💡First, determine the total number of calories in the entire package by multiplying the number of servings by the calories per serving.Next, calculate the number of calories John consumed by finding half of the total calories in the package.

💡First, I need to determine how much rope Mr. Sanchez bought this week. He bought 4 feet less than last week, so this week's purchase is 6 feet minus 4 feet, which equals 2 feet.Next, I'll add the rope bought last week and this week to find the total in feet. That's 6 feet plus 2 feet, totaling 8 feet.Since the total length of the rope is given in inches as 96 inches, I can find out how many inches are in one foot by dividing the total inches by the total feet. So, 96 inches divided by 8 feet equals 12 inches per foot.

💡Alright, so I have this problem here: Given the expansion of (2x + 1)^5, which is equal to a0 + a1x + a2x^2 + a3x^3 + a4x^4 + a5x^5, I need to find the value of a0 - a1 + a2 - a3 + a4 - a5. Hmm, okay, let's see how I can approach this.First, I remember that when dealing with polynomial expansions, sometimes substituting specific values for x can help simplify things. In this case, I need an alternating sum of the coefficients: a0 - a1 + a2 - a3 + a4 - a5. That looks like plugging in x = -1 because when x is -1, the terms with odd powers will have negative signs, and the even powers will remain positive. Let me test that idea.So, if I substitute x = -1 into the polynomial, I get:(2*(-1) + 1)^5 = a0 + a1*(-1) + a2*(-1)^2 + a3*(-1)^3 + a4*(-1)^4 + a5*(-1)^5Simplifying the left side first: 2*(-1) is -2, plus 1 is -1. So, (-1)^5 is -1. On the right side, let's compute each term:a0 remains a0.a1*(-1) is -a1.a2*(-1)^2 is a2*1, which is a2.a3*(-1)^3 is -a3.a4*(-1)^4 is a4*1, which is a4.a5*(-1)^5 is -a5.Putting it all together, the right side becomes: a0 - a1 + a2 - a3 + a4 - a5.So, from the substitution, we have:-1 = a0 - a1 + a2 - a3 + a4 - a5Therefore, the value we're looking for is -1.Wait, that seems straightforward, but let me double-check to make sure I didn't make any mistakes. Maybe I can expand (2x + 1)^5 using the binomial theorem and compute the coefficients individually to confirm.The binomial theorem states that (a + b)^n = sum from k=0 to n of (n choose k) * a^(n-k) * b^k.In this case, a = 2x, b = 1, and n = 5.So, expanding (2x + 1)^5:= C(5,0)*(2x)^5*1^0 + C(5,1)*(2x)^4*1^1 + C(5,2)*(2x)^3*1^2 + C(5,3)*(2x)^2*1^3 + C(5,4)*(2x)^1*1^4 + C(5,5)*(2x)^0*1^5Calculating each term:C(5,0) = 1, so first term is 1*(32x^5)*1 = 32x^5.C(5,1) = 5, so second term is 5*(16x^4)*1 = 80x^4.C(5,2) = 10, so third term is 10*(8x^3)*1 = 80x^3.C(5,3) = 10, so fourth term is 10*(4x^2)*1 = 40x^2.C(5,4) = 5, so fifth term is 5*(2x)*1 = 10x.C(5,5) = 1, so sixth term is 1*(1)*1 = 1.So, putting it all together:(2x + 1)^5 = 32x^5 + 80x^4 + 80x^3 + 40x^2 + 10x + 1Therefore, the coefficients are:a0 = 1a1 = 10a2 = 40a3 = 80a4 = 80a5 = 32Now, let's compute a0 - a1 + a2 - a3 + a4 - a5:= 1 - 10 + 40 - 80 + 80 - 32Let's compute step by step:1 - 10 = -9-9 + 40 = 3131 - 80 = -49-49 + 80 = 3131 - 32 = -1So, indeed, the result is -1. That matches what I got earlier by substituting x = -1. Therefore, I'm confident that the value is correct.Alternatively, another way to think about this is that evaluating the polynomial at x = -1 gives the alternating sum of the coefficients, which is exactly what we need. So, substituting x = -1 is a quick and efficient method without having to expand the entire polynomial.I think I've covered all the bases here. I first used substitution, then double-checked by expanding the polynomial and calculating each coefficient individually. Both methods led me to the same answer, so I feel pretty sure that -1 is the correct value for a0 - a1 + a2 - a3 + a4 - a5.

