💡First, I need to determine the number of letters that contain a dot but do not contain a straight line. I'll start by identifying the given information:- Total number of letters in the alphabet: 40- Number of letters containing both a dot and a straight line: 8- Number of letters containing a straight line but no dot: 24Next, I'll calculate the total number of letters that contain a straight line. This includes both the letters that have only a straight line and those that have both a dot and a straight line:Total letters with a straight line = Letters with only a straight line + Letters with both a dot and a straight lineTotal letters with a straight line = 24 + 8 = 32Now, I'll find out how many letters contain a dot. Since the total number of letters is 40 and all letters contain either a dot, a straight line, or both, the number of letters with a dot can be calculated by subtracting the number of letters with only a straight line from the total:Letters with a dot = Total letters - Letters with only a straight lineLetters with a dot = 40 - 24 = 16Finally, to find the number of letters that contain a dot but do not contain a straight line, I'll subtract the number of letters that have both a dot and a straight line from the total number of letters with a dot:Letters with only a dot = Letters with a dot - Letters with both a dot and a straight lineLetters with only a dot = 16 - 8 = 8Therefore, there are 8 letters that contain a dot but do not contain a straight line.

💡First, I'll assume the vendor starts with 100 apples for simplicity, as percentages are scalable.On the first day, the vendor sells 60% of the apples, which is 60 apples, leaving 40 apples remaining.Let ( x ) be the percentage of the remaining apples that the vendor throws away on the first day. Therefore, the number of apples thrown away on the first day is ( frac{x}{100} times 40 ).On the second day, the vendor sells 50% of the remaining apples after the first day. The remaining apples at the start of the second day are ( 40 - left( frac{x}{100} times 40 right) ).The number of apples sold on the second day is ( 0.5 times left[ 40 - left( frac{x}{100} times 40 right) right] ).The number of apples thrown away on the second day is the remaining apples after the second day's sale, which is ( left[ 40 - left( frac{x}{100} times 40 right) right] - 0.5 times left[ 40 - left( frac{x}{100} times 40 right) right] ).In total, the vendor throws away 28% of his apples, which is 28 apples out of the original 100.Setting up the equation for the total apples thrown away:[frac{x}{100} times 40 + left[ 40 - left( frac{x}{100} times 40 right) right] - 0.5 times left[ 40 - left( frac{x}{100} times 40 right) right] = 28]Simplifying the equation:[frac{x}{100} times 40 + 40 - frac{x}{100} times 40 - 0.5 times 40 + 0.5 times left( frac{x}{100} times 40 right) = 28]Combining like terms:[frac{x}{100} times 40 - 0.5 times left( frac{x}{100} times 40 right) = 28 - 40 + 0.5 times 40][0.5 times left( frac{x}{100} times 40 right) = 8]Solving for ( x ):[frac{x}{100} times 20 = 8][frac{x}{100} = frac{8}{20}][frac{x}{100} = 0.4][x = 0.4 times 100][x = 40]Therefore, the vendor throws away 40% of the remaining apples on the first day.

💡To determine how much LeRoy must give Cecilia so that they share the costs equally, follow these steps:1. **Calculate the total expenses**: The total amount spent by LeRoy, Bernardo, and Cecilia is the sum of the amounts each paid. Therefore, the total expenses, ( T ), is given by: [ T = L + B + C ]2. **Determine the amount each should pay**: Since they agreed to share the costs equally, each should pay one-third of the total expenses. Thus, the amount each should pay, ( E ), is: [ E = frac{T}{3} = frac{L + B + C}{3} ]3. **Calculate the difference LeRoy needs to pay to Cecilia**: LeRoy has already paid ( L ) dollars. To find out how much more he needs to pay to make his total contribution equal to ( E ), we calculate the difference he needs to pay to Cecilia, who has overpaid: [ text{Amount LeRoy needs to pay Cecilia} = E - L = frac{L + B + C}{3} - L ] Simplifying this expression, we get: [ frac{L + B + C - 3L}{3} = frac{B + C - 2L}{3} ]4. **Conclusion**: The amount LeRoy must give Cecilia to equalize their shares of the total expenses is (frac{B + C - 2L}{3}).The final answer is (boxed{textbf{(A)} frac{B + C - 2L}{3}}).

