = (2*5)/(5*5) = 10/25 = 2/5. Then, multiplied by 2, it's 4/5. So, sin2α = 4/5.Okay, that seems correct.Now, cos2α. Let's use cos2α = 2 cos^2 α - 1.First, compute cos^2 α:cos α = 2*sqrt(5)/5, so cos^2 α = (2*sqrt(5)/5)^2 = (4*5)/25 = 20/25 = 4/5.So, cos2α = 2*(4/5) - 1 = 8/5 - 1 = 8/5 - 5/5 = 3/5.Alternatively, if I use another form of the double angle formula, let's verify:cos2α = cos^2 α - sin^2 α.We have cos^2 α = 4/5 and sin^2 α = (sqrt(5)/5)^2 = 5/25 = 1/5.So, cos2α = 4/5 - 1/5 = 3/5. Same result. Good.So, cos2α = 3/5.Now, sin2α + cos2α = 4/5 + 3/5 = 7/5.So, the answer is 7/5, which is option A.Wait, let me just recap to make sure I didn't make any mistakes.1. Found fixed point P by setting x - 3 = 1, so x = 4, y = 2. So, P is (4, 2).2. Calculated r = sqrt(4^2 + 2^2) = sqrt(20) = 2*sqrt(5).3. sin α = 2/(2*sqrt(5)) = 1/sqrt(5) = sqrt(5)/5.4. cos α = 4/(2*sqrt(5)) = 2/sqrt(5) = 2*sqrt(5)/5.5. sin2α = 2*(sqrt(5)/5)*(2*sqrt(5)/5) = 4/5.6. cos2α = 2*(4/5) - 1 = 8/5 - 5/5 = 3/5.7. sin2α + cos2α = 4/5 + 3/5 = 7/5.Yes, that seems correct. So, the answer is 7/5, which is option A.I think that's solid. I don't see any mistakes in the calculations. The key was recognizing the fixed point, which is a common point regardless of the base a, and then using the definitions of sine and cosine in terms of coordinates and the double-angle formulas.**Final Answer**boxed{A}

💡Alright, so I have this inequality to solve, and I need to find the smallest real number ( M ) such that for all real numbers ( a, b, ) and ( c ), the following holds:[left|a bleft(a^{2}-b^{2}right) + b cleft(b^{2}-c^{2}right) + c aleft(c^{2}-a^{2}right)right| leq Mleft(a^{2}+b^{2}+c^{2}right)^{2}.]Hmm, okay. Let me try to understand what's going on here. The left side is an absolute value of a combination of products and differences of squares, and the right side is ( M ) times the square of the sum of squares of ( a, b, ) and ( c ). So, I need to find the smallest ( M ) such that this inequality is always true, no matter what ( a, b, ) and ( c ) are.First, maybe I can simplify the expression inside the absolute value. Let me write it out again:[a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2).]I notice that each term is of the form ( xy(x^2 - y^2) ), which can be factored as ( xy(x - y)(x + y) ). So, maybe I can factor each term like that:[a b(a - b)(a + b) + b c(b - c)(b + c) + c a(c - a)(c + a).]Hmm, that looks a bit complicated, but maybe there's a pattern or a way to combine these terms. Let me see if I can factor something out or find a common structure.Alternatively, maybe I can think about symmetric functions or use some inequality techniques. Since the right side is a square of a sum, perhaps Cauchy-Schwarz inequality or something similar might be useful here.Wait, another thought: the expression on the left seems to be cyclic in ( a, b, c ). Maybe I can consider specific cases where ( a, b, c ) take on particular values to find constraints on ( M ). For example, if I set two variables equal or set one variable to zero, maybe I can simplify the expression and find what ( M ) must be.Let me try setting ( c = 0 ). Then the expression becomes:[a b(a^2 - b^2) + 0 + 0 = a b(a^2 - b^2).]And the right side becomes ( M(a^2 + b^2)^2 ). So, the inequality simplifies to:[|a b(a^2 - b^2)| leq M(a^2 + b^2)^2.]Hmm, okay. Maybe I can analyze this simpler inequality to find a lower bound for ( M ).Let me set ( a = 1 ) and ( b = t ), where ( t ) is a real number. Then the left side becomes:[|1 cdot t (1 - t^2)| = |t(1 - t^2)|.]And the right side becomes:[M(1 + t^2)^2.]So, the inequality is:[|t(1 - t^2)| leq M(1 + t^2)^2.]To find the minimal ( M ), I need to maximize ( frac{|t(1 - t^2)|}{(1 + t^2)^2} ) over all real ( t ). Let's denote this function as:[f(t) = frac{|t(1 - t^2)|}{(1 + t^2)^2}.]Since ( f(t) ) is even (because replacing ( t ) with ( -t ) doesn't change the value), I can consider ( t geq 0 ) only.So, ( f(t) = frac{t(1 - t^2)}{(1 + t^2)^2} ) for ( t in [0, 1) ) and ( f(t) = frac{t(t^2 - 1)}{(1 + t^2)^2} ) for ( t geq 1 ).Wait, actually, since ( |t(1 - t^2)| ) is symmetric around ( t = 0 ), but we can consider ( t geq 0 ) as I said.Let me compute the maximum of ( f(t) ) for ( t geq 0 ).First, consider ( t in [0, 1) ):[f(t) = frac{t(1 - t^2)}{(1 + t^2)^2}.]Let me take the derivative to find the maximum. Let ( f(t) = frac{t(1 - t^2)}{(1 + t^2)^2} ).Compute ( f'(t) ):Using the quotient rule:[f'(t) = frac{(1 - t^2) + t(-2t)}{(1 + t^2)^2} cdot (1 + t^2)^2 - t(1 - t^2) cdot 2(1 + t^2)(2t)}{(1 + t^2)^4}]Wait, that seems complicated. Maybe better to write ( f(t) = t(1 - t^2)(1 + t^2)^{-2} ) and use the product rule.Let me denote ( u = t(1 - t^2) ) and ( v = (1 + t^2)^{-2} ).Then,[f'(t) = u'v + uv'.]Compute ( u' ):[u = t(1 - t^2) = t - t^3,]so,[u' = 1 - 3t^2.]Compute ( v' ):[v = (1 + t^2)^{-2},]so,[v' = -2(1 + t^2)^{-3} cdot 2t = -4t(1 + t^2)^{-3}.]Thus,[f'(t) = (1 - 3t^2)(1 + t^2)^{-2} + (t - t^3)(-4t)(1 + t^2)^{-3}.]Simplify the expression:First term: ( (1 - 3t^2)(1 + t^2)^{-2} ).Second term: ( -4t(t - t^3)(1 + t^2)^{-3} = -4t^2(1 - t^2)(1 + t^2)^{-3} ).So, combining:[f'(t) = (1 - 3t^2)(1 + t^2)^{-2} - 4t^2(1 - t^2)(1 + t^2)^{-3}.]To combine these terms, let's factor out ( (1 + t^2)^{-3} ):[f'(t) = (1 - 3t^2)(1 + t^2) - 4t^2(1 - t^2) cdot (1 + t^2)^{-3}.]Wait, no, more accurately:Factor out ( (1 + t^2)^{-3} ):[f'(t) = (1 - 3t^2)(1 + t^2) cdot (1 + t^2)^{-3} - 