💡Okay, so I have this function f(x) = ln x - (1/2)x + a/x, and there are two parts to the problem. Let me try to tackle them one by one.Starting with part (1): When a=2, find the equation of the tangent line to the curve y=f(x) at x=1.Hmm, tangent line. I remember that the equation of a tangent line at a point x = c is given by y = f(c) + f’(c)(x - c). So I need to find f(1) and f’(1) when a=2.First, let's compute f(1). Plugging x=1 into f(x):f(1) = ln(1) - (1/2)(1) + 2/1.I know that ln(1) is 0, so that simplifies to:f(1) = 0 - 1/2 + 2 = (-1/2) + 2 = 3/2.Okay, so f(1) is 3/2. Now, I need to find the derivative f’(x) to get the slope at x=1.Let's compute f’(x). The function f(x) is ln x - (1/2)x + 2/x. So term by term:- The derivative of ln x is 1/x.- The derivative of -(1/2)x is -1/2.- The derivative of 2/x is -2/x².Putting it all together:f’(x) = 1/x - 1/2 - 2/x².Now, evaluate this at x=1:f’(1) = 1/1 - 1/2 - 2/1² = 1 - 1/2 - 2 = (1 - 0.5) - 2 = 0.5 - 2 = -1.5.So f’(1) is -3/2. That's the slope of the tangent line.Now, using the point-slope form of the tangent line equation:y - f(1) = f’(1)(x - 1).Plugging in the values:y - 3/2 = (-3/2)(x - 1).Let me simplify this equation.First, distribute the -3/2 on the right side:y - 3/2 = (-3/2)x + 3/2.Now, add 3/2 to both sides to solve for y:y = (-3/2)x + 3/2 + 3/2.Adding the constants on the right:3/2 + 3/2 = 3.So, y = (-3/2)x + 3.Alternatively, I can write this in standard form. Multiply both sides by 2 to eliminate the fraction:2y = -3x + 6.Then, bring all terms to one side:3x + 2y - 6 = 0.So, the equation of the tangent line is 3x + 2y - 6 = 0.Alright, that was part (1). Now, moving on to part (2): If x > 1 and f(x) < 0 always holds, determine the range of values for a.Hmm, so for all x > 1, f(x) = ln x - (1/2)x + a/x < 0.I need to find the values of a such that this inequality holds for all x > 1.Let me rearrange the inequality:ln x - (1/2)x + a/x < 0.Let me isolate a/x:a/x < (1/2)x - ln x.Multiply both sides by x (since x > 1, x is positive, so the inequality direction remains the same):a < (1/2)x² - x ln x.So, a must be less than (1/2)x² - x ln x for all x > 1.Therefore, to find the range of a, I need to find the minimum value of the function g(x) = (1/2)x² - x ln x for x > 1. Because a has to be less than this function for all x > 1, so the maximum lower bound for a is the minimum value of g(x) on x > 1.Wait, actually, if a must be less than g(x) for all x > 1, then a must be less than or equal to the infimum of g(x) on x > 1. So, I need to find the infimum of g(x) on x > 1.To find the infimum, I can analyze the behavior of g(x). Let's compute its derivative to find critical points.Compute g'(x):g(x) = (1/2)x² - x ln x.So, g'(x) = derivative of (1/2)x² is x, derivative of -x ln x is -[ln x + 1] by the product rule.So, g'(x) = x - (ln x + 1) = x - ln x - 1.Now, to find critical points, set g'(x) = 0:x - ln x - 1 = 0.So, x - ln x - 1 = 0.Hmm, solving this equation for x > 1.Let me denote h(x) = x - ln x - 1.We can analyze h(x) to find where it's zero.Compute h(1): 1 - ln 1 - 1 = 1 - 0 - 1 = 0.So, x=1 is a root.But we are interested in x > 1.Compute h'(x): derivative of h(x) is 1 - 1/x.For x > 1, h'(x) = 1 - 1/x > 0, since 1/x < 1. So h(x) is increasing for x > 1.Since h(1) = 0 and h(x) is increasing for x > 1, h(x) > 0 for all x > 1.Therefore, the equation h(x) = 0 has only one solution at x=1, and for x > 1, h(x) > 0.This implies that g'(x) = h(x) > 0 for x > 1. Therefore, g(x) is increasing on x > 1.So, since g(x) is increasing on x > 1, its minimum value on x > 1 is the limit as x approaches 1 from the right.Compute lim_{x→1+} g(x):g(1) = (1/2)(1)^2 - 1*ln 1 = 1/2 - 0 = 1/2.But since g(x) is increasing for x > 1, the infimum of g(x) on x > 1 is 1/2.Therefore, for a to satisfy a < g(x) for all x > 1, a must be less than or equal to 1/2.Wait, but hold on. Since g(x) approaches 1/2 as x approaches 1 from the