💡Okay, so I have this problem about an arithmetic sequence. Let me try to figure it out step by step. First, the problem says that the first term, a₁, is 2, and the third term, a₃, is 6. I need to find the general term formula for the sequence {aₙ}. Hmm, arithmetic sequences have a common difference, right? So, if I can find the common difference, I can write the general formula.Let me recall the formula for the nth term of an arithmetic sequence: aₙ = a₁ + (n - 1)d, where d is the common difference. I know a₁ is 2, so I need to find d. Given that a₃ is 6, I can plug that into the formula. So, a₃ = a₁ + 2d. That becomes 6 = 2 + 2d. Let me solve for d. Subtract 2 from both sides: 6 - 2 = 2d, so 4 = 2d. Divide both sides by 2: d = 2. Okay, so the common difference is 2.Now, plugging d back into the general formula: aₙ = 2 + (n - 1) * 2. Let me simplify that. Distribute the 2: aₙ = 2 + 2n - 2. The 2 and -2 cancel out, so aₙ = 2n. That seems straightforward. So, the general term is 2n. I think that's part (1) done.Moving on to part (2). It says that {Tₙ} is the sum of the first n terms of the sequence {1/Sₙ}, and I need to find T₂₀₁₃. Hmm, okay. First, I need to figure out what Sₙ is. Sₙ is the sum of the first n terms of the arithmetic sequence {aₙ}.I remember the formula for the sum of the first n terms of an arithmetic sequence: Sₙ = n/2 * (a₁ + aₙ). I already have a₁ = 2 and aₙ = 2n from part (1). So, plugging those in: Sₙ = n/2 * (2 + 2n). Let me simplify that. Factor out a 2: Sₙ = n/2 * 2(1 + n). The 2s cancel out, so Sₙ = n(n + 1). Okay, so Sₙ is n(n + 1).Now, the sequence {1/Sₙ} would be 1/(n(n + 1)). I need to find the sum of the first 2013 terms of this sequence, which is T₂₀₁₃. So, T₂₀₁₃ = sum from k=1 to 2013 of 1/(k(k + 1)).Hmm, I think this is a telescoping series. Let me recall how to break down 1/(k(k + 1)). I can use partial fractions. Let me set 1/(k(k + 1)) = A/k + B/(k + 1). Multiplying both sides by k(k + 1), I get 1 = A(k + 1) + Bk. Let me solve for A and B.Expanding the right side: 1 = Ak + A + Bk. Combine like terms: 1 = (A + B)k + A. For this to hold for all k, the coefficients must be equal on both sides. So, A + B = 0 (since there's no k term on the left) and A = 1 (constant term). From A = 1, then B = -1.So, 1/(k(k + 1)) = 1/k - 1/(k + 1). That's a telescoping series! So, when I sum from k=1 to 2013, most terms should cancel out.Let me write out the first few terms to see the pattern:For k=1: 1/1 - 1/2For k=2: 1/2 - 1/3For k=3: 1/3 - 1/4...For k=2013: 1/2013 - 1/2014When I add all these up, the -1/2 cancels with the +1/2, the -1/3 cancels with the +1/3, and so on, all the way up to -1/2013 cancels with the +1/2013. So, what's left is the first term of the first expression and the last term of the last expression.So, T₂₀₁₃ = 1 - 1/2014. Let me compute that. 1 is 2014/2014, so 2014/2014 - 1/2014 = 2013/2014.Wait, that seems too straightforward. Let me double-check. If I have 2013 terms, each term is 1/k - 1/(k+1). So, when I sum from k=1 to 2013, it's (1 - 1/2) + (1/2 - 1/3) + ... + (1/2013 - 1/2014). Yes, all the middle terms cancel, leaving 1 - 1/2014. So, T₂₀₁₃ is indeed 2013/2014.I think that makes sense. So, the value of T₂₀₁₃ is 2013/2014.**Final Answer**(1) The general term is boxed{2n}.(2) The value of ( T_{2013} ) is boxed{dfrac{2013}{2014}}.

💡First, I recognize that the area of a rhombus can be calculated using the formula:[text{Area} = frac{d_1 times d_2}{2}]where ( d_1 ) and ( d_2 ) are the lengths of the diagonals.Given the diagonals are 30 cm and 20 cm, I substitute these values into the formula:[text{Area} = frac{30 , text{cm} times 20 , text{cm}}{2}]Next, I perform the multiplication:[30 times 20 = 600]Then, I divide by 2 to find the area:[frac{600}{2} = 300]Therefore, the area of the rhombus is 300 square centimeters.

