🧠:<think>Okay, so I have this problem about a game where players collect points based on resources x, y, and z. The points are modeled by this polynomial:[ P(x, y, z) = 3x^3 + 5y^2 + 7z + 2xy - 4yz + 6xz + 8 ]There are two sub-problems. The first one is to find the critical points by setting the partial derivatives to zero and then determining if they're maxima, minima, or saddle points. The second is to maximize P under the constraint that x + y + z = 10 using Lagrange multipliers.Starting with Sub-problem 1. Critical points occur where all the partial derivatives are zero. So I need to compute the partial derivatives with respect to x, y, and z, set each to zero, and solve the system of equations.First, let's compute the partial derivatives.Partial derivative with respect to x:[ frac{partial P}{partial x} = 9x^2 + 2y + 6z ]Wait, hold on. Let me double-check. The term 3x^3 differentiates to 9x^2. The term 2xy differentiates to 2y. The term 6xz differentiates to 6z. The rest don't involve x, so their derivatives are zero. So yes, that's correct.Partial derivative with respect to y:[ frac{partial P}{partial y} = 10y + 2x - 4z ]Similarly, 5y^2 differentiates to 10y, 2xy differentiates to 2x, -4yz differentiates to -4z. The rest don't involve y.Partial derivative with respect to z:[ frac{partial P}{partial z} = 7 - 4y + 6x ]7z differentiates to 7, -4yz differentiates to -4y, 6xz differentiates to 6x. The rest don't involve z.So now, setting each partial derivative to zero:1. ( 9x^2 + 2y + 6z = 0 ) (Equation 1)2. ( 10y + 2x - 4z = 0 ) (Equation 2)3. ( 7 - 4y + 6x = 0 ) (Equation 3)So now, we have a system of three equations:Equation 1: ( 9x^2 + 2y + 6z = 0 )Equation 2: ( 10y + 2x - 4z = 0 )Equation 3: ( 6x - 4y + 7 = 0 )Wait, Equation 3 is ( 7 - 4y + 6x = 0 ), which can be rewritten as ( 6x - 4y = -7 ).So, let me write the equations clearly:1. ( 9x^2 + 2y + 6z = 0 )2. ( 10y + 2x - 4z = 0 )3. ( 6x - 4y = -7 )So, Equations 2 and 3 can be used to solve for y and z in terms of x, perhaps.Looking at Equation 3: ( 6x - 4y = -7 ). Let's solve for y.( 6x + 7 = 4y )( y = frac{6x + 7}{4} )Okay, so y is expressed in terms of x.Now, let's substitute y into Equation 2 to solve for z.Equation 2: ( 10y + 2x - 4z = 0 )Substitute y:( 10*(6x + 7)/4 + 2x - 4z = 0 )Simplify:First, compute 10*(6x +7)/4:10/4 = 5/2, so 5/2*(6x +7) = (30x + 35)/2So, Equation 2 becomes:(30x + 35)/2 + 2x - 4z = 0Multiply all terms by 2 to eliminate the denominator:30x + 35 + 4x - 8z = 0Combine like terms:30x + 4x = 34xSo, 34x + 35 - 8z = 0Solve for z:-8z = -34x -35Divide both sides by -8:z = (34x + 35)/8So, z is expressed in terms of x as well.Now, we have expressions for y and z in terms of x. Let's plug these into Equation 1 to solve for x.Equation 1: ( 9x^2 + 2y + 6z = 0 )Substitute y = (6x +7)/4 and z = (34x +35)/8.Compute each term:2y = 2*(6x +7)/4 = (12x +14)/4 = (6x +7)/26z = 6*(34x +35)/8 = (204x +210)/8 = (102x +105)/4So, Equation 1 becomes:9x^2 + (6x +7)/2 + (102x +105)/4 = 0To combine these, let's get a common denominator of 4.Multiply each term:9x^2 = 36x^2/4(6x +7)/2 = (12x +14)/4(102x +105)/4 remains the same.So, adding them together:36x^2/4 + (12x +14)/4 + (102x +105)/4 = 0Combine numerators:36x^2 +12x +14 +102x +105 = 0Combine like terms:36x^2 + (12x +102x) + (14 +105) = 036x^2 + 114x + 119 = 0So, quadratic equation in x:36x^2 + 114x + 119 = 0Let me compute the discriminant to see if real solutions exist.Discriminant D = b² - 4ac = (114)^2 - 4*36*119Compute 114 squared:114*114: 100*100=10000, 14*100=1400, 100*14=1400, 14*14=196Wait, that's not the right way. Let me compute 114*114:114*100=11400114*14=1596So, 11400 + 1596 = 12996So, D = 12996 - 4*36*119Compute 4*36=144144*119: Let's compute 100*119=11900, 40*119=4760, 4*119=476Wait, no, 144 is 100 + 40 + 4, so:100*119=1190040*119=47604*119=476So, total is 11900 + 4760 = 16660 + 476 = 17136So, D = 12996 - 17136 = -4140Wait, discriminant is negative, which means no real solutions.Hmm, that's a problem. So, the system of equations has no real solutions.But that can't be right because the function P(x,y,z) is a polynomial, so it should have critical points.Wait, maybe I made a mistake in computing the partial derivatives or setting up the equations.Let me double-check the partial derivatives.Partial derivative with respect to x:3x^3 differentiates to 9x^2.2xy differentiates to 2y.6xz differentiates to 6z.So, yes, ∂P/∂x = 9x² + 2y + 6z.Similarly, ∂P/∂y:5y² differentiates to 10y.2xy differentiates to 2x.-4yz differentiates to -4z.So, ∂P/∂y = 10y + 2x - 4z.∂P/∂z:7z differentiates to 7.-4yz differentiates to -4y.6xz differentiates to 6x.So, ∂P/∂z = 7 - 4y + 6x.So, the partial derivatives are correct.Then, setting them to zero:1. 9x² + 2y + 6z = 02. 10y + 2x - 4z = 03. 6x - 4y + 7 = 0Wait, Equation 3 is 6x - 4y +7 =0, which is 6x -4y = -7.So, solving for y: y = (6x +7)/4.Then, plugging into Equation 2:10y + 2x -4z =010*(6x +7)/4 +2x -4z=0Which is (60x +70)/4 +2x -4z=0Simplify:15x +17.5 +2x -4z=017x +17.5 -4z=0So, 4z=17x +17.5z=(17x +17.5)/4Wait, earlier I had z=(34x +35)/8, which is the same as (17x +17.5)/4, yes.Then, plugging into Equation 1:9x² +2y +6z=0Which is 9x² +2*(6x +7)/4 +6*(17x +17.5)/4=0Compute each term:2*(6x +7)/4 = (12x +14)/4 = 3x + 3.56*(17x +17.5)/4 = (102x +105)/4 = 25.5x +26.25So, Equation 1 becomes:9x² +3x +3.5 +25.5x +26.25=0Combine like terms:9x² + (3x +25.5x) + (3.5 +26.25)=09x² +28.5x +29.75=0Multiply all terms by 4 to eliminate decimals:36x² +114x +119=0Which is the same quadratic as before. So discriminant is D=114² -4*36*119=12996 -17136= -4140.Negative discriminant, so no real solutions.Wait, so that would mean there are no critical points? But that seems odd because P is a polynomial function, which usually has critical points.But perhaps it's because the function is a cubic in x, so maybe it tends to infinity in some directions, so maybe it doesn't have local maxima or minima, but saddle points?But in three variables, critical points can be more complicated.Wait, but if the system of equations has no real solutions, that would mean there are no critical points.But that seems strange. Let me think.Wait, the function P(x,y,z) is a cubic in x, quadratic in y, and linear in z. So, as x increases, P(x,y,z) will go to infinity because of the 3x³ term. Similarly, as x decreases, it will go to negative infinity. So, maybe the function doesn't have any local maxima or minima, just saddle points.But in three variables, the concept of critical points is similar. If the gradient is zero, it's a critical point. But if the system has no real solutions, then there are no critical points.Wait, but that can't be. Because for a function of multiple variables, even if it's unbounded, it can still have critical points.Wait, let me check the partial derivatives again.Wait, maybe I made a mistake in computing the partial derivatives.Wait, P(x,y,z)=3x³ +5y² +7z +2xy -4yz +6xz +8.So, ∂P/∂x: derivative of 3x³ is 9x², derivative of 2xy is 2y, derivative of 6xz is 6z. So, 9x² +2y +6z.Similarly, ∂P/∂y: derivative of 5y² is 10y, derivative of 2xy is 2x, derivative of -4yz is -4z. So, 10y +2x -4z.∂P/∂z: derivative of 7z is 7, derivative of -4yz is -4y, derivative of 6xz is 6x. So, 7 -4y +6x.Yes, that's correct.So, the partial derivatives are correct, so the system is correct.So, if the system has no real solutions, then the function P(x,y,z) has no critical points.But that seems odd because usually, polynomials have critical points.Wait, maybe I made a mistake in solving the equations.Let me try solving the equations again.Equation 3: 6x -4y = -7 => y = (6x +7)/4.Equation 2: 10y +2x -4z=0.Substitute y:10*(6x +7)/4 +2x -4z=0Simplify:(60x +70)/4 +2x -4z=0Which is 15x +17.5 +2x -4z=0So, 17x +17.5 -4z=0 => 4z=17x +17.5 => z=(17x +17.5)/4.Then, plug into Equation 1: 9x² +2y +6z=0.Compute 2y: 2*(6x +7)/4 = (12x +14)/4 = 3x +3.5.Compute 6z:6*(17x +17.5)/4 = (102x +105)/4 =25.5x +26.25.So, Equation 1:9x² +3x +3.5 +25.5x +26.25=0.Combine like terms:9x² + (3x +25.5x)=28.5x3.5 +26.25=29.75So, 9x² +28.5x +29.75=0.Multiply by 4:36x² +114x +119=0.Discriminant D=114² -4*36*119=12996 -17136= -4140.Negative discriminant, so no real solutions.So, indeed, the system has no real solutions, meaning there are no critical points.Therefore, the function P(x,y,z) has no critical points.But that seems a bit strange, but mathematically, it's correct.So, for Sub-problem 1, the conclusion is that there are no critical points because the system of equations derived from the partial derivatives has no real solutions.Now, moving on to Sub-problem 2: maximize P(x,y,z) under the constraint x + y + z =10 using Lagrange multipliers.So, the method of Lagrange multipliers involves introducing a multiplier λ and setting up the equations:∇P = λ∇g,where g(x,y,z)=x + y + z -10=0.So, compute the gradients.First, ∇P is the vector of partial derivatives:(∂P/∂x, ∂P/∂y, ∂P/∂z) = (9x² +2y +6z, 10y +2x -4z, 7 -4y +6x)∇g is (1,1,1).So, setting up the equations:9x² +2y +6z = λ (Equation A)10y +2x -4z = λ (Equation B)7 -4y +6x = λ (Equation C)And the constraint:x + y + z =10 (Equation D)So, now we have four equations: A, B, C, D.We can set Equations A, B, C equal to each other.From Equations A and B:9x² +2y +6z =10y +2x -4zBring all terms to left side:9x² +2y +6z -10y -2x +4z=0Simplify:9x² -8y +10z -2x=0 (Equation E)Similarly, set Equations B and C equal:10y +2x -4z =7 -4y +6xBring all terms to left side:10y +2x -4z -7 +4y -6x=0Simplify:14y -4x -4z -7=0 (Equation F)Now, we have Equations E, F, and D.Equation E:9x² -8y +10z -2x=0Equation F:14y -4x -4z -7=0Equation D:x + y + z=10So, let's try to solve these.First, from Equation D: z=10 -x -y.Let's substitute z into Equations E and F.Equation E:9x² -8y +10*(10 -x -y) -2x=0Compute:9x² -8y +100 -10x -10y -2x=0Combine like terms:9x² + (-8y -10y) + (-10x -2x) +100=09x² -18y -12x +100=0 (Equation E1)Equation F:14y -4x -4*(10 -x -y) -7=0Compute:14y -4x -40 +4x +4y -7=0Simplify:14y +4y + (-4x +4x) + (-40 -7)=018y -47=0So, 18y=47 => y=47/18 ≈2.6111So, y=47/18.Now, plug y into Equation D: z=10 -x -y=10 -x -47/18.Compute 10 as 180/18, so 180/18 -47/18=133/18.Thus, z=133/18 -x.Now, plug y=47/18 and z=133/18 -x into Equation E1.Equation E1:9x² -18y -12x +100=0Substitute y=47/18:9x² -18*(47/18) -12x +100=0Simplify:9x² -47 -12x +100=0Combine constants:-47 +100=53So, 9x² -12x +53=0Quadratic equation:9x² -12x +53=0Compute discriminant D= (-12)^2 -4*9*53=144 -1908= -1764Negative discriminant, so no real solutions.Wait, that can't be. So, no real solutions for x.But that would mean that there is no maximum under the constraint.But that seems odd because P is a cubic in x, so as x increases, P increases without bound, but under the constraint x + y + z=10, x can't go to infinity.Wait, but x is constrained by x + y + z=10, so x can't be more than 10, but even so, the function P(x,y,z) is 3x³ + ... So, as x increases, P increases.But if we have a constraint x + y + z=10, then to maximize P, we should set x as large as possible, but y and z would have to be as small as possible.But if y and z can be negative, then perhaps x can be larger than 10, but the constraint is x + y + z=10, so if y and z are negative, x can be more than 10.But in the game, resources are probably non-negative, but the problem doesn't specify. It just says total resources fixed at 10.So, if resources can be negative, then x can be more than 10, but if they must be non-negative, then x is at most 10.But the problem doesn't specify, so perhaps we have to assume they can be any real numbers.But in that case, as x increases, P increases without bound, so there is no maximum.But the Lagrange multiplier method is giving us no real solutions, which suggests that the maximum is at infinity, which is not attainable.Alternatively, maybe I made a mistake in setting up the equations.Wait, let's go back.We had Equations A, B, C, D.From Equations A and B, we got Equation E.From Equations B and C, we got Equation F.From Equation F, we found y=47/18.Then, substituted into Equation D to get z=133/18 -x.Then, substituted into Equation E1, which led to 9x² -12x +53=0, which has no real solutions.So, that suggests that there is no local maximum under the constraint, which would mean that the maximum is achieved at the boundary or at infinity.But in the context of the problem, resources are probably non-negative, so x, y, z ≥0.So, perhaps the maximum occurs at the boundary of the domain.So, to find the maximum, we can consider the boundaries where one or more variables are zero.But this is getting complicated.Alternatively, maybe the function P(x,y,z) is unbounded above under the constraint x + y + z=10, so there is no maximum.But let's test this.Suppose we set y and z to be negative, so that x can be greater than 10.For example, let’s set y = -k, z = -m, with k, m ≥0.Then, x =10 +k +m.So, P(x,y,z)=3x³ +5y² +7z +2xy -4yz +6xz +8Substitute x=10 +k +m, y=-k, z=-m:P=3*(10 +k +m)^3 +5*(-k)^2 +7*(-m) +2*(10 +k +m)*(-k) -4*(-k)*(-m) +6*(10 +k +m)*(-m) +8Simplify term by term:3*(10 +k +m)^3: This is a cubic term, which will dominate as k and m increase.5k²: positive.7*(-m)= -7m.2*(10 +k +m)*(-k)= -2k*(10 +k +m)= -20k -2k² -2km.-4*(-k)*(-m)= -4km.6*(10 +k +m)*(-m)= -6m*(10 +k +m)= -60m -6km -6m².+8.So, combining all terms:3*(10 +k +m)^3 +5k² -7m -20k -2k² -2km -4km -60m -6km -6m² +8Simplify:3*(10 +k +m)^3 + (5k² -2k²) + (-7m -60m) + (-20k) + (-2km -4km -6km) + (-6m²) +8Which is:3*(10 +k +m)^3 +3k² -67m -20k -12km -6m² +8Now, as k and m increase, the term 3*(10 +k +m)^3 will dominate, which goes to infinity. Therefore, P can be made arbitrarily large by increasing k and m, which correspond to negative y and z.Therefore, under the constraint x + y + z=10, P(x,y,z) is unbounded above, so there is no maximum.But in the context of the game, resources are probably non-negative, so y and z can't be negative. So, if we assume y ≥0 and z ≥0, then x=10 - y - z ≤10.In that case, x is bounded above by 10, so P(x,y,z) is a function where x is at most 10, and y and z are non-negative.So, in that case, we can look for the maximum within the domain x,y,z ≥0 and x + y + z=10.So, perhaps we can use Lagrange multipliers with the constraints x,y,z ≥0.But the previous attempt with Lagrange multipliers didn't yield a solution because the equations led to no real solutions, suggesting that the maximum occurs on the boundary.So, we need to check the boundaries where one or more variables are zero.So, the boundaries are:1. y=0, z=0: Then x=10.2. y=0, x=10 - z: z varies from 0 to10.3. z=0, x=10 - y: y varies from 0 to10.4. x=0, y + z=10.Also, edges where two variables are zero.So, let's evaluate P at these boundaries.First, evaluate at x=10, y=0, z=0:P=3*(10)^3 +5*(0)^2 +7*(0) +2*(10)*(0) -4*(0)*(0) +6*(10)*(0) +8=3*1000 +0 +0 +0 -0 +0 +8=3000 +8=3008.Next, consider y=0, z varies from 0 to10, x=10 -z.So, P=3x³ +5*0 +7z +2x*0 -4*0*z +6x*z +8=3x³ +7z +6xz +8.But x=10 -z, so substitute:P=3*(10 -z)^3 +7z +6*(10 -z)*z +8.Compute:First, expand (10 -z)^3:(10 -z)^3=1000 -300z +30z² -z³.So, 3*(10 -z)^3=3000 -900z +90z² -3z³.Then, 7z.Then, 6*(10 -z)*z=60z -6z².So, P=3000 -900z +90z² -3z³ +7z +60z -6z² +8.Combine like terms:-3z³ + (90z² -6z²)=84z²(-900z +7z +60z)= (-900 +67)z= -833zConstants:3000 +8=3008.So, P= -3z³ +84z² -833z +3008.Now, to find the maximum of this function for z in [0,10].Take derivative with respect to z:dP/dz= -9z² +168z -833.Set to zero:-9z² +168z -833=0Multiply by -1:9z² -168z +833=0Compute discriminant D=168² -4*9*833=28224 -4*9*833.Compute 4*9=36, 36*833=36*(800 +33)=28800 +1188=29988.So, D=28224 -29988= -1764.Negative discriminant, so no real solutions. Therefore, the function is monotonic on [0,10].Compute P at z=0: P=3008.At z=10: x=0, y=0, z=10.P=3*0 +5*0 +7*10 +2*0*0 -4*0*10 +6*0*10 +8=0 +0 +70 +0 -0 +0 +8=78.So, P decreases from 3008 to78 as z increases from0 to10. So, maximum at z=0, which is 3008.Next, consider z=0, x=10 -y, y varies from0 to10.So, P=3x³ +5y² +0 +2xy -0 +0 +8=3x³ +5y² +2xy +8.With x=10 -y.Substitute:P=3*(10 -y)^3 +5y² +2*(10 -y)*y +8.Compute:(10 -y)^3=1000 -300y +30y² -y³.So, 3*(10 -y)^3=3000 -900y +90y² -3y³.5y².2*(10 -y)*y=20y -2y².So, P=3000 -900y +90y² -3y³ +5y² +20y -2y² +8.Combine like terms:-3y³ + (90y² +5y² -2y²)=93y²(-900y +20y)= -880yConstants:3000 +8=3008.So, P= -3y³ +93y² -880y +3008.Take derivative:dP/dy= -9y² +186y -880.Set to zero:-9y² +186y -880=0Multiply by -1:9y² -186y +880=0Compute discriminant D=186² -4*9*880=34596 -31680=2916.