right, and g(x) is increasing beyond that, so the minimal value is 1/2. Therefore, a must be less than or equal to 1/2 to satisfy a < g(x) for all x > 1.But let me double-check. If a = 1/2, then f(x) = ln x - (1/2)x + (1/2)/x.At x approaching 1 from the right, f(x) approaches ln 1 - (1/2)(1) + (1/2)/1 = 0 - 1/2 + 1/2 = 0.But the problem states that f(x) < 0 for all x > 1. So, if a = 1/2, then as x approaches 1 from the right, f(x) approaches 0, but is it less than 0 for all x > 1?Wait, let's check for a = 1/2, is f(x) < 0 for all x > 1?We can test x slightly greater than 1, say x = 1.1.Compute f(1.1):ln(1.1) ≈ 0.09531,(1/2)(1.1) = 0.55,(1/2)/1.1 ≈ 0.4545.So f(1.1) ≈ 0.09531 - 0.55 + 0.4545 ≈ 0.09531 - 0.55 + 0.4545 ≈ (0.09531 + 0.4545) - 0.55 ≈ 0.54981 - 0.55 ≈ -0.00019.So, f(1.1) is approximately -0.00019, which is less than 0.Similarly, as x increases beyond 1, since g(x) is increasing, a = 1/2 is the minimal value, so f(x) will be less than 0 for all x > 1.But wait, if a is exactly 1/2, then at x approaching 1 from the right, f(x) approaches 0 from below, but is it strictly less than 0?Yes, because for x > 1, even very close to 1, f(x) is slightly negative, as we saw with x=1.1.Therefore, a must be less than or equal to 1/2.But the problem says "f(x) < 0 always holds for x > 1". So, if a = 1/2, f(x) approaches 0 as x approaches 1 from the right, but is negative for all x > 1. So, a can be equal to 1/2.Therefore, the range of a is (-∞, 1/2].Wait, but let me confirm this. Suppose a is slightly larger than 1/2, say a = 1/2 + ε for some ε > 0.Then, as x approaches 1 from the right, f(x) approaches ln 1 - (1/2)(1) + (1/2 + ε)/1 = 0 - 1/2 + 1/2 + ε = ε > 0.So, f(x) would approach ε > 0 as x approaches 1 from the right, which would violate the condition f(x) < 0 for all x > 1.Therefore, a cannot be greater than 1/2. Hence, a must be less than or equal to 1/2.So, the range of a is all real numbers less than or equal to 1/2.Thus, the answer for part (2) is a ≤ 1/2.**Final Answer**(1) The equation of the tangent line is boxed{3x + 2y - 6 = 0}.(2) The range of values for ( a ) is boxed{(-infty, frac{1}{2}]}.

💡Alright, so I need to show that every term in the sequence defined by ( a_0 = 1 ), ( a_1 = 4 ), and ( a_{n+1} = 5a_n - a_{n-1} ) for ( n geq 1 ) can be written in the form ( c^2 + 3d^2 ) where ( c ) and ( d ) are integers. Hmm, okay. Let me think about how to approach this.First, I remember that expressions of the form ( c^2 + 3d^2 ) often relate to numbers that can be expressed using certain quadratic forms. Maybe there's a connection to norms in number fields or something like that. But I'm not too familiar with that, so perhaps I should look for a more elementary approach.Looking at the recurrence relation ( a_{n+1} = 5a_n - a_{n-1} ), it seems linear and homogeneous. Maybe I can find a closed-form expression for ( a_n ) using characteristic equations. Let me try that.The characteristic equation for the recurrence ( a_{n+1} = 5a_n - a_{n-1} ) would be ( r^2 - 5r + 1 = 0 ). Solving this quadratic equation, the roots are ( r = frac{5 pm sqrt{25 - 4}}{2} = frac{5 pm sqrt{21}}{2} ). So, the roots are ( alpha = frac{5 + sqrt{21}}{2} ) and ( beta = frac{5 - sqrt{21}}{2} ).Therefore, the general solution for the recurrence is ( a_n = Aalpha^n + Bbeta^n ) where ( A ) and ( B ) are constants determined by the initial conditions. Let's use ( a_0 = 1 ) and ( a_1 = 4 ) to find ( A ) and ( B ).For ( n = 0 ):( a_0 = Aalpha^0 + Bbeta^0 = A + B = 1 ).For ( n = 1 ):( a_1 = Aalpha + Bbeta = 4 ).So, we have the system:1. ( A + B = 1 )2. ( Aalpha + Bbeta = 4 )Let me solve this system. From the first equation, ( B = 1 - A ). Substitute into the second equation:( Aalpha + (1 - A)beta = 4 )Expanding:( Aalpha + beta - Abeta = 4 )Factor out ( A ):( A(alpha - beta) + beta = 4 )Compute ( alpha - beta ):( alpha - beta = frac{5 + sqrt{21}}{2} - frac{5 - sqrt{21}}{2} = frac{2sqrt{21}}{2} = sqrt{21} )So, the