💡Alright, so I've got this problem here: I need to simplify the expression (frac{sqrt{5}}{sqrt{6}} cdot frac{sqrt{7}}{sqrt{8}} cdot frac{sqrt{9}}{sqrt{10}}) and then rationalize the denominator of the resulting fraction. Hmm, okay, let's break this down step by step. I'm a bit nervous because radicals can sometimes trip me up, but I think if I take it one piece at a time, I'll be okay.First, I remember that when you multiply radicals, you can combine them under a single square root. So, for the numerators, I have (sqrt{5}), (sqrt{7}), and (sqrt{9}). If I multiply these together, it should be the same as the square root of the product of 5, 7, and 9. Similarly, for the denominators, I have (sqrt{6}), (sqrt{8}), and (sqrt{10}), which can be combined into the square root of the product of 6, 8, and 10.Let me write that out:[frac{sqrt{5} cdot sqrt{7} cdot sqrt{9}}{sqrt{6} cdot sqrt{8} cdot sqrt{10}} = frac{sqrt{5 cdot 7 cdot 9}}{sqrt{6 cdot 8 cdot 10}}]Okay, so now I need to compute those products inside the square roots. Let's start with the numerator:5 multiplied by 7 is 35, and 35 multiplied by 9 is... let me see, 35 times 9. Well, 35 times 10 is 350, so subtracting 35 gives me 315. So, the numerator becomes (sqrt{315}).Now, the denominator: 6 multiplied by 8 is 48, and 48 multiplied by 10 is 480. So, the denominator is (sqrt{480}).So now, the expression simplifies to:[frac{sqrt{315}}{sqrt{480}}]Hmm, okay. Now, I think I can simplify these square roots further by factoring out perfect squares. Let's start with the numerator, (sqrt{315}).Breaking down 315, I know that 315 divided by 5 is 63, and 63 is 9 times 7. So, 315 is 5 times 9 times 7, which is 5 times 3 squared times 7. Therefore, (sqrt{315}) can be written as (sqrt{9 cdot 35}), since 9 times 35 is 315. And since 9 is a perfect square, that comes out of the square root as 3. So, (sqrt{315} = 3sqrt{35}).Great, now the numerator is simplified. Let's move on to the denominator, (sqrt{480}).Breaking down 480, I can see that 480 divided by 16 is 30, because 16 times 30 is 480. 16 is a perfect square, so that can come out of the square root. So, (sqrt{480} = sqrt{16 cdot 30} = 4sqrt{30}).So now, substituting these back into the expression, we have:[frac{3sqrt{35}}{4sqrt{30}}]Alright, so now the expression is simplified to (frac{3sqrt{35}}{4sqrt{30}}). But the problem also asks to rationalize the denominator. Rationalizing the denominator means getting rid of the square root in the denominator. To do that, I can multiply both the numerator and the denominator by (sqrt{30}), which will eliminate the square root in the denominator.Let me write that out:[frac{3sqrt{35}}{4sqrt{30}} times frac{sqrt{30}}{sqrt{30}} = frac{3sqrt{35} cdot sqrt{30}}{4 cdot sqrt{30} cdot sqrt{30}}]Simplifying the denominator first, since (sqrt{30} cdot sqrt{30} = 30), the denominator becomes 4 times 30, which is 120.Now, the numerator is (3sqrt{35} cdot sqrt{30}). I can combine these square roots into a single square root by multiplying the radicands:[3sqrt{35 cdot 30}]Calculating 35 times 30, that's 1050. So, the numerator becomes (3sqrt{1050}).Putting it all together, the expression now is:[frac{3sqrt{1050}}{120}]Hmm, I think I can simplify this fraction further. Both the numerator and the denominator are divisible by 3. Let's divide both by 3:[frac{3sqrt{1050}}{120} = frac{sqrt{1050}}{40}]Wait, but I think I can simplify (sqrt{1050}) more. Let me check if 1050 has any perfect square factors. Breaking down 1050, I know that 1050 divided by 25 is 42, because 25 times 42 is 1050. 