√2916=54.So, y=(186 ±54)/18.Compute:y=(186 +54)/18=240/18=40/3≈13.333, which is outside the domain y≤10.y=(186 -54)/18=132/18=22/3≈7.333.So, critical point at y=22/3≈7.333.Check if this is a maximum.Second derivative: d²P/dy²= -18y +186.At y=22/3≈7.333, d²P/dy²= -18*(22/3) +186= -132 +186=54>0, so it's a minimum.Therefore, the maximum occurs at the endpoints.At y=0: P=3008.At y=10: x=0, z=0.P=3*0 +5*100 +7*0 +2*0*10 -4*10*0 +6*0*0 +8=0 +500 +0 +0 -0 +0 +8=508.So, P decreases from3008 to508 as y increases from0 to10. So, maximum at y=0, which is3008.Finally, consider x=0, y + z=10.So, P=3*0 +5y² +7z +0 -4yz +0 +8=5y² +7z -4yz +8.With z=10 -y.Substitute:P=5y² +7*(10 -y) -4y*(10 -y) +8.Compute:5y² +70 -7y -40y +4y² +8.Combine like terms:(5y² +4y²)=9y²(-7y -40y)= -47yConstants:70 +8=78.So, P=9y² -47y +78.Take derivative:dP/dy=18y -47.Set to zero:18y -47=0 => y=47/18≈2.611.Check second derivative:18>0, so it's a minimum.Therefore, maximum occurs at endpoints.At y=0: z=10.P=5*0 +7*10 -4*0*10 +8=0 +70 -0 +8=78.At y=10: z=0.P=5*100 +7*0 -4*10*0 +8=500 +0 -0 +8=508.So, maximum at y=10, P=508.Comparing all boundaries:- x=10,y=0,z=0: P=3008- y=0,z varies: max at z=0:3008- z=0,y varies: max at y=0:3008- x=0,y varies: max at y=10:508So, the maximum occurs at x=10,y=0,z=0 with P=3008.Therefore, the maximum is achieved when x=10, y=0, z=0.But wait, let's check if this is indeed the maximum.If we set x=10, y=0, z=0, P=3008.If we try to increase x beyond 10 by making y or z negative, but in the game, resources are probably non-negative, so we can't do that.Therefore, under the assumption that x,y,z ≥0, the maximum is at x=10,y=0,z=0.But wait, let me check another point.Suppose x=9, y=1, z=0.Then, P=3*729 +5*1 +0 +2*9*1 -0 +0 +8=2187 +5 +0 +18 +0 +0 +8=2218.Which is less than3008.Similarly, x=10,y=0,z=0:3008.Another point: x=10,y=1,z=-1, but z can't be negative.So, under non-negative resources, x=10,y=0,z=0 is the maximum.Therefore, the maximum is achieved at x=10,y=0,z=0 with P=3008.But wait, let me check another point.Suppose x=10,y=0,z=0: P=3008.If I set x=9,y=1,z=0: P=3*729 +5*1 +0 +2*9*1 -0 +0 +8=2187 +5 +18 +8=2218.Less than3008.x=8,y=2,z=0: P=3*512 +5*4 +0 +2*8*2 -0 +0 +8=1536 +20 +32 +8=1596.Still less.x=7,y=3,z=0: P=3*343 +5*9 +0 +2*7*3 -0 +0 +8=1029 +45 +42 +8=1124.Less.So, yes, x=10,y=0,z=0 gives the highest P.Therefore, the maximum is at x=10,y=0,z=0.But wait, let me check if the function P is increasing in x.Yes, because the leading term is 3x³, which increases as x increases.Therefore, to maximize P, set x as large as possible, which under the constraint x + y + z=10 and x,y,z ≥0, x=10,y=0,z=0.Therefore, the maximum is achieved at x=10,y=0,z=0 with P=3008.So, the answer for Sub-problem 2 is x=10,y=0,z=0.</think>

🧠:<think>Okay, so I've got this math problem here about a college student who wants to be a criminal defense attorney. They're taking a stats course, which makes sense because understanding probabilities is super important for analyzing evidence. The problem has two parts: one about calculating predictive values of a forensic test and another about determining an overall conviction rate based on how often the attorney refutes evidence. Let me try to work through each part step by step.Starting with part 1a: calculating the positive predictive value (PPV) and negative predictive value (NPV) of a forensic test. The test has a sensitivity of 98%, which means it correctly identifies 98% of the people who have the condition. Specificity is 95%, so it correctly identifies 95% of the people who don't have the condition. The prevalence of the condition is 0.5%, which is pretty low.Hmm, I remember that PPV is the probability that someone actually has the condition given that they tested positive. NPV is the opposite: the probability that someone doesn't have the condition given that they tested negative. To calculate these, I think I need to use Bayes' theorem or maybe a confusion matrix approach.Let me set up a hypothetical population to make it easier. Let's say there are 100,000 people. With a prevalence of 0.5%, that means 0.5% of 100,000 is 500 people who have the condition, and 99,500 who don't.Now, the test's sensitivity is 98%, so it correctly identifies 98% of the 500 people who have the condition. That would be 0.98 * 500 = 490 true positives. The remaining 2% of the 500 would be false negatives, so 10 people.For the specificity of 95%, the test correctly identifies 95% of the 99,500 people who don't have the condition. So that's 0.95 * 99,500 = 94,525 true negatives. The remaining 5% would be false positives, which is 0.05 * 99,500 = 4,975.Now, to find PPV, which is true positives divided by (true positives + false positives). So that's 490 / (490 + 4,975). Let me calculate that: 490 + 4,975 = 5,465. So 490 / 5,465 ≈ 0.0897 or about 8.97%.For NPV, it's true negatives divided by (true negatives + false negatives). That's 94,525 / (94,525 + 10). So 94,525 + 10 = 94,535. Then 94,525 / 94,535 ≈ 0.9999 or about 99.99%.Wait, that seems really high for PPV, but considering the low prevalence, it makes sense. Even though the test is pretty accurate, because the condition is rare, most positive results are false positives.Moving on to part 2b: calculating the overall conviction rate when the attorney refutes evidence in 60% of cases. The data says that when the attorney refutes the evidence successfully, 40% of cases result in conviction, and when they don't refute it, 85% result in conviction.So, the attorney refutes in 60% of cases. Let me denote the probability of conviction when refuting as 40%, so 0.4, and when not refuting as 85%, so 0.85.The overall conviction rate would be the weighted average of these two probabilities based on how often the attorney refutes. So, it's (0.6 * 0.4) + (0.4 * 0.85).Calculating that: 0.6 * 0.4 = 0.24 and 0.4 * 0.85 = 0.34. Adding them together gives 0.24 + 0.34 = 0.58, so 58%.Wait, let me double-check that. If 60% of cases are refuted, leading to 40% conviction rate, and 40% are not refuted, leading to 85% conviction, then yes, 0.6*0.4 is 0.24 and 0.4*0.85 is 0.34. Total is 0.58 or 58%. That seems right.I think I got both parts. For part 1a, the PPV is approximately 8.97% and NPV is approximately 99.99%. For part 2b, the overall conviction rate is 58%.Final Answera) The positive predictive value (PPV) is boxed{8.97%} and the negative predictive value (NPV) is boxed{99.99%}.b) The overall conviction rate is boxed{58%}.</think>

🧠:<think>Okay, so I need to figure out the total surface area of a Menger sponge after the 3rd iteration. Hmm, I remember that a Menger sponge is a fractal created by recursively removing parts of a cube. Let me try to break this down step by step.First, the initial cube has a side length of 3 meters. So, the surface area of a cube is 6 times the side length squared. That would be 6*(3^2) = 6*9 = 54 square meters. That's the starting point.Now, each iteration involves removing the central 1/3 of each face. I think this process is done on each cube in the structure. So, for the first iteration, we start with one cube. When we remove the central 1/3 of each face, we're essentially creating a sort of tunnel through each face. But actually, in the Menger sponge, each face is divided into a 3x3 grid, and the central square is removed. So, each face is divided into 9 smaller squares, each of side length 1 meter (since 3 meters divided by 3 is 1 meter). The central square is removed, so each face now has 8 squares instead of 9. So, the area removed per face is 1 square meter, and since there are 6 faces, the total area removed is 6*1 = 6 square meters.But wait, when we remove the central square from each face, we're not just subtracting that area; we're also creating new surfaces inside where the squares were removed. Each removed square is a hole, and the sides of that hole become new surfaces. Each hole is a square of 1x1, so each hole has 4 new edges, each of length 1 meter. So, each hole adds 4 square meters of new surface area. Since we have 6 faces, each with one hole, that's 6*4 = 24 square meters added.So, the surface area after the first iteration is the original surface area minus the area removed plus the new area added. That would be 54 - 6 + 24 = 72 square meters. Hmm, that seems right.Now, moving on to the second iteration. After the first iteration, the Menger sponge is made up of 20 smaller cubes, each of side length 1 meter. Wait, is that correct? Let me think. When you remove the central cube from each face, how many smaller cubes do you end up with?Actually, when you start with a 3x3x3 cube, you remove the central cube from each face, which is 6 cubes, but actually, in 3D, removing the central cube from each face would result in removing 20 cubes? Wait, no, that doesn't sound right. Let me visualize it.In the first iteration, you divide the cube into 27 smaller cubes (3x3x3). Then, you remove the central cube from each face, which are 6 cubes, but also the very center cube of the entire structure is removed as well. So, actually, you remove 7 cubes in total: 6 from each face and 1 from the center. So, the remaining number of cubes is 27 - 7 = 20. So, each of these 20 cubes has a side length of 1 meter.Wait, but in terms of surface area, each of these smaller cubes contributes to the overall surface area. However, some of their faces are adjacent to the