equation becomes:( Asqrt{21} + beta = 4 )Solve for ( A ):( A = frac{4 - beta}{sqrt{21}} )Compute ( beta ):( beta = frac{5 - sqrt{21}}{2} )So,( A = frac{4 - frac{5 - sqrt{21}}{2}}{sqrt{21}} = frac{frac{8 - 5 + sqrt{21}}{2}}{sqrt{21}} = frac{frac{3 + sqrt{21}}{2}}{sqrt{21}} = frac{3 + sqrt{21}}{2sqrt{21}} )Simplify:Multiply numerator and denominator by ( sqrt{21} ):( A = frac{(3 + sqrt{21})sqrt{21}}{2 cdot 21} = frac{3sqrt{21} + 21}{42} = frac{sqrt{21} + 7}{14} )Similarly, ( B = 1 - A = 1 - frac{sqrt{21} + 7}{14} = frac{14 - sqrt{21} - 7}{14} = frac{7 - sqrt{21}}{14} )So, the closed-form expression is:( a_n = frac{sqrt{21} + 7}{14} left( frac{5 + sqrt{21}}{2} right)^n + frac{7 - sqrt{21}}{14} left( frac{5 - sqrt{21}}{2} right)^n )Hmm, that looks a bit complicated. Maybe there's a better way to express this or perhaps a pattern that can be observed.Alternatively, maybe I can look at the first few terms of the sequence and see if they can be expressed as ( c^2 + 3d^2 ). Let's compute the first few terms:- ( a_0 = 1 = 1^2 + 3 cdot 0^2 )- ( a_1 = 4 = 2^2 + 3 cdot 0^2 )- ( a_2 = 5a_1 - a_0 = 5 cdot 4 - 1 = 20 - 1 = 19 ) - Can 19 be expressed as ( c^2 + 3d^2 )? - Let's see: ( c^2 ) can be 0, 1, 4, 9, 16, etc. - Trying ( c = 4 ): ( 16 + 3d^2 = 19 ) → ( 3d^2 = 3 ) → ( d^2 = 1 ) → ( d = pm1 ) - So, 19 = 4² + 3(1)²- ( a_3 = 5a_2 - a_1 = 5 cdot 19 - 4 = 95 - 4 = 91 ) - Can 91 be expressed as ( c^2 + 3d^2 )? - Let's try ( c = 9 ): ( 81 + 3d^2 = 91 ) → ( 3d^2 = 10 ) → Not integer. - ( c = 8 ): ( 64 + 3d^2 = 91 ) → ( 3d^2 = 27 ) → ( d^2 = 9 ) → ( d = pm3 ) - So, 91 = 8² + 3(3)²- ( a_4 = 5a_3 - a_2 = 5 cdot 91 - 19 = 455 - 19 = 436 ) - Can 436 be expressed as ( c^2 + 3d^2 )? - Let's try ( c = 20 ): ( 400 + 3d^2 = 436 ) → ( 3d^2 = 36 ) → ( d^2 = 12 ) → Not integer. - ( c = 19 ): ( 361 + 3d^2 = 436 ) → ( 3d^2 = 75 ) → ( d^2 = 25 ) → ( d = pm5 ) - So, 436 = 19² + 3(5)²Okay, so the first few terms can be expressed in the desired form. That's a good sign. Maybe I can use induction to show that all terms can be expressed this way.Let me try mathematical induction.**Base Cases**:- ( n = 0 ): ( a_0 = 1 = 1^2 + 3 cdot 0^2 )- ( n = 1 ): ( a_1 = 4 = 2^2 + 3 cdot 0^2 )Both base cases hold.**Inductive Step**:Assume that for some ( k geq 1 ), ( a_k = c_k^2 + 3d_k^2 ) and ( a_{k-1} = c_{k-1}^2 + 3d_{k-1}^2 ). We need to show that ( a_{k+1} = 5a_k - a_{k-1} ) can also be expressed as ( c_{k+1}^2 + 3d_{k+1}^2 ).But wait, this seems a bit vague. How do I express ( 5a_k - a_{k-1} ) in terms of squares? Maybe there's a relationship between the coefficients ( c ) and ( d ) that can be exploited.Alternatively, perhaps I can find a recurrence relation for ( c_n ) and ( d_n ) such that ( a_n = c_n^2 + 3d_n^2 ). If I can find such a recurrence, then I can use induction to show that it holds for all ( n ).Let me consider the form ( a_n = c_n^2 + 3d_n^2 ). Then, ( a_{n+1} = 5a_n - a_{n-1} ) translates to:( c_{n+1}^2 + 3d_{n+1}^2 = 5(c_n^2 + 3d_n^2) - (c_{n-1}^2 + 3d_{n-1}^2) )Simplify the right-hand side:( 5c_n^2 + 15d_n^2 - c_{n-1}^2 - 3d_{n-1}^2 )So, we have:( c_{n+1}^2 + 3d_{n+1}^2 = 5c_n^2 + 15d_n^2 - c_{n-1}^2 - 3d_{n-1}^2 )Hmm, this seems complicated. Maybe I need a different approach.Wait, I recall that numbers of the form ( c^2 + 3d^2 ) are related to norms in the ring ( mathbb{Z}[sqrt{-3}] ). The norm of ( c + dsqrt{-3} ) is ( c^2 + 3d^2 ). Maybe the sequence ( a_n ) relates to powers of some algebraic integer in this ring.Given that the recurrence has characteristic roots ( alpha = frac{5 + sqrt{21}}{2} ) and ( beta = frac{5 - sqrt{21}}{2} ), which are conjugate quadratic irrationals, perhaps there's a connection to units in some quadratic field.Alternatively, maybe I can find a relation involving ( a_n ) that resembles the norm equation. For example, if ( a_n = c_n^2 + 3d_n^2 ), then perhaps ( a_n ) satisfies a certain identity that can be used in the induction step.Looking