💡Okay, so I have this problem where I need to find how many numbers s in the set S = {0, 1, 2, ..., 25} make the function f(s) = s² + 3s + 2 a multiple of 6. Hmm, okay. Let me try to figure this out step by step.First, I remember that if a number is a multiple of 6, it must be divisible by both 2 and 3. So, for f(s) to be a multiple of 6, it needs to satisfy two conditions: f(s) must be even (divisible by 2) and f(s) must be divisible by 3.Let me write down the function again: f(s) = s² + 3s + 2. Maybe I can factor this to make it easier to analyze. Let me try factoring:f(s) = s² + 3s + 2 = (s + 1)(s + 2). Oh, that's nice! So, f(s) is the product of two consecutive integers: s + 1 and s + 2. Wait, actually, s + 1 and s + 2 are consecutive integers, right? So, their product is two consecutive integers. Hmm, that might help because in any two consecutive integers, one is even, so their product is always even. That means f(s) is always divisible by 2. So, the condition for divisibility by 2 is automatically satisfied for any s. That's good to know.So, now I just need to make sure that f(s) is divisible by 3. Since f(s) is (s + 1)(s + 2), I need to find when this product is divisible by 3. I remember that for a product to be divisible by 3, at least one of the factors must be divisible by 3. So, either (s + 1) is divisible by 3 or (s + 2) is divisible by 3. Let me write that down:Either:1. s + 1 ≡ 0 mod 3, which means s ≡ 2 mod 3, or2. s + 2 ≡ 0 mod 3, which means s ≡ 1 mod 3.So, s must be congruent to 1 or 2 modulo 3. That means s can be written as 3k + 1 or 3k + 2 for some integer k.Now, let me find all such s in the set S = {0, 1, 2, ..., 25}.First, let's find all numbers in S that are congruent to 1 mod 3:Starting from 1, adding 3 each time: 1, 4, 7, 10, 13, 16, 19, 22, 25.Wait, 25 is included because 25 = 3*8 + 1. So, that's 9 numbers.Next, numbers congruent to 2 mod 3:Starting from 2, adding 3 each time: 2, 5, 8, 11, 14, 17, 20, 23.That's 8 numbers.So, in total, 9 + 8 = 17 numbers.Wait, but hold on a second. Let me double-check. Maybe I missed something.Wait, when s = 0, let's compute f(0) = 0 + 0 + 2 = 2. 2 is not divisible by 6, so 0 is not included. Similarly, s = 1: f(1) = 1 + 3 + 2 = 6, which is divisible by 6. So, 1 is included.Wait, but according to my earlier count, s =1 is included because it's 1 mod 3. Similarly, s=2: f(2)=4 +6 +2=12, which is divisible by 6. So, 2 is included.Wait, so s=0 is not included because f(0)=2, which is not a multiple of 6. So, in my earlier count, I included s=1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25. That's 17 numbers.Wait, let me count them again:From 1 mod 3: 1,4,7,10,13,16,19,22,25: 9 numbers.From 2 mod 3: 2,5,8,11,14,17,20,23: 8 numbers.Total: 17 numbers.But wait, let me check s=25: f(25)=25² + 3*25 +2=625 +75 +2=702. 702 divided by 6 is 117, which is an integer, so yes, 25 is included.Similarly, s=23: f(23)=529 +69 +2=600, which is divisible by 6.s=22: f(22)=484 +66 +2=552, which is divisible by 6.Wait, but let me check s=25: 25 is 3*8 +1, so yes, it's 1 mod 3.Wait, but hold on, when I count s=1,4,7,10,13,16,19,22,25: that's 9 numbers.And s=2,5,8,11,14,17,20,23: that's 8 numbers.Total 17. So, the answer should be 17, which is option E.But wait, let me check if I missed any numbers or counted any extra.Wait, s=0: f(0)=2, not divisible by 6.s=1:6, yes.s=2:12, yes.s=3: f(3)=9 +9 +2=20, which is not divisible by 6.s=4:16 +12 +2=30, yes.s=5:25 +15 +2=42, yes.s=6:36 +18 +2=56, which is not divisible by 6.s=7:49 +21 +2=72, yes.s=8:64 +24 +2=90, yes.s=9:81 +27 +2=110, not divisible by 6.s=10:100 +30 +2=132, yes.s=11:121 +33 +2=156, yes.s=12:144 +36 +2=182, not divisible by 6.s=13:169 +39 +2=210, yes.s=14:196 +42 +2=240, yes.s=15:225 +45 +2=272, not divisible by 6.s=16:256 +48 +2=306, yes.s=17:289 +51 +2=342, yes.s=18:324 +54 +2=380, not divisible by 6.s=19:361 +57 +2=420, yes.s=20:400 +60 +2=462, yes.s=21:441 +63 +2=506, not divisible by 6.s=22:484 +66 +2=552, yes.s=23:529 +69 +2=600, yes.s=24:576 +72 +2=650, not divisible by 6.s=25:625 +75 +2=702, yes.So, counting the s where f(s) is divisible by 6: s=1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25. That's 17 numbers.So, the answer is 17, which is option E.