removed cubes, so those faces become internal and are not part of the exterior surface area. But in the Menger sponge, after each iteration, the surface area actually increases because we're adding more surfaces from the holes.But perhaps it's better to think in terms of the surface area formula for the Menger sponge. I recall that each iteration, the surface area changes in a specific way. Let me see if I can find a pattern or a formula.After the first iteration, the surface area was 72 square meters. Now, for the second iteration, each of the 20 smaller cubes will undergo the same process. Each of these smaller cubes has a side length of 1 meter, so their surface area is 6*(1^2) = 6 square meters. But when we apply the Menger sponge process to each, we remove the central 1/3 of each face, which is now 1/3 of 1 meter, so 1/3 meters. So, each face is divided into a 3x3 grid of smaller squares, each of side length 1/3 meters. Removing the central square would remove an area of (1/3)^2 = 1/9 square meters per face. Since each cube has 6 faces, the total area removed per small cube is 6*(1/9) = 2/3 square meters.But similar to the first iteration, removing the central square from each face not only subtracts area but also adds new surfaces. Each removed square creates a hole, and the sides of that hole add new surface area. Each hole is a square of side length 1/3 meters, so each hole has 4 edges, each of length 1/3 meters. So, each hole adds 4*(1/3) = 4/3 square meters of new surface area per face. Since each cube has 6 faces, the total new surface area added per small cube is 6*(4/3) = 8 square meters.Therefore, for each small cube, the surface area after the second iteration is the original surface area minus the area removed plus the new area added. So, 6 - 2/3 + 8 = 6 - 0.666... + 8 ≈ 13.333... square meters. But since we have 20 such cubes, the total surface area would be 20*(13.333...) ≈ 266.666... square meters.Wait, that seems too high. Let me check my calculations again.Wait, no, actually, each small cube is being processed, but the surface area calculation needs to consider that some of their faces are internal and not contributing to the exterior. Hmm, this is getting complicated.Maybe a better approach is to use the formula for the surface area of a Menger sponge after n iterations. I think the surface area after each iteration follows a specific pattern.From what I recall, the surface area of the Menger sponge after each iteration can be calculated using the formula:Surface Area = 6 * (20/9)^n * (3)^2Wait, let me think. The initial surface area is 54. After the first iteration, it's 72, which is 54 * (20/9). Then, after the second iteration, it would be 72 * (20/9) = 160, and after the third iteration, 160 * (20/9) ≈ 355.555... square meters.Wait, that seems more consistent. Let me verify this.At each iteration, the surface area is multiplied by 20/9. So, starting with 54:After 1st iteration: 54 * (20/9) = 54 * (20/9) = 6 * 20 = 120? Wait, no, 54*(20/9) = 6*20 = 120? Wait, 54 divided by 9 is 6, multiplied by 20 is 120. But earlier I calculated 72. Hmm, that's a discrepancy.Wait, maybe I was wrong earlier. Let me recast.Perhaps the surface area after each iteration is multiplied by 20/9. So, starting with 54:After 1st iteration: 54 * (20/9) = 120After 2nd iteration: 120 * (20/9) ≈ 266.666...After 3rd iteration: 266.666... * (20/9) ≈ 592.592... square meters.But wait, earlier I thought after the first iteration it was 72. So, which is correct?Let me go back to the first iteration.Original cube: surface area 54.After first iteration: each face has 8 squares instead of 9, so each face's area is 8*(1^2) = 8. So, total surface area is 6*8 = 48. But wait, that's just the exterior. However, when you remove the central cube, you also expose the inner surfaces.Each face has a hole, which is a square of 1x1. The sides of this hole are new surfaces. Each hole has 4 edges, each of length 1, so each hole adds 4 square meters. Since there are 6 faces, that's 6*4 = 24.So, total surface area is original 54 minus the area removed (6) plus the new area added (24). So, 54 - 6 + 24 = 72.But according to the formula, it should be 54*(20/9) = 120. So, which is correct?Wait, perhaps the formula is for the total surface area including both the exterior and the interior? Or maybe I'm misunderstanding the formula.Wait, let's think differently. Each iteration, the number of cubes increases, and each cube contributes to the surface area. But the surface area isn't just the sum of all individual cube surface areas because many faces are internal and not exposed.Alternatively, perhaps the surface area after each iteration can be calculated as follows:At each iteration, each face of the cube is divided into 9 smaller squares, and the central one is removed. So, for each face, instead of 1 face, we have 8 smaller faces, each of area (1/3)^2 = 1/9 of the original face area. So, the area per face becomes 8*(1/9) = 8/9 of the original face area. However, removing the central square also creates new surfaces. The hole has 4 new edges, each of length 1/3, so each hole adds 4*(1/3)^2 = 4/9 square meters per face. Wait, no, the edges are length 1/3, but the area added is the perimeter times the depth? Hmm, maybe not.Wait, perhaps it's better to consider that when you remove a cube from the center of each face, you're removing a 1x1x1 cube, but in terms of surface area, you're removing 1 square meter from each face, but adding 5 new faces (since the removed cube had 5 faces exposed inside the original cube). Wait, no, because when you remove a cube from the center of a face, you're only removing one face, but the other five faces of the removed cube become new internal surfaces.Wait, let's clarify. When you remove a cube from the center of a face, you're taking out a 1x1x1 cube. The original face had an area of 1x1, so removing the central 1/3x1/3 square (which is 1/9 of the face) would actually be a different process. Wait, no, in the first iteration, the cube is 3x3x3, so each face is 3x3. Removing the central 1x1 square from each face, which is 1/9 of the face area. So, each face loses 1 square meter, but the sides of the removed square become new surfaces.Each removed square has 4 edges, each of length 1 meter, so each hole adds 4 square meters of new surface area. Since there are 6 faces, that's 6*4 = 24. So, the total surface area becomes original 54 minus 6 (the area removed) plus 24 (the new surfaces) = 72.Now, for the second iteration, each of the 20 smaller cubes (each of side length 1) will undergo the same process. Each of these smaller cubes will have their central 1/3x1/3 square removed from each face, which is 1/9 of their face area. So, each face of the small cube loses 1/9 square meters, but each hole adds 4*(1/3)^2 = 4/9 square meters of new surface area per face. Wait, no, the edges are length 1/3, so each hole adds 4*(1/3) = 4/3 square meters per face? Wait, no, the area added is the perimeter of the hole times the depth, but in this case, the depth is 1/3 meters, so the area added per hole is 4*(1/3)*(1/3) = 4/9 square meters per face. Wait, that doesn't seem right.Wait, perhaps it's better to think in terms of the surface area change per iteration. Each iteration, the surface area is multiplied by 20/9. So, starting with 54:After 1st iteration: 54 * (20/9) = 120After 2nd iteration: 120 * (20/9) ≈ 266.666...After 3rd iteration: 266.666... * (20/9) ≈ 592.592... square meters.But earlier, I calculated after the first iteration as 72, which is different from 120. So, which is correct?Wait, perhaps the formula is considering both the exterior and interior surfaces, whereas my initial calculation only considered the exterior. Let me think.In the first iteration, the exterior surface area is 72, but the interior surfaces are also part of the total surface area. Wait, no, the Menger sponge's total surface area includes both the exterior and the interior surfaces. So, perhaps my initial calculation was only considering the exterior, but the formula accounts for both.Wait, let's recast. The initial cube has surface area 54. After the first iteration, we have 20 smaller cubes, each of surface area 6*(1^2)=6, but many of their faces are internal. However, the total surface area is not just 20*6=120 because some faces are internal and not contributing. But in reality, the surface area increases because of the new surfaces created by the holes.Wait, perhaps the correct way is to realize that each iteration, the surface area is multiplied by 20/9. So, starting with 54:After 1st iteration: 54 * (20/9) = 120After 2nd iteration: 120 * (20/9) ≈ 266.666...After 3rd iteration: 266.666... * (20/9) ≈ 592.592... square meters.But earlier, when I calculated manually, I got 72 after the first iteration. So, which is correct?Wait, perhaps the formula is correct, and my manual calculation was wrong. Let me think again.In the first iteration, the Menger sponge is created by removing 7 cubes (6 from each face and 1 from the center). Each removed cube contributes to the surface area. Each removed cube has 5 faces exposed (since one face was attached to the original cube). So, each removed cube adds 5 square meters to the surface area. Since we removed 7 cubes, that's 7*5=35 square meters added. But wait, the original cube had a surface area of 54. So, the new surface area would be 54 + 35 = 89? That doesn't match either.Wait, no, that's not right because when you remove a cube from the center of a face, you're not just adding 5 new faces; you're also removing a face from the original cube. So, for each removed cube, you lose 1 square meter from the original cube's surface area but gain 5 square meters from the removed cube's faces. So, the net change is +4 per removed cube.Since we removed 7 cubes, the total change is 7*4=28. So, the new surface area would be 54 + 28 = 82. But that still doesn't match the 72 or 120 numbers.Wait, I'm getting confused. Let me look for a reliable formula or method.Upon checking, I recall that the surface area of the Menger sponge after n iterations is given by:Surface Area = 6 * (20/9)^n * (3)^2Wait, no, that doesn't seem right. Let me think differently.Each iteration, the number of cubes increases by a factor of 20, and the surface area per cube is scaled by (1/3)^2, but also, each cube contributes new surfaces.Wait, perhaps the surface area after each iteration is multiplied by 20/9. So, starting with 54:After 1st iteration: 54 * (20/9) = 120After 2nd iteration: 120 * (20/9) ≈ 266.666...After 3rd iteration: 266.666... * (20/9) ≈ 592.592... square meters.But I'm not entirely sure. Alternatively, perhaps the surface area after n iterations is 6*(20/9)^n * (3)^2. Wait, 3^2 is 9, so 6*(20/9)^n *9 = 6*20^n.Wait, that can't be right because 6*20^3 is 48,000, which is way too high.Wait, perhaps the formula is:Surface Area = 6 * (20/9)^n * (3)^2But 3^2 is 9, so 6*(20/9)^n *9 = 6*20^n.No, that still doesn't make sense.Wait, maybe the formula is:Surface Area = 6 * (20/9)^n * (3)^2But that would be 6*(20/9)^n *9 = 54*(20/9)^n.Ah, that makes more sense. So, starting with 54, each iteration multiplies by 20/9.So, after 1st iteration: 54*(20/9) = 120After 2nd iteration: 120*(20/9) ≈ 266.666...After 3rd iteration: 266.666...*(20/9) ≈ 592.592... square meters.But earlier, when I manually calculated the first iteration, I got 72. So, which is correct?Wait, perhaps the formula is considering both the exterior and interior surfaces, whereas my manual calculation only considered the exterior. Let me think.In the first iteration, the exterior surface area is 72, but the interior surfaces are also part of the total surface area. So, the total surface area is indeed higher than 72.Wait, let me try to calculate the total surface area after the first iteration, including both exterior and interior.Original cube: 54.After removing 7 cubes (6 from each face and 1 from the center), each removed cube contributes 5 new faces. So, 7*5=35.But also, the original cube had 7 faces covered (the ones where the cubes were removed), so we lose 7 square meters from the original surface area.So, total surface area is 54 - 7 + 35 = 82.But that's different from both 72 and 120.Wait, perhaps this approach is incorrect because the removed cubes are not just adding 5 faces each. When you remove a cube from the center of a face, you're removing one face from the original cube and exposing 5 new faces from the removed cube. However, when you remove the central cube, you're removing one cube that was connected to all six faces, but actually, in the Menger sponge, the central cube is removed, but it's only connected to the center of each face, so removing it would expose 6 new faces, not 5.Wait, no, the central cube is surrounded by other cubes, so removing it would expose 6 new faces, each of area 1x1. So, removing the central cube adds 6 square meters.Similarly, removing each of the 6 face cubes adds 5 square meters each because one face was attached to the original cube.So, total added surface area is 6*5 + 6 = 30 + 6 = 36.But the original surface area was 54, and we removed 7 squares (each of 1x1), so we subtract 7.So, total surface area is 54 - 7 + 36 = 83.Hmm, that's still not matching the formula.Wait, perhaps I'm overcomplicating. Let me look for a resource or formula.Upon checking, I find that the surface area of the Menger sponge after n iterations is given by:Surface Area = 6 * (20/9)^n * (3)^2But 3^2 is 9, so it's 6*(20/9)^n *9 = 54*(20/9)^n.So, for n=0, it's 54.n=1: 54*(20/9)=120n=2: 120*(20/9)=266.666...n=3: 266.666...*(20/9)=592.592... ≈ 592.592 square meters.But earlier, when I tried to calculate manually, I got 72 after the first iteration, which is different. So, perhaps my manual calculation was incorrect because I didn't account for all the new surfaces.Wait, perhaps the formula is correct, and my manual calculation was wrong. Let me accept that the surface area after each iteration is multiplied by 20/9.So, after 3 iterations, the surface area is 54*(20/9)^3.Let me calculate that:First, (20/9)^3 = (20^3)/(9^3) = 8000/729 ≈ 10.973.Then, 54 * 10.973 ≈ 54*10 + 54*0.973 ≈ 540 + 52.542 ≈ 592.542 square meters.So, approximately 592.54 square meters.But let me confirm this with another approach.Each iteration, the number of cubes increases by a factor of 20, and each cube's surface area is scaled by (1/3)^2, but also, each cube contributes new surfaces.Wait, perhaps the total surface area after n iterations is 6*(20/9)^n * (3)^2.Wait, 3^2 is 9, so 6*(20/9)^n *9 = 54*(20/9)^n, which is the same as before.So, after 3 iterations, it's 54*(20/9)^3 ≈ 592.592 square meters.Therefore, the total surface area after the 3rd iteration is approximately 592.592 square meters.But wait, let me think again. The Menger sponge's surface area actually converges to infinity as n approaches infinity, which makes sense because it's a fractal with infinite surface area. So, each iteration increases the surface area by a factor of 20/9, which is greater than 1, so it's growing exponentially.So, after 3 iterations, it's 54*(20/9)^3.Calculating that:20/9 ≈ 2.22222.2222^3 ≈ 10.97354*10.973 ≈ 592.542So, approximately 592.54 square meters.But wait, earlier I thought it was 72 after the first iteration, but according to the formula, it's 120. So, which is correct?Wait, perhaps the formula is correct, and my manual calculation was wrong because I didn't account for all the new surfaces.Alternatively, perhaps the formula is considering both the exterior and interior surfaces, whereas my manual calculation only considered the exterior.Wait, in the first iteration, the exterior surface area is 72, but the interior surfaces are also part of the total surface area. So, the total surface area is indeed higher than 72.Wait, perhaps the formula is correct, and the total surface area after the first iteration is 120.But I'm still confused because when I manually calculated, I got 72. Let me try to reconcile this.When I removed the central cube from each face, I subtracted 6 square meters (one from each face) and added 24 square meters (4 per face). So, 54 - 6 + 24 = 72.But according to the formula, it's 54*(20/9) = 120.So, where is the discrepancy?Ah, perhaps because when you remove the central cube from each face, you're not just adding 4 per face, but also, the central cube's removal affects the overall structure.Wait, no, the central cube is removed, but it's only one cube, so it's not part of the face removals. So, in the first iteration, you remove 7 cubes: 6 from each face and 1 from the center.Each removed cube contributes 5 new faces (for the face cubes) and 6 new faces (for the center cube). So, total new faces added: 6*5 + 6 = 30 + 6 = 36.But the original surface area was 54, and we removed 7 squares (each of 1x1), so we subtract 7.So, total surface area is 54 - 7 + 36 = 83.But that's still different from both 72 and 120.Wait, perhaps the formula is correct, and my manual calculation is wrong because I'm not considering that each removed cube's faces are not all exposed. For example, the center cube's removal exposes 6 new faces, but those faces are adjacent to other cubes, so maybe not all are exposed.Wait, no, when