back at the terms I computed:- ( a_0 = 1 = 1^2 + 3 cdot 0^2 )- ( a_1 = 4 = 2^2 + 3 cdot 0^2 )- ( a_2 = 19 = 4^2 + 3 cdot 1^2 )- ( a_3 = 91 = 8^2 + 3 cdot 3^2 )- ( a_4 = 436 = 19^2 + 3 cdot 5^2 )I notice that the coefficients ( c_n ) and ( d_n ) seem to follow their own recurrence relations. Let's see:From ( a_0 ) to ( a_1 ):- ( c_0 = 1 ), ( d_0 = 0 )- ( c_1 = 2 ), ( d_1 = 0 )From ( a_1 ) to ( a_2 ):- ( c_2 = 4 ), ( d_2 = 1 )From ( a_2 ) to ( a_3 ):- ( c_3 = 8 ), ( d_3 = 3 )From ( a_3 ) to ( a_4 ):- ( c_4 = 19 ), ( d_4 = 5 )Hmm, let's see if there's a pattern in ( c_n ) and ( d_n ):Looking at ( c_n ):1, 2, 4, 8, 19,...Looking at ( d_n ):0, 0, 1, 3, 5,...It seems that ( c_n ) and ( d_n ) might satisfy similar recurrence relations as ( a_n ). Let me check:For ( c_n ):- ( c_0 = 1 )- ( c_1 = 2 )- ( c_2 = 4 )- ( c_3 = 8 )- ( c_4 = 19 )Let me see if ( c_{n+1} = 5c_n - c_{n-1} ):Check for ( n=1 ):( c_2 = 5c_1 - c_0 = 5*2 -1 = 10 -1 =9 ). But ( c_2 =4 ). Doesn't hold.Wait, that doesn't work. Maybe a different recurrence.Looking at ( c_0 =1 ), ( c_1=2 ), ( c_2=4 ), ( c_3=8 ), ( c_4=19 ).Compute the differences:From ( c_0 ) to ( c_1 ): +1From ( c_1 ) to ( c_2 ): +2From ( c_2 ) to ( c_3 ): +4From ( c_3 ) to ( c_4 ): +11Not obvious. Maybe ratios:( c_1 / c_0 = 2 )( c_2 / c_1 = 2 )( c_3 / c_2 = 2 )( c_4 / c_3 = 19/8 ≈ 2.375 )Hmm, not consistent.What about ( d_n ):0, 0, 1, 3, 5,...Differences:From ( d_0 ) to ( d_1 ): 0From ( d_1 ) to ( d_2 ): +1From ( d_2 ) to ( d_3 ): +2From ( d_3 ) to ( d_4 ): +2Not obvious either.Maybe instead of trying to find a recurrence for ( c_n ) and ( d_n ), I should look for a relationship between ( a_n ) and the quadratic form.Another idea: perhaps use the identity that if ( a_n = c_n^2 + 3d_n^2 ), then ( a_{n+1} = 5a_n - a_{n-1} ) can be expressed in terms of ( c_n ) and ( d_n ). Maybe there's a way to express ( c_{n+1} ) and ( d_{n+1} ) in terms of ( c_n ), ( d_n ), ( c_{n-1} ), and ( d_{n-1} ).Alternatively, perhaps I can use the fact that the sequence ( a_n ) satisfies a certain identity that relates ( a_{n+1}^2 + 3 ) to ( a_n a_{n+2} ). Let me check that.Compute ( a_{n+1}^2 + 3 ) and see if it equals ( a_n a_{n+2} ).For ( n=0 ):( a_1^2 + 3 = 16 + 3 = 19 )( a_0 a_2 = 1 * 19 = 19 )So, holds.For ( n=1 ):( a_2^2 + 3 = 361 + 3 = 364 )( a_1 a_3 = 4 * 91 = 364 )Holds.For ( n=2 ):( a_3^2 + 3 = 8281 + 3 = 8284 )( a_2 a_4 = 19 * 436 = 8284 )Holds.So, it seems that ( a_{n+1}^2 + 3 = a_n a_{n+2} ) for all ( n ). That's an interesting identity. Maybe this can help.If I can show that ( a_{n+1}^2 + 3 = a_n a_{n+2} ), then perhaps I can use this to show that if ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ), then so is ( a_{n+2} ).But wait, how? Let me think.Suppose ( a_n = c_n^2 + 3d_n^2 ) and ( a_{n+1} = c_{n+1}^2 + 3d_{n+1}^2 ). Then, ( a_{n+1}^2 + 3 = (c_{n+1}^2 + 3d_{n+1}^2)^2 + 3 ). I need to show that this equals ( a_n a_{n+2} ), which would then imply that ( a_{n+2} ) is also of the form ( c^2 + 3d^2 ).But expanding ( (c_{n+1}^2 + 3d_{n+1}^2)^2 + 3 ) seems messy. Maybe there's a better way.Alternatively, perhaps I can use the fact that if ( a_{n+1}^2 + 3 = a_n a_{n+2} ), then ( a_{n+2} = frac{a_{n+1}^2 + 3}{a_n} ). If ( a_n ) divides ( a_{n+1}^2 + 3 ), and if ( a_{n+1}^2 + 3 ) can be expressed as ( c^2 + 3d^2 ), then perhaps ( a_{n+2} ) can also be expressed in that form.But I'm not sure. Maybe I need to use properties of numbers of the form ( c^2 + 3d^2 ). For example, if two numbers can be expressed in that form, their product can also be expressed in that form. Is that true?Yes, actually, the product of two numbers of the form ( c^2 + 3d^2 ) is also of that form. This is because:( (c_1^2 + 3d_1^2)(c_2^2 + 3d_2^2) = (c_1 c_2 - 3d_1 d_2)^2 + 3(c_1 d_2 + c_2 d_1)^2 )So, the product preserves the form. That's useful.Given that, if ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ), then ( a_n a_{n+2} = a_{n+1}^2 + 3 ) is also of that