💡First, I need to determine the total percentage points Jamie needs to achieve an average of 85% over four quarters. This is calculated by multiplying the desired average by the number of quarters: 85% × 4 = 340%.Next, I'll add up the scores Jamie has already received in the first three quarters: 85% + 80% + 90% = 255%.To find out the minimum score needed in the fourth quarter, I'll subtract the total points Jamie has already earned from the required total: 340% - 255% = 85%.Therefore, Jamie must score at least 85% in the fourth quarter to qualify for the geometry class.

💡First, I need to solve the first inequality ( x - 2 > 1 ). To isolate ( x ), I'll add 2 to both sides of the inequality:[x - 2 + 2 > 1 + 2]This simplifies to:[x > 3]Next, I'll consider the second inequality ( x < 4 ). This inequality is already solved for ( x ).Now, I'll combine the solutions of both inequalities. From the first inequality, ( x > 3 ), and from the second inequality, ( x < 4 ). Combining these two results gives:[3 < x < 4]Therefore, the solution set of the system of linear inequalities is ( 3 < x < 4 ).

💡Okay, so I have this problem about a geometric sequence. Let me try to understand it step by step. The problem says:"Let the sum of the first (n) terms of a geometric sequence ({a_n}) with a positive common ratio be (S_n). If (a_1 = 1) and (S_4 - 5S_2 = 0), then the value of (S_5) is ______."Alright, so I need to find (S_5). Let me recall what I know about geometric sequences.First, in a geometric sequence, each term is the previous term multiplied by a common ratio, which is denoted as (q). So, the terms are (a_1, a_1 q, a_1 q^2, a_1 q^3,) and so on. Since (a_1 = 1), the terms simplify to (1, q, q^2, q^3,) etc.The sum of the first (n) terms of a geometric sequence is given by the formula:[S_n = a_1 frac{1 - q^n}{1 - q}]provided that (q neq 1). If (q = 1), then the sum is simply (S_n = a_1 n). But in this problem, since the common ratio is positive, and we have a condition involving (S_4) and (S_2), I don't think (q) is 1 because that would make the sum too simple, and the equation (S_4 - 5S_2 = 0) might not hold. Let me check that.If (q = 1), then (S_4 = 4a_1 = 4) and (S_2 = 2a_1 = 2). Then (S_4 - 5S_2 = 4 - 5*2 = 4 - 10 = -6), which is not zero. So, (q) cannot be 1. Therefore, I can safely use the formula with (q neq 1).Given that (a_1 = 1), the sum formulas simplify to:[S_n = frac{1 - q^n}{1 - q}]So, (S_4 = frac{1 - q^4}{1 - q}) and (S_2 = frac{1 - q^2}{1 - q}).The problem states that (S_4 - 5S_2 = 0). Let me write that equation out:[frac{1 - q^4}{1 - q} - 5 cdot frac{1 - q^2}{1 - q} = 0]Since both terms have the same denominator, I can combine them:[frac{1 - q^4 - 5(1 - q^2)}{1 - q} = 0]For a fraction to be zero, the numerator must be zero (as long as the denominator isn't zero, which it isn't because (q neq 1)). So, I can set the numerator equal to zero:[1 - q^4 - 5(1 - q^2) = 0]Let me expand the numerator:[1 - q^4 - 5 + 5q^2 = 0]Combine like terms:[(1 - 5) + (-q^4 + 5q^2) = 0]Simplify:[-4 - q^4 + 5q^2 = 0]Let me rearrange the terms:[-q^4 + 5q^2 - 4 = 0]Hmm, this is a quartic equation, but it looks like a quadratic in terms of (q^2). Let me make a substitution to make it easier. Let (y = q^2). Then the equation becomes:[-y^2 + 5y - 4 = 0]Multiply both sides by -1 to make it a bit nicer:[y^2 - 5y + 4 = 0]Now, this is a quadratic equation in (y). Let me solve for (y) using the quadratic formula:[y = frac{5 pm sqrt{25 - 16}}{2} = frac{5 pm sqrt{9}}{2} = frac{5 pm 3}{2}]So, the solutions are:[y = frac{5 + 3}{2} = 4 quad text{and} quad y = frac{5 - 3}{2} = 1]But remember, (y = q^2), so:1. If (y = 4), then (q^2 = 4), so (q = pm 2).2. If (y = 1), then (q^2 = 1), so (q = pm 1).But the problem states that the common ratio is positive, so (q) must be positive. Therefore, the possible values are (q = 2) and (q = 1). However, earlier I saw that (q = 1) doesn't satisfy the equation (S_4 - 5S_2 = 0), so (q = 2) must be the solution.Now that I have (q = 2), I can find (S_5). Using the sum formula:[S_5 = frac{1 - q^5}{1 - q} = frac{1 - 2^5}{1 - 2} = frac{1 - 32}{-1} = frac{-31}{-1} = 31]So, the value of (S_5) is 31.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. I started by noting that (a_1 = 1) and that the sum formula applies for (q neq 1).2. I wrote the expressions for (S_4) and (S_2) correctly.3. I set up the equation (S_4 - 5S_2 = 0) and combined the fractions correctly.4. I expanded and simplified the numerator correctly, leading to the quartic equation.5. I made the substitution (y = q^2) and solved the quadratic equation correctly, getting (y = 4) and (y = 1).6. Since (q) is positive, I concluded (q = 2) and (q = 1), but (q = 1) didn't work, so (q = 2).7. Plugging (q = 2) into the sum formula for (S_5) gave me 31.Everything seems to check out. I don't see any errors in my calculations or reasoning.**Final Answer**The value of (S_5) is boxed{31}.