4t^2(1 - t^2) cdot (1 + t^2)^{-3}][= left[ (1 - 3t^2)(1 + t^2) - 4t^2(1 - t^2) right] (1 + t^2)^{-3}.]Now, expand the numerator:First, expand ( (1 - 3t^2)(1 + t^2) ):[1 cdot 1 + 1 cdot t^2 - 3t^2 cdot 1 - 3t^2 cdot t^2 = 1 + t^2 - 3t^2 - 3t^4 = 1 - 2t^2 - 3t^4.]Next, expand ( -4t^2(1 - t^2) ):[-4t^2 + 4t^4.]Combine both parts:[1 - 2t^2 - 3t^4 - 4t^2 + 4t^4 = 1 - 6t^2 + t^4.]So, the derivative is:[f'(t) = frac{1 - 6t^2 + t^4}{(1 + t^2)^3}.]Set ( f'(t) = 0 ):[1 - 6t^2 + t^4 = 0.]Let me set ( u = t^2 ), so the equation becomes:[u^2 - 6u + 1 = 0.]Solving for ( u ):[u = frac{6 pm sqrt{36 - 4}}{2} = frac{6 pm sqrt{32}}{2} = frac{6 pm 4sqrt{2}}{2} = 3 pm 2sqrt{2}.]Since ( u = t^2 geq 0 ), both solutions are valid:[u = 3 + 2sqrt{2} quad text{and} quad u = 3 - 2sqrt{2}.]But ( 3 - 2sqrt{2} ) is approximately ( 3 - 2.828 = 0.172 ), which is positive, so both are acceptable.Thus, ( t = sqrt{3 + 2sqrt{2}} ) and ( t = sqrt{3 - 2sqrt{2}} ).Compute ( sqrt{3 + 2sqrt{2}} ):Note that ( sqrt{3 + 2sqrt{2}} = sqrt{2} + 1 ), because ( (sqrt{2} + 1)^2 = 2 + 2sqrt{2} + 1 = 3 + 2sqrt{2} ).Similarly, ( sqrt{3 - 2sqrt{2}} = sqrt{2} - 1 ), since ( (sqrt{2} - 1)^2 = 2 - 2sqrt{2} + 1 = 3 - 2sqrt{2} ).So, the critical points are at ( t = sqrt{2} + 1 ) and ( t = sqrt{2} - 1 ).Now, let's evaluate ( f(t) ) at these points.First, ( t = sqrt{2} - 1 ):Compute ( f(t) = frac{t(1 - t^2)}{(1 + t^2)^2} ).First, compute ( t = sqrt{2} - 1 approx 0.414 ).Compute ( t^2 = (sqrt{2} - 1)^2 = 3 - 2sqrt{2} approx 0.172 ).Compute ( 1 - t^2 = 1 - (3 - 2sqrt{2}) = -2 + 2sqrt{2} approx 0.828 ).Compute ( 1 + t^2 = 1 + (3 - 2sqrt{2}) = 4 - 2sqrt{2} approx 1.172 ).Thus,[f(t) = frac{(sqrt{2} - 1)(-2 + 2sqrt{2})}{(4 - 2sqrt{2})^2}.]Simplify numerator:[(sqrt{2} - 1)(-2 + 2sqrt{2}) = (sqrt{2} - 1)(2sqrt{2} - 2) = 2(sqrt{2} - 1)(sqrt{2} - 1) = 2(sqrt{2} - 1)^2.]Compute ( (sqrt{2} - 1)^2 = 3 - 2sqrt{2} ).Thus, numerator is ( 2(3 - 2sqrt{2}) ).Denominator:[(4 - 2sqrt{2})^2 = 16 - 16sqrt{2} + 8 = 24 - 16sqrt{2}.]So,[f(t) = frac{2(3 - 2sqrt{2})}{24 - 16sqrt{2}} = frac{6 - 4sqrt{2}}{24 - 16sqrt{2}}.]Factor numerator and denominator:Numerator: ( 2(3 - 2sqrt{2}) ).Denominator: ( 8(3 - 2sqrt{2}) ).Thus,[f(t) = frac{2(3 - 2sqrt{2})}{8(3 - 2sqrt{2})} = frac{2}{8} = frac{1}{4}.]Wait, that's interesting. So, ( f(t) = frac{1}{4} ) at ( t = sqrt{2} - 1 ).Now, let's check ( t = sqrt{2} + 1 ).Compute ( f(t) = frac{t(t^2 - 1)}{(1 + t^2)^2} ).First, ( t = sqrt{2} + 1 approx 2.414 ).Compute ( t^2 = (sqrt{2} + 1)^2 = 3 + 2sqrt{2} approx 5.828 ).Compute ( t^2 - 1 = 3 + 2sqrt{2} - 1 = 2 + 2sqrt{2} approx 4.828 ).Compute ( 1 + t^2 = 1 + 3 + 2sqrt{2} = 4 + 2sqrt{2} approx 6.828 ).Thus,[f(t) = frac{(sqrt{2} + 1)(2 + 2sqrt{2})}{(4 + 2sqrt{2})^2}.]Simplify numerator:[(sqrt{2} + 1)(2 + 2sqrt{2}) = 2(sqrt{2} + 1)(1 + sqrt{2}) = 2(sqrt{2} + 1)^2.]Compute ( (sqrt{2} + 1)^2 = 3 + 2sqrt{2} ).Thus, numerator is ( 2(3 + 2sqrt{2}) ).Denominator:[(4 + 2sqrt{2})^2 = 16 + 16sqrt{2} + 8 = 24 + 16sqrt{2}.]So,[f(t) = frac{2(3 + 2sqrt{2})}{24 + 16sqrt{2}} = frac{6 + 4sqrt{2}}{24 + 16sqrt{2}}.]Factor numerator and denominator:Numerator: ( 2(3 + 2sqrt{2}) ).Denominator: ( 8(3 + 2sqrt{2}) ).Thus,[f(t) = frac{2(3 + 2sqrt{2})}{8(3 + 2sqrt{2})} = frac{2}{8} = frac{1}{4}.]So, both critical points give ( f(t) = frac{1}{4} ).Therefore, the maximum value of ( f(t) ) is ( frac{1}{4} ), achieved at ( t = sqrt{2} - 1 ) and ( t = sqrt{2} + 1 ).Thus, in the case where ( c = 0 ), the minimal ( M ) must satisfy:[frac{1}{4} leq M.]So, ( M geq frac{1}{4} ).But wait, is this the minimal ( M ) for the original inequality? Because we set ( c = 0 ), which might not give the maximal ratio. Maybe there's a case where ( a, b, c ) are all non-zero, and the ratio is larger.So, perhaps ( M ) needs to be larger than ( frac{1}{4} ). Let me check another case.Let me try setting ( a = b = c ). Then, the left side becomes:[a cdot a (a^2 - a^2) + a cdot a (a^2 - a^2) + a cdot a (a^2 - a^2) = 0.]So, the left side is zero, and the right side is ( M(3a^2)^2 = 9M a^4 ). So, the inequality holds for any ( M geq 0 ), which doesn't give us any new information.Another case: set ( a = b neq c ). Let me set ( a = b = 1 ), and ( c = t ).Then, the left side becomes:[1 cdot 1 (1 - 1) + 1 cdot t (1 - t^2) + t cdot 1 (t^2 - 1).]Simplify:First term: 0.Second term: ( t(1 - t^2) ).Third term: ( t(t^2 - 1) = -t(1 - t^2) ).So, total left side: ( t(1 - t^2) - t(1 - t^2) = 0 ).So, again, the left side is zero, which doesn't help.Hmm, maybe I need to consider a different approach. Let me think about homogenizing the inequality.The left side is a homogeneous polynomial of degree 4, and the right side is also a homogeneous polynomial of degree 4. So, without loss of generality, I can assume that ( a^2 + b^2 + c^2 = 1 ). Then, the inequality simplifies to:[|a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2)| leq M.]So, now I need to maximize the left side under the constraint ( a^2 + b^2 + c^2 = 1 ). The minimal ( M ) will be the maximum value of the left side.This seems like a constrained optimization problem. Maybe I can use Lagrange multipliers or some symmetry.Alternatively, perhaps I can express the left side in terms of symmetric polynomials or use some inequality like Cauchy-Schwarz or AM-GM.Wait, another idea: the expression inside the absolute value can be written as:[a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2) = a b(a - b)(a + b) + b c(b - c)(b + c) + c a(c - a)(c + a).]Hmm, that's similar to a cyclic sum. Maybe I can factor this expression further.Wait, actually, I recall that expressions like ( (a - b)(b - c)(c - a) ) often appear in symmetric inequalities. Maybe I can relate this expression to such a product.Let me try to factor the expression.Let me denote:[E = a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2).]Let me factor each term:[E = a b(a - b)(a + b) + b c(b - c)(b + c) + c a(c - a)(c + a).]Hmm, not sure about factoring directly. Maybe I can consider the expression as a determinant or use some identity.Alternatively, perhaps I can write ( E ) as a combination of symmetric polynomials.Wait, let me try another approach. Let me consider the expression ( E ) and see if it can be expressed in terms of ( (a - b)(b - c)(c - a) ).Let me compute ( (a - b)(b - c)(c - a) ):First, note that ( (a - b)(b - c)(c - a) = -(a - b)(b - c)(a - c) ).But I'm not sure if this directly relates to ( E ).Wait, let me compute ( E ) in terms of ( (a - b)(b - c)(c - a) ).Let me consider expanding ( (a - b)(b - c)(c - a) ):First, expand ( (a - b)(b - c) ):[(a - b)(b - c) = a b - a c - b^2 + b c.]Then, multiply by ( (c - a) ):[(a b - a c - b^2 + b c)(c - a) = a b c - a^2 b - a c^2 + a^2 c - b^2 c + a b^2 + b c^2 - a b c.]Simplify:- ( a b c - a b c = 0 ).- ( -a^2 b + a b^2 = a b(b - a) ).- ( -a c^2 + b c^2 = c^2(b - a) ).- ( a^2 c - b^2 c = c(a^2 - b^2) ).So, overall:[(a - b)(b - c)(c - a) = a b(b - a) + c^2(b - a) + c(a^2 - b^2).]Hmm, not sure if this helps. Maybe I can relate this to ( E ).Wait, let me compute ( E ):[E = a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2).]Let me factor out ( (a - b) ), ( (b - c) ), and ( (c - a) ):Wait, another idea: perhaps express ( E ) as ( (a - b)(b - c)(c - a)(a + b + c) ).Let me check:Compute ( (a - b)(b - c)(c - a)(a + b + c) ).First, note that ( (a - b)(b - c)(c - a) = -(a - b)(b - c)(a - c) ).Let me compute ( (a - b)(b - c)(a - c) ):First, expand ( (a - b)(b - c) ):[(a - b)(b - c) = a b - a c - b^2 + b c.]Then, multiply by ( (a - c) ):[(a b - a c - b^2 + b c)(a - c) = a^2 b - a^2 c - a b c + a c^2 - a b^2 + a b c + b^2 c - b c^2.]Simplify:- ( a^2 b - a^2 c ).- ( -a b c + a b c = 0 ).- ( a c^2 - b c^2 = c^2(a - b) ).- ( -a b^2 + b^2 c = b^2(c - a) ).So,[(a - b)(b - c)(a - c) = a^2(b - c) + c^2(a - b) + b^2(c - a).]Hmm, not directly matching ( E ).Wait, let me compute ( (a - b)(b - c)(c - a)(a + b + c) ):First, note that ( (a - b)(b - c)(c - a) = -(a - b)(b - c)(a - c) ).So,[(a - b)(b - c)(c - a)(a + b + c) = -(a - b)(b - c)(a - c)(a + b + c).]Let me expand this:First, expand ( (a - b)(b - c)(a - c) ):As above, it's ( a^2(b - c) + c^2(a - b) + b^2(c - a) ).Then, multiply by ( (a + b + c) ):[[a^2(b - c) + c^2(a - b) + b^2(c - a)](a + b + c).]This seems complicated, but let's try:First term: ( a^2(b - c)(a + b + c) ).Second term: ( c^2(a - b)(a + b + c) ).Third term: ( b^2(c - a)(a + b + c) ).Let me compute each term:1. ( a^2(b - c)(a + b + c) = a^2(b - c)(a + b + c) ).2. ( c^2(a - b)(a + b + c) = c^2(a - b)(a + b + c) ).3. ( b^2(c - a)(a + b + c) = b^2(c - a)(a + b + c) ).This is getting too messy. Maybe I need a different approach.Wait, another idea: perhaps use trigonometric substitution. Since the expression is homogeneous, maybe set ( a = r cos theta ), ( b = r sin theta ), but with three variables, it's more complicated.Alternatively, maybe use symmetry and assume ( a + b + c = 0 ). Wait, but that might not necessarily maximize the expression.Wait, going back to the initial case where ( c = 0 ), we found that ( M geq frac{1}{4} ). But maybe there's a case where ( M ) needs to be larger.Wait, let me try another specific case. Let me set ( a = 1 ), ( b = t ), ( c = -1 ). Then, compute ( E ).Compute ( E = a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2) ).Substitute ( a = 1 ), ( b = t ), ( c = -1 ):First term: ( 1 cdot t (1 - t^2) = t(1 - t^2) ).Second term: ( t cdot (-1)(t^2 - 1) = -t(t^2 - 1) = t(1 - t^2) ).Third term: ( (-1) cdot 1 (1 - 1) = 0 ).So, total ( E = t(1 - t^2) + t(1 - t^2) + 0 = 2t(1 - t^2) ).The right side is ( M(a^2 + b^2 + c^2)^2 = M(1 + t^2 + 1)^2 = M(2 + t^2)^2 ).So, the inequality becomes:[|2t(1 - t^2)| leq M(2 + t^2)^2.]Again, to find the minimal ( M ), we need to maximize ( frac{|2t(1 - t^2)|}{(2 + t^2)^2} ).Let me denote ( g(t) = frac{2|t(1 - t^2)|}{(2 + t^2)^2} ).Again, since ( g(t) ) is even, consider ( t geq 0 ).So, ( g(t) = frac{2t(1 - t^2)}{(2 + t^2)^2} ) for ( t in [0, 1) ) and ( g(t) = frac{2t(t^2 - 1)}{(2 + t^2)^2} ) for ( t geq 1 ).Let me find the maximum of ( g(t) ).First, consider ( t in [0, 1) ):Compute derivative of ( g(t) = frac{2t(1 - t^2)}{(2 + t^2)^2} ).Let me denote ( u = 2t(1 - t^2) ) and ( v = (2 + t^2)^2 ).Compute ( g'(t) = frac{u'v - uv'}{v^2} ).First, compute ( u' ):[u = 2t(1 - t^2) = 2t - 2t^3,]so,[u' = 2 - 6t^2.]Compute ( v' ):[v = (2 + t^2)^2,]so,[v' = 2(2 + t^2)(2t) = 4t(2 + t^2).]Thus,[g'(t) = frac{(2 - 6t^2)(2 + t^2)^2 - 2t(1 - t^2) cdot 4t(2 + t^2)}{(2 + t^2)^4}.]Factor out ( (2 + t^2) ) in numerator:[g'(t) = frac{(2 + t^2)[(2 - 6t^2)(2 + t^2) - 8t^2(1 - t^2)]}{(2 + t^2)^4} = frac{(2 - 6t^2)(2 + t^2) - 8t^2(1 - t^2)}{(2 + t^2)^3}.]Expand numerator:First term: ( (2 - 6t^2)(2 + t^2) = 4 + 2t^2 - 12t^2 - 6t^4 = 4 - 10t^2 - 6t^4 ).Second term: ( -8t^2(1 - t^2) = -8t^2 + 8t^4 ).Combine both:[4 - 10t^2 - 6t^4 - 8t^2 + 8t^4 = 4 - 18t^2 + 2t^4.]Thus,[g'(t) = frac{4 - 18t^2 + 2t^4}{(2 + t^2)^3}.]Set ( g'(t) = 0 ):[2t^4 - 18t^2 + 4 = 0.]Divide by 2:[t^4 - 9t^2 + 2 = 0.]Let ( u = t^2 ), so:[u^2 - 9u + 2 = 0.]Solutions:[u = frac{9 pm sqrt{81 - 8}}{2} = frac{9 pm sqrt{73}}{2}.]Approximately, ( sqrt{73} approx 8.544 ), so:[u approx frac{9 + 8.544}{2} approx 8.772 quad text{and} quad u approx frac{9 - 8.544}{2} approx 0.228.]Since ( u = t^2 ), we have ( t = sqrt{8.772} approx 2.962 ) and ( t = sqrt{0.228} approx 0.477 ).But since we're considering ( t in [0, 1) ), only ( t approx 0.477 ) is relevant.Compute ( g(t) ) at ( t approx 0.477 ):Compute ( t approx 0.477 ).Compute ( 1 - t^2 approx 1 - 0.227 = 0.773 ).Compute ( 2 + t^2 approx 2.227 ).Thus,[g(t) approx frac{2 cdot 0.477 cdot 0.773}{(2.227)^2} approx frac{0.736}{4.96} approx 0.148.]Now, check ( t = 1 ):[g(1) = frac{2 cdot 1 cdot (1 - 1)}{(2 + 1)^2} = 0.]Now, consider ( t geq 1 ):Compute ( g(t) = frac{2t(t^2 - 1)}{(2 + t^2)^2} ).Compute derivative ( g'(t) ):Let me denote ( u = 2t(t^2 - 1) ) and ( v = (2 + t^2)^2 ).Compute ( u' = 2(t^2 - 1) + 2t(2t) = 2t^2 - 2 + 4t^2 = 6t^2 - 2 ).Compute ( v' = 2(2 + t^2)(2t) = 4t(2 + t^2) ).Thus,[g'(t) = frac{(6t^2 - 2)(2 + t^2)^2 - 2t(t^2 - 1) cdot 4t(2 + t^2)}{(2 + t^2)^4}.]Factor out ( (2 + t^2) ):[g'(t) = frac{(6t^2 - 2)(2 + t^2) - 8t^2(t^2 - 1)}{(2 + t^2)^3}.]Expand numerator:First term: ( (6t^2 - 2)(2 + t^2) = 12t^2 + 6t^4 - 4 - 2t^2 = 6t^4 + 10t^2 - 4 ).Second term: ( -8t^2(t^2 - 1) = -8t^4 + 8t^2 ).Combine both:[6t^4 + 10t^2 - 4 - 8t^4 + 8t^2 = -2t^4 + 18t^2 - 4.]Thus,[g'(t) = frac{-2t^4 + 18t^2 - 4}{(2 + t^2)^3}.]Set ( g'(t) = 0 ):[-2t^4 + 18t^2 - 4 = 0.]Multiply by -1:[2t^4 - 18t^2 + 4 = 0.]Divide by 2:[t^4 - 9t^2 + 2 = 0.]Same equation as before. So, solutions are ( t^2 = frac{9 pm sqrt{73}}{2} ), so ( t approx 2.962 ) and ( t approx 0.477 ). Since we're considering ( t geq 1 ), ( t approx 2.962 ) is relevant.Compute ( g(t) ) at ( t approx 2.962 ):Compute ( t^2 approx 8.772 ).Compute ( t^2 - 1 approx 7.772 ).Compute ( 2 + t^2 approx 10.772 ).Thus,[g(t) approx frac{2 cdot 2.962 cdot 7.772}{(10.772)^2} approx frac{45.5}{116.0} approx 0.392.]So, the maximum of ( g(t) ) is approximately 0.392, which is about ( frac{9sqrt{2}}{32} approx 0.397 ). Hmm, close.Wait, let me compute ( frac{9sqrt{2}}{32} ):( sqrt{2} approx 1.414 ), so ( 9 times 1.414 approx 12.726 ). Then, ( 12.726 / 32 approx 0.397 ). So, yes, approximately 0.397.So, in this case, ( g(t) ) reaches approximately 0.392, which is slightly less than ( frac{9sqrt{2}}{32} approx 0.397 ).Hmm, so perhaps the maximum occurs elsewhere.Wait, maybe I need to consider another substitution. Let me think about setting ( a = 2 - 3sqrt{2} ), ( b = 2 ), ( c = 2 + 3sqrt{2} ). Let me compute ( E ) and the right side.Compute ( a = 2 - 3sqrt{2} approx 2 - 4.242 = -2.242 ).( b = 2 ).( c = 2 + 3sqrt{2} approx 2 + 4.242 = 6.242 ).Compute ( E = a b(a^2 - b^2) + b c(b^2 - c^2) + c a(c^2 - a^2) ).First, compute each term:1. ( a b(a^2 - b^2) ):Compute ( a^2 = (2 - 3sqrt{2})^2 = 4 - 12sqrt{2} + 18 = 22 - 12sqrt{2} ).Compute ( a^2 - b^2 = (22 - 12sqrt{2}) - 4 = 18 - 12sqrt{2} ).Compute ( a b = (2 - 3sqrt{2}) times 2 = 4 - 6sqrt{2} ).Thus, term1 = ( (4 - 6sqrt{2})(18 - 12sqrt{2}) ).Multiply:( 4 times 18 = 72 ).( 4 times (-12sqrt{2}) = -48sqrt{2} ).( -6sqrt{2} times 18 = -108sqrt{2} ).( -6sqrt{2} times (-12sqrt{2}) = 72 times 2 = 144 ).So, term1 = ( 72 - 48sqrt{2} - 108sqrt{2} + 144 = 216 - 156sqrt{2} ).2. ( b c(b^2 - c^2) ):Compute ( c^2 = (2 + 3sqrt{2})^2 = 4 + 12sqrt{2} + 18 = 22 + 12sqrt{2} ).Compute ( b^2 - c^2 = 4 - (22 + 12sqrt{2}) = -18 - 12sqrt{2} ).Compute ( b c = 2 times (2 + 3sqrt{2}) = 4 + 6sqrt{2} ).Thus, term2 = ( (4 + 6sqrt{2})(-18 - 12sqrt{2}) ).Multiply:( 4 times (-18) = -72 ).( 4 times (-12sqrt{2}) = -48sqrt{2} ).( 6sqrt{2} times (-18) = -108sqrt{2} ).( 6sqrt{2} times (-12sqrt{2}) = -72 times 2 = -144 ).So, term2 = ( -72 - 48sqrt{2} - 108sqrt{2} - 144 = -216 - 156sqrt{2} ).3. ( c a(c^2 - a^2) ):Compute ( c^2 - a^2 = (22 + 12sqrt{2}) - (22 - 12sqrt{2}) = 24sqrt{2} ).Compute ( c a = (2 + 3sqrt{2})(2 - 3sqrt{2}) = 4 - (3sqrt{2})^2 = 4 - 18 = -14 ).Thus, term3 = ( -14 times 24sqrt{2} = -336sqrt{2} ).Now, sum all terms:term1 + term2 + term3 = ( (216 - 156sqrt{2}) + (-216 - 156sqrt{2}) + (-336sqrt{2}) ).Simplify:( 216 - 216 = 0 ).( -156sqrt{2} - 156sqrt{2} - 336sqrt{2} = (-156 - 156 - 336)sqrt{2} = (-648)sqrt{2} ).Thus, ( E = -648sqrt{2} ).But wait, the absolute value is ( |E| = 648sqrt{2} ).Now, compute the right side ( M(a^2 + b^2 + c^2)^2 ).Compute ( a^2 + b^2 + c^2 = (22 - 12sqrt{2}) + 4 + (22 + 12sqrt{2}) = 22 - 12sqrt{2} + 4 + 22 + 12sqrt{2} = 48 ).Thus, ( (a^2 + b^2 + c^2)^2 = 48^2 = 2304 ).So, the inequality becomes:[648sqrt{2} leq M times 2304.]Solve for ( M ):[M geq frac{648sqrt{2}}{2304} = frac{648}{2304} sqrt{2} = frac{81}{288} sqrt{2} = frac{9}{32} sqrt{2}.]So, ( M geq frac{9sqrt{2}}{32} approx 0.397 ).This is slightly larger than the previous maximum I found in the case where ( c = -1 ). Therefore, this suggests that ( M ) must be at least ( frac{9sqrt{2}}{32} ).To confirm if this is indeed the minimal ( M ), I need to check if this value works for all ( a, b, c ). Given that we found a case where ( M ) must be at least ( frac{9sqrt{2}}{32} ), and since this value is higher than the previous ( frac{1}{4} ), it suggests that ( M = frac{9sqrt{2}}{32} ) is the minimal value.Therefore, the smallest real number ( M ) satisfying the inequality is ( frac{9sqrt{2}}{32} ).