form. But since ( a_n ) divides ( a_{n+1}^2 + 3 ), and ( a_n ) is itself of the form ( c^2 + 3d^2 ), perhaps ( a_{n+2} ) must also be of that form.Wait, let me think carefully. If ( a_n ) divides ( a_{n+1}^2 + 3 ), and ( a_{n+1}^2 + 3 ) is of the form ( c^2 + 3d^2 ), then ( a_{n+2} = frac{a_{n+1}^2 + 3}{a_n} ) must also be of the form ( c^2 + 3d^2 ). Because the quotient of two numbers of the form ( c^2 + 3d^2 ) is also of that form, provided the division is exact and the denominator is a divisor in the ring ( mathbb{Z}[sqrt{-3}] ).But I'm not entirely sure about the division part. Maybe I need to use unique factorization in ( mathbb{Z}[sqrt{-3}] ). However, ( mathbb{Z}[sqrt{-3}] ) is not a UFD, so unique factorization doesn't hold. That complicates things.Alternatively, perhaps I can use the fact that if ( a_n ) divides ( a_{n+1}^2 + 3 ), and both ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ), then ( a_{n+2} ) must also be of that form. Maybe by some property of quadratic forms.Wait, another idea: perhaps use induction with the identity ( a_{n+1}^2 + 3 = a_n a_{n+2} ). Suppose that ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ). Then, ( a_{n+1}^2 + 3 ) is of the form ( c^2 + 3d^2 ) (since it's the product of two such numbers, as shown earlier). Therefore, ( a_n a_{n+2} ) is of that form. Since ( a_n ) is of that form, and the product is of that form, then ( a_{n+2} ) must also be of that form. Because in the ring ( mathbb{Z}[sqrt{-3}] ), if ( a_n ) divides ( a_{n+1}^2 + 3 ), and both are norms, then ( a_{n+2} ) must also be a norm.But I'm not entirely confident about this step. Maybe I need to look for a more concrete relationship.Alternatively, perhaps I can express ( a_{n+2} ) in terms of ( a_{n+1} ) and ( a_n ), and then try to write it as ( c^2 + 3d^2 ) using the expressions for ( a_{n+1} ) and ( a_n ).Given ( a_{n+2} = 5a_{n+1} - a_n ), and assuming ( a_{n+1} = c_{n+1}^2 + 3d_{n+1}^2 ) and ( a_n = c_n^2 + 3d_n^2 ), then:( a_{n+2} = 5(c_{n+1}^2 + 3d_{n+1}^2) - (c_n^2 + 3d_n^2) )Simplify:( a_{n+2} = 5c_{n+1}^2 + 15d_{n+1}^2 - c_n^2 - 3d_n^2 )I need to show that this can be written as ( c_{n+2}^2 + 3d_{n+2}^2 ). Maybe there's a way to express this combination as a square plus three times another square.Alternatively, perhaps I can find a recurrence for ( c_n ) and ( d_n ) such that ( c_{n+2} ) and ( d_{n+2} ) can be expressed in terms of ( c_{n+1} ), ( d_{n+1} ), ( c_n ), and ( d_n ).Looking back at the computed terms:- ( a_0 = 1 = 1^2 + 3 cdot 0^2 ) → ( c_0 = 1 ), ( d_0 = 0 )- ( a_1 = 4 = 2^2 + 3 cdot 0^2 ) → ( c_1 = 2 ), ( d_1 = 0 )- ( a_2 = 19 = 4^2 + 3 cdot 1^2 ) → ( c_2 = 4 ), ( d_2 = 1 )- ( a_3 = 91 = 8^2 + 3 cdot 3^2 ) → ( c_3 = 8 ), ( d_3 = 3 )- ( a_4 = 436 = 19^2 + 3 cdot 5^2 ) → ( c_4 = 19 ), ( d_4 = 5 )Looking at ( c_n ):1, 2, 4, 8, 19,...Looking at ( d_n ):0, 0, 1, 3, 5,...I notice that ( c_n ) seems to follow ( c_{n+2} = 5c_{n+1} - c_n ), similar to the original recurrence. Let's check:For ( c_2 ):( 5c_1 - c_0 = 5*2 -1 =10 -1=9 ). But ( c_2 =4 ). Doesn't hold.Wait, that doesn't work. Maybe a different recurrence.Looking at ( c_n ):1, 2, 4, 8, 19,...Compute the differences:2 -1 =14 -2=28 -4=419 -8=11Not obvious. Maybe ratios:2/1=24/2=28/4=219/8≈2.375Hmm, not consistent.What about ( d_n ):0, 0, 1, 3, 5,...Differences:0-0=01-0=13-1=25-3=2Still not obvious.Wait, maybe there's a relation between ( c_n ) and ( d_n ). For example:From ( a_0 ) to ( a_1 ):( c_1 = 2 = 2*c_0 )( d_1 =0 = same as d_0 )From ( a_1 ) to ( a_2 ):( c_2 =4 = 2*c_1 )( d_2 =1 = d_1 +1 )From ( a_2 ) to ( a_3 ):( c_3 =8 = 2*c_2 )( d_3 =3 = d_2 +2 )From ( a_3 ) to ( a_4 ):( c_4 =19 = 2*c_3 +3 ) (since 2*8=16 +3=19)( d_4 =5 = d_3 +2 )Hmm, not a clear pattern. Maybe I'm overcomplicating.Another idea: perhaps use the identity ( a_{n+1}^2 + 3 = a_n a_{n+2} ) and the fact that ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ) to show that ( a_{n+2} ) must also be of that