you remove the center cube, all 6 of its faces become new surfaces because they were previously internal.Similarly, when you remove a cube from the center of a face, you expose 5 new faces, but one of those faces is adjacent to another cube, so it's not all exposed.Wait, no, when you remove a cube from the center of a face, you're removing it from the original cube, so the 5 new faces are all exposed because they were previously internal.Wait, perhaps I'm overcomplicating. Let me accept that the formula is correct and that the surface area after each iteration is multiplied by 20/9.So, after 3 iterations, the surface area is 54*(20/9)^3 ≈ 592.592 square meters.Therefore, the answer to the first question is approximately 592.59 square meters.Now, moving on to the second question. The number of projectors needed is proportional to the interior surface area, with a proportionality constant of 1 projector per 6 square meters.Wait, but the total surface area includes both the exterior and interior surfaces. So, if the total surface area is 592.59 square meters, and we need to find the interior surface area, we need to subtract the exterior surface area.Wait, but in the Menger sponge, the exterior surface area after n iterations is 6*(8/9)^n * (3)^2.Wait, let me think. The exterior surface area decreases by a factor of 8/9 each iteration because each face is divided into 9, and 8 remain.So, starting with 54:After 1st iteration: 54*(8/9) = 48After 2nd iteration: 48*(8/9) ≈ 42.666...After 3rd iteration: 42.666...*(8/9) ≈ 37.333... square meters.So, the exterior surface area after 3 iterations is approximately 37.333 square meters.Therefore, the interior surface area is the total surface area minus the exterior surface area: 592.592 - 37.333 ≈ 555.259 square meters.Then, the number of projectors needed is 555.259 / 6 ≈ 92.543, which we can round up to 93 projectors.But wait, let me verify this approach.Alternatively, perhaps the interior surface area is the total surface area minus the exterior surface area. So, if the total surface area is 592.592 and the exterior is 37.333, then interior is 592.592 - 37.333 ≈ 555.259.Then, projectors needed: 555.259 /6 ≈ 92.543, so 93 projectors.But let me check if the exterior surface area formula is correct.Yes, the exterior surface area after n iterations is 6*(8/9)^n * (3)^2.So, for n=3:6*(8/9)^3 *9 = 6*(512/729)*9 = 6*(512/81) ≈ 6*6.320987654 ≈ 37.92592593 square meters.So, approximately 37.926.Therefore, interior surface area is 592.592 - 37.926 ≈ 554.666 square meters.Then, projectors needed: 554.666 /6 ≈ 92.444, which we can round up to 93 projectors.Alternatively, if we use exact fractions:Total surface area after 3 iterations: 54*(20/9)^3 = 54*(8000/729) = (54*8000)/729 = (54/729)*8000 = (6/81)*8000 = (2/27)*8000 = 16000/27 ≈ 592.5925926.Exterior surface area: 6*(8/9)^3*9 = 6*(512/729)*9 = 6*(512/81) = 3072/81 = 1024/27 ≈ 37.92592593.Interior surface area: 16000/27 - 1024/27 = (16000 - 1024)/27 = 14976/27 = 554.6666667.Projectors needed: 554.6666667 /6 = 554.6666667 ÷6 ≈ 92.44444445.Since you can't have a fraction of a projector, you'd need 93 projectors.Therefore, the answers are:1. Total surface area after 3rd iteration: 16000/27 ≈ 592.59 square meters.2. Number of projectors: 93.</think>

🧠:<think>Okay, so I have this problem about a composer creating a soundtrack with frequencies defined by a recursive sequence. There are two parts: first, finding the general form of the frequency sequence, and second, determining the base frequency which is the greatest common divisor (GCD) of the first 10 terms and identifying which terms are multiples of this base frequency.Starting with part 1: The frequencies ( f_n ) follow the recursive relationship ( f_{n+2} = f_{n+1} + f_n ) with initial conditions ( f_1 = 440 ) Hz and ( f_2 = 466.16 ) Hz. Hmm, this looks a lot like the Fibonacci sequence, where each term is the sum of the two preceding ones. But instead of starting with 1 and 1, it starts with 440 and 466.16. So, I think the general form will be similar to the Fibonacci sequence but scaled by some constants.In the Fibonacci sequence, the general term is given by Binet's formula: ( F_n = frac{phi^n - psi^n}{sqrt{5}} ), where ( phi = frac{1 + sqrt{5}}{2} ) is the golden ratio and ( psi = frac{1 - sqrt{5}}{2} ). Since our sequence is similar but with different starting values, I believe we can express ( f_n ) in terms of ( phi ) and ( psi ) as well.Let me denote the general solution as ( f_n = A phi^n + B psi^n ), where A and B are constants determined by the initial conditions. So, I need to solve for A and B using ( f_1 ) and ( f_2 ).Given:( f_1 = 440 = A phi + B psi )( f_2 = 466.16 = A phi^2 + B psi^2 )I can write these as a system of equations:1. ( A phi + B psi = 440 )2. ( A phi^2 + B psi^2 = 466.16 )I need to solve for A and B. Let me recall that ( phi^2 = phi + 1 ) and ( psi^2 = psi + 1 ) because they satisfy the quadratic equation ( x^2 = x + 1 ). So, substituting these into equation 2:( A (phi + 1) + B (psi + 1) = 466.16 )Expanding:( A phi + A + B psi + B = 466.16 )But from equation 1, ( A phi + B psi = 440 ), so substituting:( 440 + A + B = 466.16 )Therefore:( A + B = 466.16 - 440 = 26.16 )So now I have:1. ( A phi + B psi = 440 )2. ( A + B = 26.16 )I can solve this system. Let me denote equation 2 as ( A = 26.16 - B ) and substitute into equation 1:( (26.16 - B) phi + B psi = 440 )Expanding:( 26.16 phi - B phi + B psi = 440 )Factor out B:( 26.16 phi + B (psi - phi) = 440 )I know that ( psi - phi = frac{1 - sqrt{5}}{2} - frac{1 + sqrt{5}}{2} = -sqrt{5} ). So:( 26.16 phi - B sqrt{5} = 440 )Let me compute ( 26.16 phi ). First, ( phi approx 1.61803 ). So:( 26.16 * 1.61803 ≈ 26.16 * 1.618 ≈ Let's compute 26 * 1.618 = 42.068, and 0.16 * 1.618 ≈ 0.25888. So total ≈ 42.068 + 0.25888 ≈ 42.32688 )So, ( 42.32688 - B sqrt{5} = 440 )Therefore, ( -B sqrt{5} = 440 - 42.32688 ≈ 397.67312 )Thus, ( B = -397.67312 / sqrt{5} ≈ -397.67312 / 2.23607 ≈ -177.66 )So, B ≈ -177.66Then, from equation 2, ( A = 26.16 - B ≈ 26.16 - (-177.66) ≈ 26.16 + 177.66 ≈ 203.82 )So, A ≈ 203.82 and B ≈ -177.66Therefore, the general form is:( f_n = A phi^n + B psi^n ≈ 203.82 phi^n - 177.66 psi^n )But let me check if these values make sense. Let's compute ( f_1 ):( f_1 ≈ 203.82 * 1.61803 - 177.66 * (-0.61803) )Compute each term:203.82 * 1.61803 ≈ 203.82 * 1.618 ≈ Let's compute 200 * 1.618 = 323.6, and 3.82 * 1.618 ≈ 6.18, so total ≈ 323.6 + 6.18 ≈ 329.78177.66 * (-0.61803) ≈ -177.66 * 0.618 ≈ Let's compute 177.66 * 0.6 = 106.596, 177.66 * 0.018 ≈ 3.198, so total ≈ 106.596 + 3.198 ≈ 109.794, so negative is ≈ -109.794Therefore, ( f_1 ≈ 329.78 - (-109.794) ≈ 329.78 + 109.794 ≈ 439.574 ) Hz, which is approximately 440 Hz. Close enough, considering rounding errors.Similarly, let's compute ( f_2 ):( f_2 ≈ 203.82 * phi^2 - 177.66 * psi^2 )But ( phi^2 = phi + 1 ≈ 2.61803 ), and ( psi^2 = psi + 1 ≈ 0.38197 )So:203.82 * 2.61803 ≈ Let's compute 200 * 2.618 ≈ 523.6, 3.82 * 2.618 ≈ 9.99, so total ≈ 523.6 + 9.99 ≈ 533.59177.66 * 0.38197 ≈ 177.66 * 0.38 ≈ 67.51, 177.66 * 0.00197 ≈ 0.35, so total ≈ 67.51 + 0.35 ≈ 67.86Therefore, ( f_2 ≈ 533.59 - 67.86 ≈ 465.73 ) Hz, which is close to 466.16 Hz. Again, considering rounding, it's accurate.So, the general form is ( f_n = A phi^n + B psi^n ) with A ≈ 203.82 and B ≈ -177.66. But perhaps we can express A and B more precisely without rounding.Let me try to compute A and B symbolically.We had:1. ( A phi + B psi = 440 )2. ( A + B = 26.16 )Let me express A and B in terms of the initial conditions.Let me denote equation 1 as:( A phi + B psi = 440 ) ...(1)Equation 2 as:( A + B = 26.16 ) ...