💡Okay, so I have this problem about a quadratic function f(x) = ax² + (b - 8)x - a - ab. The problem has three parts, and I need to solve each one step by step. Let me start with part (1).**Problem (1): Find the expression of f(x).**Alright, the problem states that f(x) is positive when x is between -3 and 2, and negative otherwise. That means the quadratic function crosses the x-axis at x = -3 and x = 2. So, these are the roots of the equation f(x) = 0. Since it's a quadratic equation, it can be written in the form f(x) = a(x + 3)(x - 2). But wait, the given function is f(x) = ax² + (b - 8)x - a - ab. Hmm, maybe I can use the roots to find the coefficients a and b.Let me recall that for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. In this case, the roots are -3 and 2, so:Sum of roots = (-3) + 2 = -1 = -(b - 8)/aProduct of roots = (-3)(2) = -6 = (-a - ab)/aSo, I can set up two equations:1. (-3 + 2) = -1 = -(b - 8)/a2. (-3)(2) = -6 = (-a - ab)/aLet me solve the first equation:-1 = -(b - 8)/aMultiply both sides by a:- a = -(b - 8)Multiply both sides by -1:a = b - 8So, equation 1 gives me a = b - 8.Now, equation 2:-6 = (-a - ab)/aSimplify the numerator:- a - ab = -a(1 + b)So, the equation becomes:-6 = (-a(1 + b))/aThe a in the numerator and denominator cancels out (assuming a ≠ 0, which it can't be because it's a quadratic):-6 = -(1 + b)Multiply both sides by -1:6 = 1 + bSo, b = 6 - 1 = 5.Now, substitute b = 5 into equation 1:a = 5 - 8 = -3.So, a = -3 and b = 5.Therefore, the function f(x) is:f(x) = -3x² + (5 - 8)x - (-3) - (-3)(5)Simplify:f(x) = -3x² - 3x + 3 + 15f(x) = -3x² - 3x + 18Wait, let me double-check that:- The coefficient of x is (b - 8) = 5 - 8 = -3.- The constant term is -a - ab = -(-3) - (-3)(5) = 3 + 15 = 18.Yes, that seems correct. So, f(x) = -3x² - 3x + 18.**Problem (2): If the solution set of the inequality ax² + bx + c ≤ 0 is ℝ, find the range of values for c.**Hmm, okay. So, the inequality is -3x² + 5x + c ≤ 0, since a = -3 and b = 5 from part (1). The solution set is all real numbers, which means the quadratic is always less than or equal to zero. For a quadratic to be always negative or zero, it must open downwards (which it does, since a = -3 < 0) and have no real roots or a repeated real root. So, the discriminant must be less than or equal to zero.The discriminant D of ax² + bx + c is D = b² - 4ac.So, D = 5² - 4*(-3)*c = 25 + 12c.For the quadratic to be always ≤ 0, D ≤ 0:25 + 12c ≤ 0Solving for c:12c ≤ -25c ≤ -25/12So, c must be less than or equal to -25/12.Wait, let me check the calculation:D = 25 - 4*(-3)*c = 25 + 12c.Yes, correct. So, 25 + 12c ≤ 0 => 12c ≤ -25 => c ≤ -25/12.So, the range of c is (-∞, -25/12].**Problem (3): When x > -1, find the maximum value of y = (f(x) - 21)/(x + 1).**Alright, so first, let's write down f(x):f(x) = -3x² - 3x + 18So, f(x) - 21 = -3x² - 3x + 18 - 21 = -3x² - 3x - 3Therefore, y = (-3x² - 3x - 3)/(x + 1)Simplify numerator:Factor out -3:y = -3(x² + x + 1)/(x + 1)Hmm, can I perform polynomial division or factor the numerator?Let me try to divide x² + x + 1 by x + 1.Divide x² + x + 1 by x + 1:x + 1 ) x² + x + 1First term: x² / x = xMultiply (x + 1) by x: x² + xSubtract: (x² + x + 1) - (x² + x) = 0 + 0 + 1So, the division gives x with a remainder of 1. Therefore,x² + x + 1 = (x + 1)(x) + 1So, y = -3[(x + 1)(x) + 1]/(x + 1) = -3[x + 1/(x + 1)]Wait, let me write that again:y = -3[(x + 1)(x) + 1]/(x + 1) = -3[x + 1/(x + 1)]Wait, no:Wait, (x + 1)(x) + 1 = x(x + 1) + 1, so when you divide by (x + 1), it becomes x + 1/(x + 1). So, yes:y = -3[x + 1/(x + 1)]So, y = -3x - 3/(x + 1)But maybe another approach is better. Let me consider substitution.Let t = x + 1. Since x > -1, t > 0.Express y in terms of t:x = t - 1So, y = (-3x² - 3x - 3)/tLet me compute the numerator:-3x² - 3x - 3 = -3(x² + x + 1)Substitute x = t - 1:x² = (t - 1)² = t² - 2t + 1x = t - 1So, x² + x + 1 = (t² - 2t + 1) + (t - 1) + 1 = t² - 2t + 1 + t - 1 + 1 = t² - t + 1Therefore, numerator is -3(t² - t + 1)So, y = -3(t² - t + 1)/t = -3(t - 1 + 1/t)So, y = -3(t + 1/t - 1)Hmm, interesting. So, y = -3(t + 1/t - 1) = -3t - 3/t + 3Wait, but I think I made a miscalculation:Wait, t² - t + 1 divided by t is t - 1 + 1/t. So, y = -3(t - 1 + 1/t) = -3t + 3 - 3/tSo, y = -3t - 3/t + 3Wait, that seems correct.So, y = -3t - 3/t + 3We need to find the maximum value of y when t > 0.Let me consider the function y(t) = -3t - 3/t + 3To find its maximum, take derivative with respect to t:dy/dt = -3 + 3/t²Set derivative equal to zero:-3 + 3/t² = 0=> 3/t² = 3=> 1/t² = 1=> t² = 1=> t = 1 (since t > 0)So, critical point at t = 1.Now, check second derivative to confirm if it's a maximum or minimum.Second derivative:d²y/dt² = -6/t³At t = 1, d²y/dt² = -6 < 0, so it's a local maximum.So, the maximum occurs at t = 1.Compute y(1):y(1) = -3(1) - 3/1 + 3 = -3 - 3 + 3 = -3So, the maximum value of y is -3.Alternatively, since y = -3(t + 1/t - 1), and t + 1/t ≥ 2 by AM ≥ GM, so t + 1/t -1 ≥ 1, so y = -3*(something ≥1) ≤ -3.Thus, the maximum value is -3, achieved when t = 1, which corresponds to x = t - 1 = 0.So, when x = 0, y = -3.Therefore, the maximum value is -3.**Final Answer**(1) boxed{-3x^2 - 3x + 18}(2) boxed{(-infty, -dfrac{25}{12}]}(3) boxed{-3}