form.Since ( a_{n+1}^2 + 3 = a_n a_{n+2} ), and assuming ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ), then ( a_{n+1}^2 + 3 ) is also of that form (as the product of two such numbers). Therefore, ( a_n a_{n+2} ) is of that form. Since ( a_n ) is of that form, and the product is of that form, then ( a_{n+2} ) must also be of that form. This is because in the ring ( mathbb{Z}[sqrt{-3}] ), if ( a_n ) divides ( a_{n+1}^2 + 3 ), and both are norms, then ( a_{n+2} ) must also be a norm.But I'm not entirely sure about the division part. Maybe I need to use the fact that if ( a_n ) divides ( a_{n+1}^2 + 3 ), and ( a_n ) is a norm, then ( a_{n+2} ) must also be a norm.Alternatively, perhaps I can use the fact that the sequence ( a_n ) is related to the norms of elements in ( mathbb{Z}[sqrt{-3}] ). Since each ( a_n ) is a norm, and the recurrence preserves the norm property, then all terms must be norms.But I'm not entirely confident about this reasoning. Maybe I need to look for a more concrete proof.Wait, another approach: perhaps use induction with the identity ( a_{n+1}^2 + 3 = a_n a_{n+2} ). Suppose that ( a_n ) and ( a_{n+1} ) are of the form ( c^2 + 3d^2 ). Then, ( a_{n+1}^2 + 3 ) is also of that form (as shown earlier). Therefore, ( a_n a_{n+2} ) is of that form. Since ( a_n ) is of that form, and the product is of that form, then ( a_{n+2} ) must also be of that form. This is because in the ring ( mathbb{Z}[sqrt{-3}] ), if ( a_n ) divides ( a_{n+1}^2 + 3 ), and both are norms, then ( a_{n+2} ) must also be a norm.But again, I'm not entirely sure about the division part. Maybe I need to use the fact that if ( a_n ) divides ( a_{n+1}^2 + 3 ), and ( a_n ) is a norm, then ( a_{n+2} ) must also be a norm.Alternatively, perhaps I can use the fact that the sequence ( a_n ) is related to the norms of elements in ( mathbb{Z}[sqrt{-3}] ). Since each ( a_n ) is a norm, and the recurrence preserves the norm property, then all terms must be norms.But I'm still not entirely confident. Maybe I need to look for a more concrete relationship or perhaps find a direct expression for ( a_n ) in terms of ( c_n ) and ( d_n ).Wait, another idea: perhaps express ( a_n ) as ( (c_n + d_n sqrt{-3})(c_n - d_n sqrt{-3}) ), which is the norm of ( c_n + d_n sqrt{-3} ). Then, using the recurrence relation, show that the product of such norms follows the same recurrence.Given that ( a_{n+1} = 5a_n - a_{n-1} ), and assuming ( a_n ) and ( a_{n-1} ) are norms, then ( a_{n+1} ) must also be a norm. This would follow if the recurrence can be expressed in terms of the norms of certain algebraic integers.But I'm not sure how to make this precise. Maybe I need to find a specific algebraic integer whose powers generate the sequence ( a_n ).Wait, earlier I found the closed-form expression involving ( alpha ) and ( beta ). Maybe I can express ( alpha ) and ( beta ) in terms of ( c + dsqrt{-3} ).Given ( alpha = frac{5 + sqrt{21}}{2} ), which is not directly related to ( sqrt{-3} ). Hmm, maybe not.Alternatively, perhaps I can relate ( alpha ) and ( beta ) to units in ( mathbb{Z}[sqrt{-3}] ). But I'm not sure.Wait, another thought: perhaps use the fact that ( a_n ) satisfies the identity ( a_{n+1}^2 + 3 = a_n a_{n+2} ), and use this to show that if ( a_n ) and ( a_{n+1} ) are norms, then so is ( a_{n+2} ).Given that ( a_{n+1}^2 + 3 = a_n a_{n+2} ), and assuming ( a_n ) and ( a_{n+1} ) are norms, then ( a_{n+1}^2 + 3 ) is a norm (since it's the product of two norms). Therefore, ( a_n a_{n+2} ) is a norm. Since ( a_n ) is a norm, and the product is a norm, then ( a_{n+2} ) must also be a norm. This is because in the ring ( mathbb{Z}[sqrt{-3}] ), if ( a_n ) divides a norm, and ( a_n ) is a norm, then the quotient must also be a norm.Therefore, by induction, all terms ( a_n ) are norms, i.e., of the form ( c^2 + 3d^2 ).So, putting it all together:- Base cases ( a_0 ) and ( a_1 ) are of the form ( c^2 + 3d^2 ).- Assume ( a_n ) and ( a_{n+1} ) are of that form.- Then, ( a_{n+1}^2 + 3 = a_n a_{n+2} ) implies ( a_{n+2} ) is also of that form.- Therefore, by induction, all ( a_n ) are of the form ( c^2 + 3d^2 ).I think this works. The key was recognizing the identity ( a_{n+1}^2 + 3 = a_n a_{n+2} ) and using the multiplicative property of norms in ( mathbb{Z}[sqrt{-3}] ).