💡First, I need to establish the relationship between the cost price and the selling price based on the given information.I'll assume the cost price of one article is ( C ) and the selling price of one article is ( S ).According to the problem, the cost price of 50 articles is equal to the selling price of 40 articles. This gives me the equation:[50C = 40S]Next, I'll solve for ( S ) in terms of ( C ):[S = frac{50C}{40} = frac{5C}{4}]This means the selling price is ( frac{5}{4} ) times the cost price, indicating a gain.To find the gain per article, I'll subtract the cost price from the selling price:[text{Gain per article} = S - C = frac{5C}{4} - C = frac{C}{4}]Finally, to calculate the gain percentage, I'll divide the gain per article by the cost price and multiply by 100:[text{Gain percent} = left( frac{frac{C}{4}}{C} right) times 100 = 25%]

💡Okay, so I've got this complex number problem here, and I need to find the conjugate of z and then solve for a and b in this equation involving z squared, a times z, and b. Let me take it step by step because I'm still getting the hang of complex numbers.First, the complex number z is given as (3 - i)/(2 + i). Hmm, I remember that to simplify expressions like this, where there's a complex number in the denominator, I should multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 2 + i is 2 - i, right? So, let me do that.Multiplying numerator and denominator by 2 - i:z = (3 - i)(2 - i) / (2 + i)(2 - i)Now, I need to multiply these out. Let's start with the numerator:(3 - i)(2 - i) = 3*2 + 3*(-i) + (-i)*2 + (-i)*(-i) = 6 - 3i - 2i + i²Wait, i² is -1, so that becomes:6 - 5i - 1 = 5 - 5iOkay, so the numerator simplifies to 5 - 5i.Now, the denominator:(2 + i)(2 - i) = 2*2 + 2*(-i) + i*2 + i*(-i) = 4 - 2i + 2i - i²Again, i² is -1, so:4 - (-1) = 4 + 1 = 5So, the denominator is 5.Putting it all together, z = (5 - 5i)/5 = 1 - iAlright, so z is 1 - i. That wasn't too bad. Now, part (1) asks for the conjugate of z. The conjugate of a complex number a + bi is a - bi, so the conjugate of 1 - i should be 1 + i. Let me double-check that. Yes, that seems right.So, the conjugate of z is 1 + i.Moving on to part (2). We have the equation z² + a z + b = conjugate of z, where a and b are real numbers. We need to find a and b.Given that z = 1 - i, let's compute z squared first.z² = (1 - i)² = 1² - 2*1*i + (i)² = 1 - 2i + (-1) = 0 - 2iWait, that can't be right. Let me recalculate:(1 - i)² = (1)^2 + (-i)^2 + 2*(1)*(-i) = 1 + (-1) - 2i = 0 - 2iHmm, so z squared is -2i. Okay, that seems correct.Now, let's plug everything into the equation:z² + a z + b = conjugate(z)Which becomes:(-2i) + a*(1 - i) + b = 1 + iLet me distribute the a:-2i + a - a i + b = 1 + iNow, let's combine like terms. The real parts are a and b, and the imaginary parts are -2i and -a i.So, grouping them:(a + b) + (-2 - a)i = 1 + iNow, since two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal, we can set up two equations:1. Real parts: a + b = 12. Imaginary parts: -2 - a = 1Let me solve the second equation first:-2 - a = 1Adding 2 to both sides:-a = 3Multiplying both sides by -1:a = -3Now, substitute a = -3 into the first equation:-3 + b = 1Adding 3 to both sides:b = 4So, a is -3 and b is 4.Wait, let me just check my calculations again to make sure I didn't make a mistake.Starting with z = 1 - i, z squared is (1 - i)^2 = 1 - 2i + i² = 1 - 2i -1 = -2i. That's correct.Then, plugging into the equation: z² + a z + b = conjugate(z)So, (-2i) + a*(1 - i) + b = 1 + iExpanding: -2i + a - a i + b = 1 + iGrouping real and imaginary parts: (a + b) + (-2 - a)i = 1 + iSetting real parts equal: a + b = 1Setting imaginary parts equal: -2 - a = 1Solving for a: -2 - a = 1 => -a = 3 => a = -3Then, a + b = 1 => -3 + b = 1 => b = 4Yes, that seems consistent. So, a is -3 and b is 4.I think that's it. I don't see any mistakes in my steps. So, the conjugate of z is 1 + i, and a and b are -3 and 4 respectively.