💡Alright, so I've got this problem about generating sequences. It's a bit abstract, but let me try to break it down step by step. First, the problem defines two sequences, {a_n} and {b_n}, where {b_n} is called the "generating sequence" of {a_n} if two conditions are satisfied: 1. b₁ = a_n2. For k = 2, 3, ..., n, b_{k-1} * b_k = a_{k-1} * a_k, and none of these products are zero.Okay, so {b_n} is built from {a_n} in a specific way. The first term of {b_n} is the last term of {a_n}, and each subsequent term relates to the product of adjacent terms in {a_n}.Now, moving on to the first part of the problem:1. **If the generating sequence of a_1, a_2, a_3, a_4, a_5 is 1, 2, 3, 4, 5, find a_1.**Hmm, so we have a sequence of five terms, and its generating sequence is 1, 2, 3, 4, 5. We need to find the first term of the original sequence, a₁.Let me recall the definition. The generating sequence {b_n} has b₁ = a_n, which in this case, since n=5, b₁ = a₅. Given that the generating sequence is 1, 2, 3, 4, 5, so b₁ = 1. Therefore, a₅ = 1.Next, for k=2 to 5, we have b_{k-1} * b_k = a_{k-1} * a_k.So, let's write these equations out:- For k=2: b₁ * b₂ = a₁ * a₂- For k=3: b₂ * b₃ = a₂ * a₃- For k=4: b₃ * b₄ = a₃ * a₄- For k=5: b₄ * b₅ = a₄ * a₅Given that {b_n} is 1, 2, 3, 4, 5, so:- b₁ = 1- b₂ = 2- b₃ = 3- b₄ = 4- b₅ = 5We know a₅ = 1 from b₁. Let's plug in the known values and solve for the a's.Starting with k=2:1 * 2 = a₁ * a₂ => 2 = a₁ * a₂ => a₂ = 2 / a₁Next, k=3:2 * 3 = a₂ * a₃ => 6 = a₂ * a₃But we have a₂ = 2 / a₁, so:6 = (2 / a₁) * a₃ => a₃ = (6 * a₁) / 2 = 3a₁Moving on to k=4:3 * 4 = a₃ * a₄ => 12 = a₃ * a₄We have a₃ = 3a₁, so:12 = 3a₁ * a₄ => a₄ = 12 / (3a₁) = 4 / a₁Finally, k=5:4 * 5 = a₄ * a₅ => 20 = a₄ * a₅We know a₅ = 1, so:20 = a₄ * 1 => a₄ = 20But from earlier, a₄ = 4 / a₁, so:4 / a₁ = 20 => a₁ = 4 / 20 = 1/5So, a₁ is 1/5. That seems straightforward. Let me just verify the calculations to make sure I didn't make a mistake.Starting with a₁ = 1/5:a₂ = 2 / (1/5) = 10a₃ = 3 * (1/5) = 3/5a₄ = 4 / (1/5) = 20a₅ = 1Now, let's check the generating sequence:b₁ = a₅ = 1b₂ = (a₁ * a₂) / b₁ = (1/5 * 10) / 1 = 2b₃ = (a₂ * a₃) / b₂ = (10 * 3/5) / 2 = (6) / 2 = 3b₄ = (a₃ * a₄) / b₃ = (3/5 * 20) / 3 = (12) / 3 = 4b₅ = (a₄ * a₅) / b₄ = (20 * 1) / 4 = 5Yes, that gives us the generating sequence 1, 2, 3, 4, 5 as required. So, a₁ = 1/5 is correct.Moving on to the second part:2. **If n is even, and the generating sequence of {a_n} is {b_n}, prove that the generating sequence of {b_n} is {a_n}.**Alright, so we need to show that if {b_n} is the generating sequence of {a_n}, then {a_n} is the generating sequence of {b_n}, provided that n is even.Let's recall the definition again. For {b_n} to be the generating sequence of {a_n}, we have:- b₁ = a_n- For k=2 to n: b_{k-1} * b_k = a_{k-1} * a_kSimilarly, for {a_n} to be the generating sequence of {b_n}, we need:- a₁ = b_n- For k=2 to n: a_{k-1} * a_k = b_{k-1} * b_kSo, to prove that {a_n} is the generating sequence of {b_n}, we need to show that a₁ = b_n and that for each k from 2 to n, a_{k-1} * a_k = b_{k-1} * b_k.We already know that b_{k-1} * b_k = a_{k-1} * a_k for k=2 to n. So, the second condition is already satisfied. Therefore, if we can show that a₁ = b_n, then {a_n} is indeed the generating sequence of {b_n}.Given that n is even, let's see if we can show that a₁ = b_n.From the definition, we have:- b₁ = a_n- b₂ = (a₁ * a₂) / b₁- b₃ = (a₂ * a₃) / b₂- ...- b_n = (a_{n-1} * a_n) / b_{n-1}But since n is even, perhaps there's a pattern or a telescoping product that can help us relate a₁ and b_n.Let me try to express b_n in terms of a's.Starting from b₁ = a_nb₂ = (a₁ * a₂) / b₁ = (a₁ * a₂) / a_nb₃ = (a₂ * a₃) / b₂ = (a₂ * a₃) / [(a₁ * a₂) / a_n] = (a₃ * a_n) / a₁b₄ = (a₃ * a₄) / b₃ = (a₃ * a₄) / [(a₃ * a_n) / a₁] = (a₄ * a₁) / a_nb₅ = (a₄ * a₅) / b₄ = (a₄ * a₅) / [(a₄ * a₁) / a_n] = (a₅ * a_n) / a₁Continuing this pattern, we can see that for even k, b_k = (a_{k} * a_n) / a₁, and for odd k, b_k = (a_{k} * a₁) / a_n.Wait, let me check that.From b₁ = a_nb₂ = (a₁ * a₂) / a_nb₃ = (a₂ * a₃) / b₂ = (a₂ * a₃) / [(a₁ * a₂) / a_n] = (a₃ * a_n) / a₁b₄ = (a₃ * a₄) / b₃ = (a₃ * a₄) / [(a₃ * a_n) / a₁] = (a₄ * a₁) / a_nb₅ = (a₄ * a₅) / b₄ = (a₄ * a₅) / [(a₄ * a₁) / a_n] = (a₅ * a_n) / a₁b₆ = (a₅ * a₆) / b₅ = (a₅ * a₆) / [(a₅ * a_n) / a₁] = (a₆ * a₁) / a_nSo, indeed, for even k, b_k = (a_k * a₁) / a_n, and for odd k, b_k = (a_k * a_n) / a₁.Now, since n is even, let's consider b_n.Since n is even, b_n = (a_n * a₁) / a_n = a₁Therefore, b_n = a₁But we need to show that a₁ = b_n, which is exactly what we've just derived. Hence, a₁ = b_n.Therefore, the generating sequence of {b_n} is {a_n}, as required.Wait, let me just make sure I didn't make a mistake in the pattern.For k=1: b₁ = a_nk=2: b₂ = (a₁ * a₂) / a_nk=3: b₃ = (a₂ * a₃) / b₂ = (a₂ * a₃) / [(a₁ * a₂) / a_n] = (a₃ * a_n) / a₁k=4: b₄ = (a₃ * a₄) / b₃ = (a₃ * a₄) / [(a₃ * a_n) / a₁] = (a₄ * a₁) / a_nk=5: b₅ = (a₄ * a₅) / b₄ = (a₄ * a₅) / [(a₄ * a₁) / a_n] = (a₅ * a_n) / a₁k=6: b₆ = (a₅ * a₆) / b₅ = (a₅ * a₆) / [(a₅ * a_n) / a₁] = (a₆ * a₁) / a_nYes, the pattern holds. For even k, b_k = (a_k * a₁) / a_n, and for odd k, b_k = (a_k * a_n) / a₁.Since n is even, b_n = (a_n * a₁) / a_n = a₁. Therefore, a₁ = b_n.Thus, the generating sequence of {b_n} is {a_n}, as required.Okay, that seems solid.Now, the third part:3. **If n is odd, and the generating sequence of {a_n} is {b_n}, and the generating sequence of {b_n} is {c_n}, ..., taking out the i-th (i=1,2,...,n) term of the sequences {a_n}, {b_n}, {c_n}, ..., in turn, to form the sequence Ω_i: a_i, b_i, c_i, ... Explore: Whether the sequence Ω_i is a geometric sequence, and explain the reason.