(2)From equation (2), ( A = 26.16 - B ). Substitute into equation (1):( (26.16 - B) phi + B psi = 440 )Expanding:( 26.16 phi - B phi + B psi = 440 )Factor B:( 26.16 phi + B (psi - phi) = 440 )We know that ( psi - phi = -sqrt{5} ), so:( 26.16 phi - B sqrt{5} = 440 )Therefore:( -B sqrt{5} = 440 - 26.16 phi )Hence,( B = frac{26.16 phi - 440}{sqrt{5}} )Similarly, ( A = 26.16 - B = 26.16 - frac{26.16 phi - 440}{sqrt{5}} )Let me compute B:First, compute ( 26.16 phi ):( 26.16 * phi = 26.16 * frac{1 + sqrt{5}}{2} = 26.16 * frac{1}{2} + 26.16 * frac{sqrt{5}}{2} = 13.08 + 13.08 sqrt{5} )So,( B = frac{13.08 + 13.08 sqrt{5} - 440}{sqrt{5}} = frac{-426.92 + 13.08 sqrt{5}}{sqrt{5}} )Simplify:( B = frac{-426.92}{sqrt{5}} + frac{13.08 sqrt{5}}{sqrt{5}} = frac{-426.92}{sqrt{5}} + 13.08 )Similarly, ( A = 26.16 - B = 26.16 - left( frac{-426.92}{sqrt{5}} + 13.08 right ) = 26.16 - 13.08 + frac{426.92}{sqrt{5}} = 13.08 + frac{426.92}{sqrt{5}} )So, we can write A and B as:( A = 13.08 + frac{426.92}{sqrt{5}} )( B = 13.08 - frac{426.92}{sqrt{5}} )But perhaps we can rationalize the denominators:( A = 13.08 + frac{426.92}{sqrt{5}} = 13.08 + frac{426.92 sqrt{5}}{5} = 13.08 + 85.384 sqrt{5} )Similarly,( B = 13.08 - frac{426.92}{sqrt{5}} = 13.08 - frac{426.92 sqrt{5}}{5} = 13.08 - 85.384 sqrt{5} )Wait, let me compute 426.92 / 5:426.92 / 5 = 85.384So, yes, that's correct.Therefore, ( A = 13.08 + 85.384 sqrt{5} ) and ( B = 13.08 - 85.384 sqrt{5} )Therefore, the general term is:( f_n = A phi^n + B psi^n = left(13.08 + 85.384 sqrt{5}right) phi^n + left(13.08 - 85.384 sqrt{5}right) psi^n )But perhaps we can write this more neatly. Let me factor out 13.08:( f_n = 13.08 (phi^n + psi^n) + 85.384 sqrt{5} (phi^n - psi^n) )But ( phi^n + psi^n ) is actually related to the Fibonacci sequence. Specifically, ( phi^n + psi^n = F_n sqrt{5} ) where ( F_n ) is the nth Fibonacci number. Wait, let me recall:Actually, Binet's formula is ( F_n = frac{phi^n - psi^n}{sqrt{5}} ). So, ( phi^n - psi^n = F_n sqrt{5} ). Also, ( phi^n + psi^n ) is another sequence, sometimes denoted as Lucas numbers. Indeed, Lucas numbers satisfy ( L_n = phi^n + psi^n ).So, Lucas numbers: ( L_n = phi^n + psi^n ). So, we can write:( f_n = 13.08 L_n + 85.384 sqrt{5} F_n )But let me compute 85.384 * sqrt(5):sqrt(5) ≈ 2.23607, so 85.384 * 2.23607 ≈ Let's compute 85 * 2.23607 ≈ 190.065, and 0.384 * 2.23607 ≈ 0.859, so total ≈ 190.065 + 0.859 ≈ 190.924So, 85.384 sqrt(5) ≈ 190.924Therefore, ( f_n ≈ 13.08 L_n + 190.924 F_n )But let me check if this is correct. Let me compute ( f_1 ):For n=1, ( L_1 = 1 ), ( F_1 = 1 ). So,( f_1 ≈ 13.08 * 1 + 190.924 * 1 = 13.08 + 190.924 = 204.004 ) Hz. Wait, that's not 440 Hz. Hmm, something's wrong here.Wait, perhaps I made a mistake in the substitution. Let me go back.We had:( f_n = A phi^n + B psi^n )Where:( A = 13.08 + 85.384 sqrt{5} )( B = 13.08 - 85.384 sqrt{5} )So, when n=1:( f_1 = A phi + B psi )But we already know that ( f_1 = 440 ), so that's consistent. But when I tried to express it in terms of Lucas and Fibonacci numbers, I must have messed up.Wait, perhaps it's better to leave it in terms of A and B without trying to express it in terms of known sequences, unless necessary.Alternatively, perhaps I can write the general term as:( f_n = alpha phi^n + beta psi^n )Where ( alpha ) and ( beta ) are constants determined by the initial conditions.Given that, and having found A and B as approximately 203.82 and -177.66, perhaps it's acceptable to present the general form as:( f_n = A phi^n + B psi^n ) with A ≈ 203.82 and B ≈ -177.66But maybe we can express A and B more precisely.Wait, let's see. From earlier, we had:( A = frac{f_2 - psi f_1}{phi - psi} )Similarly,( B = frac{phi f_1 - f_2}{phi - psi} )Because in the standard linear recurrence solution, the constants are given by:( A = frac{f_2 - psi f_1}{phi - psi} )( B = frac{phi f_1 - f_2}{phi - psi} )Given that ( phi - psi = sqrt{5} ), so:( A = frac{f_2 - psi f_1}{sqrt{5}} )( B = frac{phi f_1 - f_2}{sqrt{5}} )So, let's compute A and B precisely.Given ( f_1 = 440 ), ( f_2 = 466.16 ), ( phi = frac{1 + sqrt{5}}{2} ), ( psi = frac{1 - sqrt{5}}{2} )Compute A:( A = frac{466.16 - psi * 440}{sqrt{5}} )First, compute ( psi * 440 ):( psi = frac{1 - sqrt{5}}{2} ≈ frac{1 - 2.23607}{2} ≈ frac{-1.23607}{2} ≈ -0.61803 )So, ( psi * 440 ≈ -0.61803 * 440 ≈ -271.933 )Therefore, numerator ≈ 466.16 - (-271.933) ≈ 466.16 + 271.933 ≈ 738.093Thus, ( A ≈ 738.093 / 2.23607 ≈ 330 )Similarly, compute B:( B = frac{phi * 440 - 466.16}{sqrt{5}} )Compute ( phi * 440 ≈ 1.61803 * 440 ≈ 711.933 )So, numerator ≈ 711.933 - 466.16 ≈ 245.773Thus, ( B ≈ 245.773 / 2.23607 ≈ 109.8 )Wait, but earlier when I solved numerically, I got A ≈ 203.82 and B ≈ -177.66. There's a discrepancy here. Hmm, perhaps I made a mistake in the earlier substitution.Wait, let's re-examine. The standard solution for a linear recurrence ( f_{n+2} = f_{n+1} + f_n ) is indeed ( f_n = A phi^n + B psi^n ). The constants A and B are found by solving the system:( f_1 = A phi + B psi )( f_2 = A phi^2 + B psi^2 )But since ( phi^2 = phi + 1 ) and ( psi^2 = psi + 1 ), we can write:( f_2 = A (phi + 1) + B (psi + 1) = A phi + A + B psi + B = (A phi + B psi) + (A + B) = f_1 + (A + B) )Therefore, ( A + B = f_2 - f_1 = 466.16 - 440 = 26.16 )So, we have:1. ( A phi + B psi = 440 )2. ( A + B = 26.16 )So, as before, solving for A and B.Let me write this as a linear system:Equation 1: ( A phi + B psi = 440 )Equation 2: ( A + B = 26.16 )Express in matrix form:[ φ ψ ] [A] = [440][1 1 ] [B] [26.16]To solve this, we can use Cramer's rule or substitution.Compute determinant of the coefficient matrix:Determinant D = φ*1 - ψ*1 = φ - ψ = sqrt(5)Then,A = (440*1 - ψ*26.16)/DB = (φ*26.16 - 440*1)/DCompute A:A = (440 - ψ*26.16)/sqrt(5)Similarly, B = (φ*26.16 - 440)/sqrt(5)Compute ψ*26.16:ψ ≈ -0.61803, so ψ*26.16 ≈ -16.16Thus, A ≈ (440 - (-16.16))/sqrt(5) ≈ (456.16)/2.23607 ≈ 203.82Similarly, compute φ*26.16 ≈ 1.61803*26.16 ≈ 42.326Thus, B ≈ (42.326 - 440)/sqrt(5) ≈ (-397.674)/2.23607 ≈ -177.66So, A ≈ 203.82 and B ≈ -177.66, as before.Therefore, the general form is:( f_n = 203.82 phi^n - 177.66 psi^n )But to express this more precisely, perhaps we can write A and B in terms of exact expressions.Given that:( A = frac{440 - psi * 26.16}{sqrt{5}} )But ( psi = frac{1 - sqrt{5}}{2} ), so:( A = frac{440 - frac{1 - sqrt{5}}{2} * 26.16}{sqrt{5}} )Compute ( frac{1 - sqrt{5}}{2} * 26.16 ):= ( frac{26.16 - 26.16 sqrt{5}}{2} )= ( 13.08 - 13.08 sqrt{5} )Thus,( A = frac{440 - (13.08 - 13.08 sqrt{5})}{sqrt{5}} )= ( frac{440 - 13.08 + 13.08 sqrt{5}}{sqrt{5}} )= ( frac{426.92 + 13.08 sqrt{5}}{sqrt{5}} )= ( frac{426.92}{sqrt{5}} + frac{13.08 sqrt{5}}{sqrt{5}} )= ( frac{426.92}{sqrt{5}} + 13.08 )Similarly, ( B = frac{phi * 26.16 - 440}{sqrt{5}} )( phi = frac{1 + sqrt{5}}{2} ), so:= ( frac{frac{1 + sqrt{5}}{2} * 26.16 - 440}{sqrt{5}} )= ( frac{13.08 + 13.08 sqrt{5} - 440}{sqrt{5}} )= ( frac{-426.92 + 13.08 sqrt{5}}{sqrt{5}} )= ( frac{-426.92}{sqrt{5}} + frac{13.08 sqrt{5}}{sqrt{5}} )= ( frac{-426.92}{sqrt{5}} + 13.08 )So, A and B can be written as:( A = 13.08 + frac{426.92}{sqrt{5}} )( B = 13.08 - frac{426.92}{sqrt{5}} )But perhaps we can rationalize the denominators:( A = 13.08 + frac{426.92}{sqrt{5}} = 13.08 + frac{426.92 sqrt{5}}{5} = 13.08 + 85.384 sqrt{5} )Similarly,( B = 13.08 - frac{426.92}{sqrt{5}} = 13.08 - frac{426.92 sqrt{5}}{5} = 13.08 - 85.384 sqrt{5} )Therefore, the general form is:( f_n = left(13.08 + 85.384 sqrt{5}right) phi^n + left(13.08 - 85.384 sqrt{5}right) psi^n )But perhaps we can factor out 13.08 and 85.384:( f_n = 13.08 (phi^n + psi^n) + 85.384 sqrt{5} (phi^n - psi^n) )As I thought earlier, ( phi^n + psi^n = L_n ) (Lucas numbers) and ( phi^n - psi^n = sqrt{5} F_n ) (Fibonacci numbers). Therefore,( f_n = 13.08 L_n + 85.384 * 5 F_n ) because ( sqrt{5} * sqrt{5} = 5 )Wait, no:Wait, ( phi^n - psi^n = sqrt{5} F_n ), so ( 85.384 sqrt{5} (phi^n - psi^n) = 85.384 * sqrt{5} * sqrt{5} F_n = 85.384 * 5 F_n = 426.92 F_n )Therefore, ( f_n = 13.08 L_n + 426.92 F_n )So, that's another way to express it.But let me verify with n=1:( L_1 = 1 ), ( F_1 = 1 )So, ( f_1 = 13.08 * 1 + 426.92 * 1 = 13.08 + 426.92 = 440 ), which is correct.Similarly, n=2:( L_2 = 3 ), ( F_2 = 1 )( f_2 = 13.08 * 3 + 426.92 * 1 = 39.24 + 426.92 = 466.16 ), which matches.So, this seems correct.Therefore, the general form can be expressed as:( f_n = 13.08 L_n + 426.92 F_n )Where ( L_n ) is the nth Lucas number and ( F_n ) is the nth Fibonacci number.Alternatively, since Lucas and Fibonacci numbers are integer sequences, but our coefficients are decimal, perhaps it's better to leave it in terms of A and B with the exact expressions.But for the purpose of this problem, I think expressing it as ( f_n = A phi^n + B psi^n ) with A and B computed as above is sufficient.So, summarizing part 1: The general form of ( f_n ) is a linear combination of ( phi^n ) and ( psi^n ) with coefficients A ≈ 203.82 and B ≈ -177.66, or more precisely, ( A = 13.08 + 85.384 sqrt{5} ) and ( B = 13.08 - 85.384 sqrt{5} ).Moving on to part 2: The composer wants to use only frequencies that are integer multiples of a base frequency ( b ) Hz, which is the greatest common divisor (GCD) of the sequence terms ( f_1, f_2, ldots, f_{10} ). We need to calculate ( b ) and identify which terms are multiples of ( b ).First, let's note that the frequencies are defined by a linear recurrence relation with real coefficients. However, the initial terms ( f_1 = 440 ) Hz and ( f_2 = 466.16 ) Hz are not integers. Wait, 440 is an integer, but 466.16 is not. Hmm, that complicates things because the GCD is typically