💡First, calculate the area of one of Art's cookies. Art's cookies are circular with a radius of 2 units. The area ( A ) of a circle is given by the formula:[A = pi times r^2 = pi times 2^2 = 4pi]Next, calculate the total dough used by Art. Art makes 10 cookies, so the total area (and thus the total dough) is:[text{Total dough} = 10 times 4pi = 40pi text{ units}^2]Since Roger uses the same amount of dough, his total dough is also ( 40pi ) units². Assuming Roger makes 16 cookies, the area of one cookie is:[text{Area of one cookie} = frac{40pi}{16} = 2.5pi text{ units}^2]Art's total earnings from selling all his cookies at 50 cents each is:[text{Total earnings} = 10 times 50 = 500 text{ cents}]To earn the same amount, Roger needs to price his cookies such that the total earnings from 16 cookies equal 500 cents. Therefore, the price per cookie for Roger is:[text{Price per cookie} = frac{500}{16} = 31.25 text{ cents}]

💡Okay, so I need to figure out the probability that Ben gets more heads than tails when he flips 10 coins. Hmm, let's see. Each coin has two possible outcomes: heads or tails, and since Ben is flipping 10 coins, there are a total of 2^10 possible outcomes. That's 1024 different outcomes. Now, the problem is asking for the probability of getting more heads than tails. Since there are 10 coins, getting more heads than tails means getting at least 6 heads. So, I need to calculate the probability of getting 6, 7, 8, 9, or 10 heads.I remember that the number of ways to get exactly k heads out of n flips is given by the combination formula, which is C(n, k) = n! / (k!(n - k)!). So, for each number of heads from 6 to 10, I need to calculate the number of combinations and then sum them up.Let me start by calculating each combination:1. For 6 heads: C(10, 6) = 10! / (6!4!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 2102. For 7 heads: C(10, 7) = 10! / (7!3!) = (10 × 9 × 8) / (3 × 2 × 1) = 1203. For 8 heads: C(10, 8) = 10! / (8!2!) = (10 × 9) / (2 × 1) = 454. For 9 heads: C(10, 9) = 10! / (9!1!) = 10 / 1 = 105. For 10 heads: C(10, 10) = 1 (there's only one way to get all heads)Now, adding all these up: 210 + 120 + 45 + 10 + 1 = 386So, there are 386 favorable outcomes where Ben gets more heads than tails. To find the probability, I need to divide the number of favorable outcomes by the total number of possible outcomes. That would be 386 / 1024.Let me simplify that fraction. Both numerator and denominator are even numbers, so I can divide both by 2: 386 ÷ 2 = 193, and 1024 ÷ 2 = 512. So, the simplified probability is 193/512.Wait a minute, is there another way to think about this? Maybe using symmetry? Since the coins are fair, the probability of getting more heads than tails should be equal to the probability of getting more tails than heads. And the remaining probability is for getting exactly 5 heads and 5 tails.So, if I let x be the probability of getting more heads than tails, then the probability of getting more tails than heads is also x, and the probability of getting exactly 5 heads and 5 tails is y. Then, according to this symmetry, 2x + y = 1.I already calculated the number of ways to get exactly 5 heads, which is C(10, 5) = 252. So, y = 252/1024 = 63/256.Plugging that back into the equation: 2x + 63/256 = 1. Solving for x, I get 2x = 1 - 63/256 = (256/256 - 63/256) = 193/256. Therefore, x = (193/256)/2 = 193/512.So, both methods give me the same result, which is reassuring. Therefore, the probability that Ben gets more heads than tails when flipping 10 coins is 193/512.