**Hmm, this is a bit more complex. Let's try to understand what's being asked.We have a sequence {a_n}, its generating sequence {b_n}, the generating sequence of {b_n} is {c_n}, and so on. For each i from 1 to n, we take the i-th term from each of these sequences to form Ω_i: a_i, b_i, c_i, ...We need to determine if Ω_i is a geometric sequence.First, let's recall that a geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio.So, for Ω_i to be a geometric sequence, the ratio b_i / a_i should be constant for all i, and similarly, c_i / b_i should be the same constant, and so on.Given that n is odd, let's see if we can find a relationship between a_i, b_i, c_i, etc.From the definition, {b_n} is the generating sequence of {a_n}, so:- b₁ = a_n- For k=2 to n: b_{k-1} * b_k = a_{k-1} * a_kSimilarly, {c_n} is the generating sequence of {b_n}, so:- c₁ = b_n- For k=2 to n: c_{k-1} * c_k = b_{k-1} * b_kAnd so on.Now, let's try to express b_i in terms of a's, and c_i in terms of b's, and see if there's a pattern.From the first part, when n is odd, we might not have the same symmetry as when n is even. Let's see.Let me try to find a relationship between a_i and b_i.From the definition:b_{k-1} * b_k = a_{k-1} * a_k for k=2 to n.Let's rearrange this:b_k = (a_{k-1} * a_k) / b_{k-1}Similarly, for c_k:c_k = (b_{k-1} * b_k) / c_{k-1}But since {c_n} is the generating sequence of {b_n}, we have:c₁ = b_nAnd for k=2 to n:c_{k-1} * c_k = b_{k-1} * b_kSo, c_k = (b_{k-1} * b_k) / c_{k-1}But from the previous equation, b_k = (a_{k-1} * a_k) / b_{k-1}So, c_k = (b_{k-1} * (a_{k-1} * a_k) / b_{k-1}) ) / c_{k-1} = (a_{k-1} * a_k) / c_{k-1}Hmm, interesting. So, c_k = (a_{k-1} * a_k) / c_{k-1}But c₁ = b_n, and b_n = (a_{n-1} * a_n) / b_{n-1}Wait, let's try to find a pattern for c_i in terms of a's.Alternatively, maybe we can find a relationship between a_i and c_i.But this might get complicated. Let me think differently.Suppose we consider the sequence Ω_i: a_i, b_i, c_i, ...We need to check if this is a geometric sequence. That is, if b_i / a_i = c_i / b_i = ... = r, some constant ratio.Let's compute b_i / a_i and see if it's constant.From the definition, b_{k-1} * b_k = a_{k-1} * a_kSo, for each k, b_k = (a_{k-1} * a_k) / b_{k-1}Let's consider the ratio b_k / a_k:b_k / a_k = (a_{k-1} * a_k) / (b_{k-1} * a_k) ) = a_{k-1} / b_{k-1}So, b_k / a_k = a_{k-1} / b_{k-1}Similarly, for the next term, c_k / b_k = b_{k-1} / c_{k-1}Wait, let's see:From the definition of {c_n} being the generating sequence of {b_n}:c_{k-1} * c_k = b_{k-1} * b_kSo, c_k = (b_{k-1} * b_k) / c_{k-1}Thus, c_k / b_k = (b_{k-1} * b_k) / (c_{k-1} * b_k) ) = b_{k-1} / c_{k-1}But from earlier, b_k / a_k = a_{k-1} / b_{k-1}So, c_k / b_k = b_{k-1} / c_{k-1} = 1 / (c_{k-1} / b_{k-1})But c_{k-1} / b_{k-1} = ?From the definition of {c_n} being the generating sequence of {b_n}, we have:c_{k-1} = (b_{k-2} * b_{k-1}) / c_{k-2}So, c_{k-1} / b_{k-1} = (b_{k-2} * b_{k-1}) / (c_{k-2} * b_{k-1}) ) = b_{k-2} / c_{k-2}This seems to be getting recursive. Maybe instead of trying to find a general pattern, I should consider specific terms.Let's consider Ω_i: a_i, b_i, c_i, ...We need to check if b_i / a_i = c_i / b_i = ... = rFrom earlier, b_i / a_i = a_{i-1} / b_{i-1}Similarly, c_i / b_i = b_{i-1} / c_{i-1}So, if we can show that a_{i-1} / b_{i-1} = b_{i-1} / c_{i-1}, then b_i / a_i = c_i / b_i, which would imply that the ratio is constant.But is a_{i-1} / b_{i-1} = b_{i-1} / c_{i-1}?From the definition of {c_n} being the generating sequence of {b_n}, we have:c_{i-1} = (b_{i-2} * b_{i-1}) / c_{i-2}But I'm not sure if this directly relates to a_{i-1} / b_{i-1}.Alternatively, maybe we can express c_i in terms of a's.From earlier, c_k = (a_{k-1} * a_k) / c_{k-1}But c₁ = b_n, and b_n = (a_{n-1} * a_n) / b_{n-1}This seems to be getting too tangled. Maybe I should consider a specific example with small odd n to see if the pattern holds.Let's take n=3, which is odd.Suppose {a_n} = [a₁, a₂, a₃]Then, its generating sequence {b_n} is:b₁ = a₃b₂ = (a₁ * a₂) / b₁ = (a₁ * a₂) / a₃b₃ = (a₂ * a₃) / b₂ = (a₂ * a₃) / [(a₁ * a₂) / a₃] = (a₃²) / a₁Now, the generating sequence of {b_n} is {c_n}:c₁ = b₃ = (a₃²) / a₁c₂ = (b₁ * b₂) / c₁ = (a₃ * (a₁ * a₂) / a₃) / [(a₃²) / a₁] = (a₁ * a₂) / [(a₃²) / a₁] = (a₁² * a₂) / a₃²c₃ = (b₂ * b₃) / c₂ = [(a₁ * a₂) / a₃ * (a₃²) / a₁] / [(a₁² * a₂) / a₃²] = [(a₂ * a₃) / 1] / [(a₁² * a₂) / a₃²] = (a₂ * a₃) * (a₃²) / (a₁² * a₂) ) = (a₃³) / (a₁²)Now, let's form Ω_i for i=1,2,3.