defined for integers. So, perhaps we need to consider the frequencies as real numbers and find a common divisor in terms of their decimal representations? Or maybe the problem expects us to treat the frequencies as exact decimals and find a common divisor in terms of their fractional parts.Wait, 440 is an integer, 466.16 is 46616/100 = 11654/25. Let me check if all terms are rational numbers. Since the recurrence is linear with rational coefficients (if we consider 466.16 as a rational number), then all terms will be rational.Indeed, ( f_1 = 440 = 440/1 ), ( f_2 = 466.16 = 46616/100 = 11654/25 ). Let me compute the next few terms to see if they are rational.Compute ( f_3 = f_2 + f_1 = 466.16 + 440 = 906.16 ) Hz( f_4 = f_3 + f_2 = 906.16 + 466.16 = 1372.32 ) Hz( f_5 = f_4 + f_3 = 1372.32 + 906.16 = 2278.48 ) Hz( f_6 = f_5 + f_4 = 2278.48 + 1372.32 = 3650.8 ) Hz( f_7 = f_6 + f_5 = 3650.8 + 2278.48 = 5929.28 ) Hz( f_8 = f_7 + f_6 = 5929.28 + 3650.8 = 9580.08 ) Hz( f_9 = f_8 + f_7 = 9580.08 + 5929.28 = 15509.36 ) Hz( f_{10} = f_9 + f_8 = 15509.36 + 9580.08 = 25089.44 ) HzSo, all terms up to ( f_{10} ) are rational numbers with denominators as powers of 10, since 466.16 is 46616/100, and adding integers (which are /1) to it will result in fractions with denominators dividing 100, 1000, etc.But to find the GCD of these terms, we need to express them as fractions and find the GCD of their numerators when expressed with a common denominator.Alternatively, since all terms are multiples of 0.01 Hz, perhaps the base frequency ( b ) is 0.01 Hz, but let's check.Wait, 440 Hz is 440.00, which is 44000/100, 466.16 is 46616/100, 906.16 is 90616/100, etc. So, all terms can be expressed as multiples of 0.01 Hz. Therefore, the GCD of all these terms would be at least 0.01 Hz. But is there a larger common divisor?To find the GCD, let's express each term as a fraction over 100:( f_1 = 440 = 44000/100 )( f_2 = 466.16 = 46616/100 )( f_3 = 906.16 = 90616/100 )( f_4 = 1372.32 = 137232/100 )( f_5 = 2278.48 = 227848/100 )( f_6 = 3650.8 = 365080/100 )( f_7 = 5929.28 = 592928/100 )( f_8 = 9580.08 = 958008/100 )( f_9 = 15509.36 = 1550936/100 )( f_{10} = 25089.44 = 2508944/100 )Now, the GCD of the numerators: 44000, 46616, 90616, 137232, 227848, 365080, 592928, 958008, 1550936, 2508944We need to find the GCD of these numbers.First, note that all numerators are even, so 2 is a common divisor. Let's divide each by 2:22000, 23308, 45308, 68616, 113924, 182540, 296464, 479004, 775468, 1254472Still even, divide by 2 again:11000, 11654, 22654, 34308, 56962, 91270, 148232, 239502, 387734, 627236Now, check if all are divisible by 2 again:11000 is even, 11654 is even, 22654 even, 34308 even, 56962 even, 91270 even, 148232 even, 239502 even, 387734 even, 627236 even.Divide by 2 again:5500, 5827, 11327, 17154, 28481, 45635, 74116, 119751, 193867, 313618Now, check divisibility:5500: factors include 5, 11, etc.5827: Let's check if it's prime. 5827 ÷ 7 = 832.428... no. 5827 ÷ 13 = 448.23... no. 5827 ÷ 17 = 342.76... no. 5827 ÷ 19 = 306.68... no. 5827 ÷ 23 = 253.347... no. 5827 ÷ 29 = 200.931... no. 5827 ÷ 31 = 187.967... no. 5827 ÷ 37 = 157.486... no. 5827 ÷ 43 = 135.511... no. 5827 ÷ 47 = 123.978... no. 5827 ÷ 53 = 110.0... Wait, 53*110 = 5830, which is 3 more, so no. So, 5827 might be prime.Similarly, 11327: Let's check divisibility. 11327 ÷ 7 = 1618.142... no. ÷11=1029.727... no. ÷13=871.307... no. ÷17=666.294... no. ÷19=596.157... no. ÷23=492.478... no. ÷29=390.586... no. ÷31=365.387... no. ÷37=306.135... no. ÷43=263.418... no. ÷47=240.999... ≈241, 47*241=11327? Let's check: 47*240=11280, plus 47=11327. Yes! So, 5827=47*124 (Wait, no, 47*124=5828, which is 1 more). Wait, no, earlier I thought 5827 ÷47≈123.978, which is not exact. Wait, 47*241=11327, which is another number. So, 5827 is prime? Or maybe not. Let me check 5827 ÷ 7=832.428, no. ÷13=448.23, no. ÷17=342.76, no. ÷19=306.68, no. ÷23=253.347, no. ÷29=200.931, no. ÷31=187.967, no. ÷37=157.486, no. ÷43=135.511, no. ÷47=123.978, no. ÷53=110.0, but 53*110=5830, which is 3 more. So, 5827 is prime.Similarly, 17154: even, so divisible by 2.28481: Let's check divisibility. 28481 ÷7=4068.714... no. ÷11=2589.181... no. ÷13=2190.846... no. ÷17=1675.352... no. ÷19=1499.0... Wait, 19*1499=28481? Let's compute 19*1500=28500, so 19*1499=28500-19=28481. Yes! So, 28481=19*1499.45635: ends with 5, divisible by 5: 45635=5*9127.74116: even.119751: sum of digits 1+1+9+7+5+1=24, divisible by 3: 119751 ÷3=39917.387734: even.313618: even.So, looking at the reduced numerators after dividing by 8 (since we divided by 2 three times):5500, 5827, 11327, 17154, 28481, 45635, 74116, 119751, 387734, 313618We need to find the GCD of these numbers. Since 5827 is prime, and 5500 is 5^3 * 2^2 * 11, 5827 is prime, so GCD is 1 because 5827 doesn't divide 5500.Wait, but 5827 is a prime number, and 5500 is 5^3 * 2^2 * 11, so they share no common factors. Therefore, the GCD of all these numerators is 1.But wait, let me check if all numerators are divisible by 1, which they are, but is there a larger common divisor?Wait, 5500 and 5827: GCD(5500,5827). Let's compute GCD(5500,5827).Using Euclidean algorithm:5827 ÷5500=1 with remainder 3275500 ÷327=16 with remainder 268 (since 327*16=5232, 5500-5232=268)327 ÷268=1 with remainder 59268 ÷59=4 with remainder 4259 ÷42=1 with remainder 1742 ÷17=2 with remainder 817 ÷8=2 with remainder 18 ÷1=8 with remainder 0So, GCD is 1.Therefore, the GCD of all numerators is 1.But wait, the numerators were divided by 8, so the original numerators had a GCD of 8, because we divided by 2 three times. Wait, no, the original numerators were 44000, 46616, etc., and we divided each by 8 to get 5500, 5827, etc. So, the GCD of the original numerators is 8 * GCD(5500,5827,...) = 8*1=8.But wait, let me think again. The original numerators are multiples of 8? No, we divided by 2 three times, so the original GCD was 8 times the GCD of the reduced numerators. Since the reduced numerators have GCD 1, the original numerators have GCD 8.But wait, let me confirm:If we have numbers N1, N2, ..., N10, and we divide each by 8 to get M1, M2, ..., M10, then GCD(N1, N2, ..., N10) = 8 * GCD(M1, M2, ..., M10). Since GCD(M1,...,M10)=1, then GCD(N1,...,N10)=8.Therefore, the GCD of the numerators is 8, and since all terms are expressed over 100, the GCD of the fractions is 8/100 = 0.08 Hz.Wait, but 8/100 simplifies to 2/25, which is 0.08 Hz.But let me verify:If all terms are multiples of 0.08 Hz, then each numerator must be divisible by 8, which they are, as we saw. Therefore, the base frequency ( b = 0.08 ) Hz.But let's check with the first few terms:( f_1 = 440 = 440.00 = 0.08 * 5500 )( f_2 = 466.16 = 0.08 * 5827 )( f_3 = 906.16 = 0.08 * 11327 )( f_4 = 1372.32 = 0.08 * 17154 )Yes, all these are multiples of 0.08 Hz.Therefore, the base frequency ( b = 0.08 ) Hz.Now, we need to identify which terms in the sequence up to ( f_{10} ) are integer multiples of ( b = 0.08 ) Hz. But wait, since we've established that all terms are multiples of 0.08 Hz, because their numerators are multiples of 8, then all terms ( f_1 ) to ( f_{10} ) are integer multiples of 0.08 Hz.Wait, but 0.08 Hz is a decimal, and the terms are given in Hz with two decimal places. So, when we say integer multiples, we mean that ( f_n / b ) is an integer.Given ( b = 0.08 ), then ( f_n / 0.08 ) must be integer.Since ( f_n = text{numerator}/100 ), then ( f_n / 0.08 = (text{numerator}/100) / (8/100) = text{numerator}/8 ), which is integer because numerator is divisible by 8.Therefore, all terms ( f_1 ) to ( f_{10} ) are integer multiples of ( b = 0.08 ) Hz.Wait, but let me check ( f_2 = 466.16 ). 466.16 / 0.08 = 5827, which is integer.Similarly, ( f_3 = 906.16 / 0.08 = 11327, integer.Yes, all terms are integer multiples of 0.08 Hz.Therefore, the base frequency ( b = 0.08 ) Hz, and all terms from ( f_1 ) to ( f_{10} ) are multiples of ( b ).But wait, the problem says "the base frequency ( b ) Hz, which is the greatest common divisor of the sequence terms ( f_1, f_2, ldots, f_n )". Since we're dealing with real numbers, the GCD is typically defined for integers, but in this case, since all terms are rational numbers with denominator 100, we can consider their numerators and find the GCD in terms of 1/100 Hz. But as we saw, the GCD of the numerators is 8, so the GCD of the terms is 8/100 = 0.08 Hz.Therefore, the base frequency ( b = 0.08 ) Hz, and all terms ( f_1 ) to ( f_{10} ) are multiples of ( b ).So, summarizing part 2: The base frequency ( b = 0.08 ) Hz, and all terms from ( f_1 ) to ( f_{10} ) are integer multiples of ( b ).</think>