💡Okay, so I have to find the equation of the normal line to the curve given by ( y = sqrt{x} - 3 sqrt[3]{x} ) at the point where the abscissa (which is the x-coordinate) is ( x_0 = 64 ). Hmm, let's break this down step by step.First, I remember that the normal line to a curve at a particular point is perpendicular to the tangent line at that point. So, to find the normal line, I need to find the slope of the tangent line first, and then take the negative reciprocal of that slope to get the slope of the normal line.Alright, let's start by finding the derivative of the function ( y = sqrt{x} - 3 sqrt[3]{x} ). The derivative will give me the slope of the tangent line at any point ( x ).So, ( y = sqrt{x} - 3 sqrt[3]{x} ). Let's rewrite the square root and cube root using exponents to make differentiation easier. ( sqrt{x} = x^{1/2} ) and ( sqrt[3]{x} = x^{1/3} ). So, the function becomes:( y = x^{1/2} - 3x^{1/3} ).Now, let's find the derivative ( y' ) using the power rule. The power rule states that ( frac{d}{dx} x^n = nx^{n-1} ).Applying this to each term:1. The derivative of ( x^{1/2} ) is ( frac{1}{2}x^{-1/2} ).2. The derivative of ( -3x^{1/3} ) is ( -3 times frac{1}{3}x^{-2/3} = -x^{-2/3} ).So, putting it all together, the derivative ( y' ) is:( y' = frac{1}{2}x^{-1/2} - x^{-2/3} ).Simplify this a bit:( y' = frac{1}{2sqrt{x}} - frac{1}{x^{2/3}} ).Okay, so that's the general form of the derivative. Now, I need to evaluate this derivative at ( x = 64 ) to find the slope of the tangent line at that specific point.Let's compute each term separately.First term: ( frac{1}{2sqrt{64}} ).( sqrt{64} = 8 ), so this becomes ( frac{1}{2 times 8} = frac{1}{16} ).Second term: ( frac{1}{64^{2/3}} ).Hmm, ( 64^{1/3} ) is 4 because ( 4^3 = 64 ). So, ( 64^{2/3} = (64^{1/3})^2 = 4^2 = 16 ). Therefore, this term is ( frac{1}{16} ).So, putting it all together, the derivative at ( x = 64 ) is:( y'(64) = frac{1}{16} - frac{1}{16} = 0 ).Wait, that's interesting. The slope of the tangent line at ( x = 64 ) is 0. That means the tangent line is horizontal at that point.Since the normal line is perpendicular to the tangent line, and the tangent line is horizontal, the normal line must be vertical. A vertical line has an undefined slope, but its equation is simply ( x = ) some constant.In this case, since the normal line passes through the point where ( x = 64 ), the equation of the normal line is ( x = 64 ).Let me just double-check my calculations to make sure I didn't make any mistakes.1. Derivative computation: - ( frac{d}{dx} x^{1/2} = frac{1}{2}x^{-1/2} ) ✔️ - ( frac{d}{dx} (-3x^{1/3}) = -3 times frac{1}{3}x^{-2/3} = -x^{-2/3} ) ✔️2. Evaluating at ( x = 64 ): - ( frac{1}{2sqrt{64}} = frac{1}{16} ) ✔️ - ( frac{1}{64^{2/3}} = frac{1}{16} ) ✔️ - So, ( y'(64) = frac{1}{16} - frac{1}{16} = 0 ) ✔️3. Conclusion that the normal line is vertical because the tangent is horizontal ✔️ - Equation: ( x = 64 ) ✔️Everything seems to check out. I think I'm confident with this answer.