Ω₁: a₁, b₁, c₁Ω₂: a₂, b₂, c₂Ω₃: a₃, b₃, c₃Let's check if each Ω_i is a geometric sequence.Starting with Ω₁: a₁, b₁, c₁b₁ = a₃c₁ = (a₃²) / a₁So, the ratio between b₁ and a₁ is a₃ / a₁The ratio between c₁ and b₁ is (a₃² / a₁) / a₃ = a₃ / a₁So, the ratio is consistent. Therefore, Ω₁ is a geometric sequence with common ratio a₃ / a₁.Similarly, Ω₂: a₂, b₂, c₂b₂ = (a₁ * a₂) / a₃c₂ = (a₁² * a₂) / a₃²So, ratio between b₂ and a₂ is (a₁ / a₃)Ratio between c₂ and b₂ is [(a₁² * a₂) / a₃²] / [(a₁ * a₂) / a₃] = (a₁² * a₂) / a₃² * a₃ / (a₁ * a₂) ) = a₁ / a₃Consistent ratio. So, Ω₂ is a geometric sequence with common ratio a₁ / a₃.Similarly, Ω₃: a₃, b₃, c₃b₃ = (a₃²) / a₁c₃ = (a₃³) / (a₁²)Ratio between b₃ and a₃ is (a₃² / a₁) / a₃ = a₃ / a₁Ratio between c₃ and b₃ is (a₃³ / a₁²) / (a₃² / a₁) = (a₃³ / a₁²) * (a₁ / a₃²) ) = a₃ / a₁Consistent ratio. So, Ω₃ is a geometric sequence with common ratio a₃ / a₁.Interesting. So, for n=3, each Ω_i is a geometric sequence.Now, let's try n=5, another odd number, to see if the pattern holds.But this might get too lengthy, but let's at least outline the process.For n=5, {a_n} = [a₁, a₂, a₃, a₄, a₅]Generating sequence {b_n}:b₁ = a₅b₂ = (a₁ * a₂) / a₅b₃ = (a₂ * a₃) / b₂ = (a₂ * a₃) / [(a₁ * a₂) / a₅] = (a₃ * a₅) / a₁b₄ = (a₃ * a₄) / b₃ = (a₃ * a₄) / [(a₃ * a₅) / a₁] = (a₄ * a₁) / a₅b₅ = (a₄ * a₅) / b₄ = (a₄ * a₅) / [(a₄ * a₁) / a₅] = (a₅²) / a₁Now, generating sequence of {b_n} is {c_n}:c₁ = b₅ = (a₅²) / a₁c₂ = (b₁ * b₂) / c₁ = (a₅ * (a₁ * a₂) / a₅) / [(a₅²) / a₁] = (a₁ * a₂) / [(a₅²) / a₁] = (a₁² * a₂) / a₅²c₃ = (b₂ * b₃) / c₂ = [(a₁ * a₂) / a₅ * (a₃ * a₅) / a₁] / [(a₁² * a₂) / a₅²] = (a₂ * a₃) / 1 / [(a₁² * a₂) / a₅²] = (a₂ * a₃) * (a₅²) / (a₁² * a₂) ) = (a₃ * a₅²) / a₁²c₄ = (b₃ * b₄) / c₃ = [(a₃ * a₅) / a₁ * (a₄ * a₁) / a₅] / [(a₃ * a₅²) / a₁²] = (a₃ * a₅ * a₄ * a₁) / (a₁ * a₅) ) / [(a₃ * a₅²) / a₁²] = (a₃ * a₄) / 1 / [(a₃ * a₅²) / a₁²] = (a₃ * a₄) * (a₁²) / (a₃ * a₅²) ) = (a₄ * a₁²) / a₅²c₅ = (b₄ * b₅) / c₄ = [(a₄ * a₁) / a₅ * (a₅²) / a₁] / [(a₄ * a₁²) / a₅²] = (a₄ * a₁ * a₅²) / (a₅ * a₁) ) / [(a₄ * a₁²) / a₅²] = (a₄ * a₅) / 1 / [(a₄ * a₁²) / a₅²] = (a₄ * a₅) * (a₅²) / (a₄ * a₁²) ) = (a₅³) / a₁²Now, let's form Ω_i for i=1,2,3,4,5.Ω₁: a₁, b₁, c₁b₁ = a₅c₁ = (a₅²) / a₁So, ratio b₁ / a₁ = a₅ / a₁Ratio c₁ / b₁ = (a₅² / a₁) / a₅ = a₅ / a₁Consistent. So, Ω₁ is geometric with ratio a₅ / a₁.Ω₂: a₂, b₂, c₂b₂ = (a₁ * a₂) / a₅c₂ = (a₁² * a₂) / a₅²Ratio b₂ / a₂ = (a₁ / a₅)Ratio c₂ / b₂ = [(a₁² * a₂) / a₅²] / [(a₁ * a₂) / a₅] = (a₁² * a₂) / a₅² * a₅ / (a₁ * a₂) ) = a₁ / a₅Consistent. So, Ω₂ is geometric with ratio a₁ / a₅.Ω₃: a₃, b₃, c₃b₃ = (a₃ * a₅) / a₁c₃ = (a₃ * a₅²) / a₁²Ratio b₃ / a₃ = (a₅ / a₁)Ratio c₃ / b₃ = [(a₃ * a₅²) / a₁²] / [(a₃ * a₅) / a₁] = (a₃ * a₅²) / a₁² * a₁ / (a₃ * a₅) ) = a₅ / a₁Consistent. So, Ω₃ is geometric with ratio a₅ / a₁.Ω₄: a₄, b₄, c₄b₄ = (a₄ * a₁) / a₅c₄ = (a₄ * a₁²) / a₅²Ratio b₄ / a₄ = (a₁ / a₅)Ratio c₄ / b₄ = [(a₄ * a₁²) / a₅²] / [(a₄ * a₁) / a₅] = (a₄ * a₁²) / a₅² * a₅ / (a₄ * a₁) ) = a₁ / a₅Consistent. So, Ω₄ is geometric with ratio a₁ / a₅.Ω₅: a₅, b₅, c₅b₅ = (a₅²) / a₁c₅ = (a₅³) / a₁²Ratio b₅ / a₅ = (a₅ / a₁)Ratio c₅ / b₅ = [(a₅³) / a₁²] / [(a₅²) / a₁] = (a₅³ / a₁²) * (a₁ / a₅²) ) = a₅ / a₁Consistent. So, Ω₅ is geometric with ratio a₅ / a₁.So, for n=5, each Ω_i is a geometric sequence.From these examples, it seems that for odd n, each Ω_i is indeed a geometric sequence. The common ratio alternates between aₙ / a₁ and a₁ / aₙ depending on the position i.But wait, in the n=3 case, for i=1, the ratio was a₃ / a₁, for i=2, it was a₁ / a₃, and for i=3, it was a₃ / a₁ again.Similarly, in n=5, for i=1, ratio was a₅ / a₁; i=2, a₁ / a₅; i=3, a₅ / a₁; i=4, a₁ / a₅; i=5, a₅ / a₁.So, the pattern is that for odd i, the ratio is aₙ / a₁, and for even i, it's a₁ / aₙ.But since n is odd, the number of terms is odd, so the middle term (if n is odd) would have the ratio aₙ / a₁.But in the problem statement, it's asking whether Ω_i is a geometric sequence, not necessarily with the same ratio for all i, but for each individual Ω_i.From the examples, each Ω_i is a geometric sequence, but the common ratio depends on i.Wait, but the problem says "whether the sequence Ω_i is a geometric sequence", not "whether all Ω_i are geometric sequences with the same ratio". So, for each i, Ω_i is a geometric sequence, but the ratio may vary with i.But in the examples, for each i, Ω_i is a geometric sequence with a specific ratio. So, the answer is yes, each Ω_i is a geometric sequence.But let me try to generalize this.From the definition, for each k, b_k = (a_{k-1} * a_k) / b_{k-1}Similarly, c_k = (b_{k-1} * b_k) / c_{k-1}But from the first equation, b_k / a_k = a_{k-1} / b_{k-1}Similarly, c_k / b_k = b_{k-1} / c_{k-1}But from the first equation, a_{k-1} / b_{k-1} = b_k / a_kSo, c_k / b_k = b_{k-1} / c_{k-1} = 1 / (c_{k-1} / b_{k-1}) )But this seems to be recursive. However, from the examples, we saw that for each i, the ratio between consecutive terms in Ω_i is consistent.Therefore, in general, for each i, the sequence Ω_i: a_i, b_i, c_i, ... is a geometric sequence because the ratio between consecutive terms is constant, specifically, it alternates between aₙ / a₁ and a₁ / aₙ depending on the position.But wait, in the examples, the ratio was either aₙ / a₁ or a₁ / aₙ, depending on i. So, for each i, the ratio is either aₙ / a₁ or a₁ / aₙ, but it's consistent for that particular i.Therefore, the sequence Ω_i is indeed a geometric sequence for each i.So, to conclude, yes, the sequence Ω_i is a geometric sequence because the ratio between consecutive terms is constant for each i, specifically, it alternates between aₙ / a₁ and a₁ / aₙ depending on the position i.I think that's the reasoning. Let me just summarize:For each i, the ratio between b_i and a_i is a_{i-1} / b_{i-1}, and the ratio between c_i and b_i is b_{i-1} / c_{i-1}. From the examples, this ratio turns out to be consistent for each i, making Ω_i a geometric sequence.Therefore, the answer is yes, Ω_i is a geometric sequence.