25 is a perfect square, so I can take that out of the square root.So, (sqrt{1050} = sqrt{25 cdot 42} = 5sqrt{42}).Substituting that back into the expression, we get:[frac{5sqrt{42}}{40}]Now, I can simplify this fraction by dividing both numerator and denominator by 5:[frac{5sqrt{42}}{40} = frac{sqrt{42}}{8}]Wait a second, that doesn't seem right. Let me double-check my steps. When I simplified (sqrt{1050}), I said it was (sqrt{25 cdot 42}), which is correct because 25 times 42 is 1050. So, (sqrt{1050} = 5sqrt{42}). Then, substituting back, we have:[frac{3 cdot 5sqrt{42}}{120} = frac{15sqrt{42}}{120}]Ah, I see where I went wrong earlier. I forgot that the 3 in the numerator was multiplied by the 5 from the square root. So, it's actually 15sqrt{42} over 120, not 5sqrt{42} over 40. Let me correct that.So, (frac{15sqrt{42}}{120}) can be simplified by dividing numerator and denominator by 15:[frac{15sqrt{42}}{120} = frac{sqrt{42}}{8}]Wait, 15 divided by 15 is 1, and 120 divided by 15 is 8. So, yes, that simplifies to (frac{sqrt{42}}{8}).But hold on, earlier I had (frac{3sqrt{1050}}{120}), which I simplified to (frac{sqrt{1050}}{40}), and then further to (frac{sqrt{42}}{8}). Let me verify if that's correct.Starting from (frac{3sqrt{1050}}{120}), dividing numerator and denominator by 3 gives (frac{sqrt{1050}}{40}). Then, simplifying (sqrt{1050}) as (5sqrt{42}), we get (frac{5sqrt{42}}{40}), which simplifies to (frac{sqrt{42}}{8}) by dividing numerator and denominator by 5.Yes, that seems correct. So, the final simplified expression with a rationalized denominator is (frac{sqrt{42}}{8}).But wait, let me go back to the step where I had (frac{3sqrt{35}}{4sqrt{30}}). When I multiplied numerator and denominator by (sqrt{30}), I got (frac{3sqrt{1050}}{120}). Then, simplifying (sqrt{1050}) as (5sqrt{42}), that becomes (frac{15sqrt{42}}{120}), which simplifies to (frac{sqrt{42}}{8}).Yes, that makes sense. So, the final answer is (frac{sqrt{42}}{8}).Wait, but in my initial steps, I thought of simplifying (sqrt{1050}) as (5sqrt{42}), but I also considered whether 42 can be broken down further. Let me check: 42 is 6 times 7, and neither 6 nor 7 are perfect squares, so (sqrt{42}) cannot be simplified further. Therefore, (frac{sqrt{42}}{8}) is indeed the simplest form with a rationalized denominator.Just to recap, the steps were:1. Combine the radicals in the numerator and denominator.2. Simplify each square root by factoring out perfect squares.3. Rationalize the denominator by multiplying numerator and denominator by the radical in the denominator.4. Simplify the resulting expression by reducing any common factors.I think I followed all those steps correctly, and each time I checked my work, it seemed to hold up. So, I feel confident that (frac{sqrt{42}}{8}) is the correct simplified form with a rationalized denominator.

💡First, I recognize that Mike currently has 74 baseball cards after selling 13 to Sam.To find out how many baseball cards Mike had initially, I need to determine the original number before the sale.Let ( x ) represent the initial number of baseball cards Mike had.According to the problem, after selling 13 cards, Mike has 74 left. This can be expressed as:[ x - 13 = 74 ]To solve for ( x ), I will add 13 to both sides of the equation:[ x - 13 + 13 = 74 + 13 ][ x = 87 ]Therefore, Mike initially had 87 baseball cards.

💡First, I recognize that Arun and Tarun together can complete the work in 10 days. This means their combined work rate is ( frac{1}{10} ) of the work per day.Arun alone can complete the work in 60 days, so his individual work rate is ( frac{1}{60} ) of the work per day.To find Tarun's work rate, I subtract Arun's rate from their combined rate:[frac{1}{10} - frac{1}{60} = frac{6}{60} - frac{1}{60} = frac{5}{60} = frac{1}{12}]So, Tarun's work rate is ( frac{1}{12} ) of the work per day.Next, I calculate the work completed by both Arun and Tarun in the first 4 days:[4 times left( frac{1}{10} right) = frac{4}{10} = frac{2}{5}]This means that ( frac{2}{5} ) of the work is completed, leaving ( frac{3}{5} ) of the work remaining.Finally, I determine how many days Arun alone will need to complete the remaining ( frac{3}{5} ) of the work:[frac{3}{5} div frac{1}{60} = frac{3}{5} times 60 = 36 text{ days}]