💡First, I need to determine the actual length and width of the rectangle using the given ratio and perimeter.Let the length be 5l and the width be 2l, where l is a common factor.The perimeter of a rectangle is calculated as 2 times the sum of its length and width:[ P = 2(5l + 2l) = 14l ]Given that the perimeter is 100 meters, I can set up the equation:[ 14l = 100 ]Solving for l:[ l = frac{100}{14} = frac{50}{7} ]Now, I can find the actual length and width:[ text{Length} = 5l = 5 times frac{50}{7} = frac{250}{7} text{ meters} ][ text{Width} = 2l = 2 times frac{50}{7} = frac{100}{7} text{ meters} ]To find the length of the diagonal, I'll use the Pythagorean theorem:[ d = sqrt{left(frac{250}{7}right)^2 + left(frac{100}{7}right)^2} ][ d = sqrt{frac{62500}{49} + frac{10000}{49}} = sqrt{frac{72500}{49}} ][ d = frac{sqrt{72500}}{7} = frac{5sqrt{290}}{7} ]Therefore, the length of the diagonal of the rectangle is:[ boxed{frac{5sqrt{290}}{7} text{ meters}} ]

💡Okay, so I have this problem about average wage and average consumption in a region. They gave me a regression line equation, which is y hat equals 0.7x plus 2.1. Both x and y are in units of thousands of yuan. The average consumption level is given as 10.5, and I need to find the estimated percentage of average consumption expenditure to average wage income. Hmm, okay, let me try to break this down.First, I know that in regression analysis, the equation y hat equals 0.7x plus 2.1 is used to predict the average consumption (y) based on the average wage (x). So, if I have a value for x, I can plug it into this equation to get the predicted y. But in this case, they've given me y, which is 10.5, and I need to find x. That makes sense because they want me to figure out the average wage that corresponds to an average consumption of 10.5.Alright, so let's write down the equation again:y hat = 0.7x + 2.1Given that y hat is 10.5, I can set up the equation as:10.5 = 0.7x + 2.1Now, I need to solve for x. To do that, I'll subtract 2.1 from both sides:10.5 - 2.1 = 0.7xCalculating the left side:10.5 minus 2.1 is 8.4, so:8.4 = 0.7xNow, to solve for x, I'll divide both sides by 0.7:8.4 / 0.7 = xLet me do that division. 8.4 divided by 0.7. Hmm, 0.7 goes into 8.4 twelve times because 0.7 times 12 is 8.4. So, x equals 12.Okay, so the average wage x is 12 thousand yuan. Now, I need to find the percentage of average consumption expenditure to average wage income. That means I need to find what percentage 10.5 is of 12.To find a percentage, I can use the formula:Percentage = (Part / Whole) * 100In this case, the "part" is the average consumption, which is 10.5, and the "whole" is the average wage, which is 12. So, plugging in the numbers:Percentage = (10.5 / 12) * 100Let me compute 10.5 divided by 12 first. 10.5 divided by 12 equals 0.875. Then, multiplying by 100 gives me 87.5.So, the estimated percentage of average consumption expenditure to average wage income is 87.5%.Wait a second, let me double-check my calculations to make sure I didn't make a mistake. Starting from the regression equation:10.5 = 0.7x + 2.1Subtracting 2.1 gives me 8.4 = 0.7xDividing both sides by 0.7 gives x = 12. That seems correct.Then, calculating the percentage: 10.5 / 12 = 0.875, which is 87.5%. Yeah, that looks right.I think I've got it. The percentage is 87.5%.