💡First, I recognize that the sequence is defined by the general term ( a_n = 3n + 1 ), where ( n ) is a positive integer.To find out which term corresponds to the value 40, I set up the equation ( 3n + 1 = 40 ).Next, I solve for ( n ) by subtracting 1 from both sides of the equation, resulting in ( 3n = 39 ).Finally, I divide both sides by 3 to find ( n = 13 ), indicating that 40 is the 13th term in the sequence.

💡Okay, so I need to figure out how many clips Natalia sold altogether in April and May. Let me start by understanding the problem step by step.First, it says that Natalia sold clips to 48 of her friends in April. That means in April, she sold 48 clips. Okay, that's straightforward.Then, it mentions that she sold half as many clips in May. So, in May, she sold half the number of clips she sold in April. Since she sold 48 clips in April, I need to find half of 48 to know how many she sold in May.Let me write that down: Half of 48 is 48 divided by 2, which is 24. So, she sold 24 clips in May.Now, to find the total number of clips she sold in both months, I need to add the number of clips sold in April and May together. That would be 48 clips in April plus 24 clips in May.So, 48 plus 24 equals 72. Therefore, Natalia sold a total of 72 clips in April and May combined.Wait, let me double-check to make sure I didn't make any mistakes. April sales: 48 clips. May sales: half of April, which is 24. Total: 48 + 24 = 72. Yep, that seems correct.I think I've got it. The total number of clips Natalia sold in April and May is 72.