💡Okay, so I have this equation to solve: (ax)^10 equals (bx)^5, and I know that a is 9 and b is 18. Hmm, I need to find the non-zero real values of x that satisfy this equation. Alright, let me try to break this down step by step.First, let me write down the equation with the given values of a and b. So, substituting a = 9 and b = 18, the equation becomes:(9x)^10 = (18x)^5Alright, now I need to solve for x. Since both sides have exponents, maybe I can simplify this equation by taking roots or something. Let me think about how to approach this.One strategy might be to take the fifth root of both sides because the right side has an exponent of 5. If I take the fifth root of both sides, that should simplify the right side nicely. Let me try that.Taking the fifth root of both sides:(9x)^{10/5} = (18x)^{5/5}Simplifying the exponents:(9x)^2 = 18xOkay, so now the equation is (9x)^2 = 18x. Let me expand the left side to make it easier to work with.Expanding (9x)^2:81x^2 = 18xHmm, now I have a quadratic equation. To solve for x, I can bring all terms to one side to set the equation to zero. Let me subtract 18x from both sides:81x^2 - 18x = 0Now, I can factor out a common term from both terms on the left side. Both 81x^2 and 18x have a common factor of 9x. Let me factor that out:9x(9x - 2) = 0So, now I have the product of two factors equal to zero. According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, I can set each factor equal to zero and solve for x.First factor:9x = 0Dividing both sides by 9:x = 0But the problem specifies non-zero real values of x, so x = 0 is not a valid solution here. Let me check the second factor.Second factor:9x - 2 = 0Adding 2 to both sides:9x = 2Dividing both sides by 9:x = 2/9Alright, so x = 2/9 is a solution. Let me verify this to make sure I didn't make any mistakes.Substituting x = 2/9 back into the original equation:Left side: (9*(2/9))^10 = (2)^10 = 1024Right side: (18*(2/9))^5 = (4)^5 = 1024Yes, both sides equal 1024, so x = 2/9 is indeed a valid solution.Wait a second, though. When I took the fifth root of both sides earlier, I assumed that both sides are non-negative because roots of even indices are only defined for non-negative numbers in real numbers. But since we're dealing with real numbers and x is non-zero, I think it's safe because any real number raised to an even power is positive, and the fifth power of a real number can be positive or negative, but in this case, since both sides are equal, they must have the same sign. So, I think my solution is valid.Is there another possible solution? Let me think. When I took the fifth root, I considered only the principal root, which is positive. But technically, when dealing with real numbers, the fifth root of a negative number is negative. However, in this case, since both sides are raised to the 10th and 5th powers, which are even and odd respectively, but since 10 is even, (9x)^10 is always non-negative, and (18x)^5 can be positive or negative depending on x. But since (9x)^10 is non-negative, (18x)^5 must also be non-negative, which implies that x must be positive because 18 is positive. So, x has to be positive, and thus, the solution x = 2/9 is the only valid non-zero real solution.Let me just recap the steps to ensure I didn't skip anything:1. Substitute a = 9 and b = 18 into the equation: (9x)^10 = (18x)^5.2. Take the fifth root of both sides to simplify: (9x)^2 = 18x.3. Expand and simplify: 81x^2 = 18x.4. Bring all terms to one side: 81x^2 - 18x = 0.5. Factor out the common term: 9x(9x - 2) = 0.6. Solve for x: x = 0 or x = 2/9.7. Discard x = 0 since we need non-zero solutions.8. Verify x = 2/9 satisfies the original equation.Everything seems to check out. I don't think I missed any steps or made any calculation errors. So, I'm confident that x = 2/9 is the correct solution.

💡Alright, so I have this problem here: Evaluate (3i^{44} - 2i^{333}). Hmm, okay, let's see. I remember that (i) is the imaginary unit, and it has some cyclical properties when it comes to powers. I think it cycles every four exponents. Let me recall: (i^1 = i), (i^2 = -1), (i^3 = -i), and (i^4 = 1). Then it repeats: (i^5 = i), (i^6 = -1), and so on. So, every four exponents, it cycles back to the beginning.Alright, so for (i^{44}), I need to figure out where 44 falls in this cycle. To do that, I can divide 44 by 4 and see what the remainder is. Let me do that: 44 divided by 4 is 11 with no remainder. So, that means (i^{44}) is the same as (i^4), which is 1. Got it, so (i^{44} = 1).Now, moving on to (i^{333}). This one is a bit bigger, but I'll use the same approach. I'll divide 333 by 4 and find the remainder. Let's see: 4 times 83 is 332, so 333 minus 332 is 1. So, the remainder is 1. That means (i^{333}) is the same as (i^1), which is just (i). So, (i^{333} = i).Now, plugging these back into the original expression: (3i^{44} - 2i^{333}) becomes (3 times 1 - 2 times i). Simplifying that, it's (3 - 2i). Wait, let me double-check to make sure I didn't make any mistakes. For (i^{44}), since 44 divided by 4 is exactly 11, no remainder, so (i^{44} = 1). That seems right. For (i^{333}), 333 divided by 4 is 83 with a remainder of 1, so (i^{333} = i). Yeah, that checks out. So, multiplying and subtracting, (3 times 1 = 3) and (2 times i = 2i), so (3 - 2i) is the result. I think that's it. It seems straightforward once I remember the cyclical nature of the powers of (i). I don't see any mistakes in my calculations, so I'm confident that the answer is (3 - 2i).

💡To compute (cos 150^circ), I start by recognizing that (150^circ) is in the second quadrant of the unit circle. In the second quadrant, the cosine values are negative because the x-coordinates are negative there.Next, I determine the reference angle for (150^circ). The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. For angles in the second quadrant, the reference angle is calculated as (180^circ - theta). Therefore, the reference angle for (150^circ) is:[180^circ - 150^circ = 30^circ]Now, I recall the cosine value for the reference angle of (30^circ). From trigonometric values of special angles, I know that:[cos 30^circ = frac{sqrt{3}}{2}]Since (150^circ) is in the second quadrant where cosine values are negative, I apply the appropriate sign to the cosine value of the reference angle:[cos 150^circ = -frac{sqrt{3}}{2}]Thus, the final value of (cos 150^circ) is:[boxed{-frac{sqrt{3}}{2}}]