💡Alright, so I have this problem about sequences. It's divided into two parts. Let me take it step by step.**Part 1: Geometric Sequence**Okay, the first part says that {aₙ} is a geometric sequence with a₂ = 1 and the sum of the first n terms is Sₙ. They tell me that S₂ = 3, and I need to find the limit of Sₙ as n approaches infinity.Hmm, let's recall what a geometric sequence is. In a geometric sequence, each term is multiplied by a common ratio, q. So, if a₁ is the first term, then a₂ = a₁ * q, a₃ = a₁ * q², and so on.Given that a₂ = 1, I can write that as:a₂ = a₁ * q = 1.Also, the sum of the first 2 terms, S₂, is given as 3. So, S₂ = a₁ + a₂ = 3.Since a₂ is 1, that means a₁ + 1 = 3, so a₁ must be 2.Now, from a₂ = a₁ * q, we have 1 = 2 * q, so q = 1/2.Alright, so the common ratio is 1/2, and the first term is 2.Now, I need to find the limit of Sₙ as n approaches infinity. For a geometric series, the sum Sₙ is given by:Sₙ = a₁ * (1 - qⁿ) / (1 - q).Plugging in the values we have:Sₙ = 2 * (1 - (1/2)ⁿ) / (1 - 1/2) = 2 * (1 - (1/2)ⁿ) / (1/2) = 4 * (1 - (1/2)ⁿ).So, as n approaches infinity, (1/2)ⁿ approaches 0 because any number between -1 and 1 raised to an infinite power approaches 0.Therefore, the limit of Sₙ as n approaches infinity is:limₙ→∞ Sₙ = 4 * (1 - 0) = 4.Okay, that seems straightforward. I think that's the answer for part 1.**Part 2: Arithmetic Sequence**Now, the second part says that {aₙ} is an arithmetic sequence with a₂ = 1, and the sum of the first 2n terms is S_{2n} ≥ n. I need to find the range of values for the common difference d.Alright, let's recall what an arithmetic sequence is. In an arithmetic sequence, each term is obtained by adding a common difference, d, to the previous term. So, a₁ is the first term, a₂ = a₁ + d, a₃ = a₁ + 2d, etc.Given that a₂ = 1, we can write:a₂ = a₁ + d = 1.So, a₁ = 1 - d.Now, the sum of the first 2n terms, S_{2n}, is given by the formula:S_{2n} = (2n)/2 * [2a₁ + (2n - 1)d] = n * [2a₁ + (2n - 1)d].We are told that S_{2n} ≥ n, so:n * [2a₁ + (2n - 1)d] ≥ n.We can divide both sides by n (assuming n ≠ 0, which it isn't since n is a positive integer):2a₁ + (2n - 1)d ≥ 1.Now, substitute a₁ = 1 - d into the inequality:2(1 - d) + (2n - 1)d ≥ 1.Let's expand that:2 - 2d + 2n d - d ≥ 1.Combine like terms:2 - 3d + 2n d ≥ 1.Subtract 2 from both sides:-3d + 2n d ≥ -1.Factor out d:d(-3 + 2n) ≥ -1.So, d(2n - 3) ≥ -1.Now, let's solve for d. We have:d ≥ -1 / (2n - 3).But wait, the direction of the inequality depends on the sign of (2n - 3). Let's analyze that.For n = 1:2n - 3 = 2 - 3 = -1. So, d ≥ -1 / (-1) = 1.But 2n - 3 is negative when n = 1, so dividing both sides by a negative number reverses the inequality:d ≤ 1.Wait, that seems conflicting. Let me double-check.Starting from:d(2n - 3) ≥ -1.When n = 1:d(-1) ≥ -1.Divide both sides by -1 (inequality reverses):d ≤ 1.Okay, so for n = 1, d ≤ 1.For n = 2:2n - 3 = 4 - 3 = 1. So, d ≥ -1 / 1 = -1.Since 2n - 3 is positive when n ≥ 2, the inequality remains as is:d ≥ -1.But wait, for n = 2, we have d ≥ -1.But for n = 1, d ≤ 1.So, combining these, we have d ≤ 1 and d ≥ -1.But wait, let's check for larger n.For n = 3:2n - 3 = 6 - 3 = 3. So, d ≥ -1 / 3 ≈ -0.333.Similarly, for n = 4:d ≥ -1 / 5 = -0.2.As n increases, -1 / (2n - 3) approaches 0 from below.So, for larger n, d must be greater than or equal to a number approaching 0.Therefore, the most restrictive condition is d ≥ -1 / (2n - 3) for each n, but as n increases, this lower bound approaches 0.But since the inequality must hold for all n, the most restrictive lower bound is d ≥ 0.Wait, let me think again.If for each n, d must satisfy d ≥ -1 / (2n - 3), and as n increases, -1 / (2n - 3) approaches 0 from below, meaning that d must be greater than or equal to numbers approaching 0.Therefore, to satisfy the inequality for all n, d must be greater than or equal to 0.But for n = 1, we have d ≤ 1.So, combining both, d must be between 0 and 1, inclusive.Wait, but let's verify this.If d = 0, then the sequence is constant, with all terms equal to a₁ = 1 - d = 1.Then, S_{2n} = 2n * 1 = 2n, which is certainly greater than or equal to n.If d = 1, then a₁ = 1 - 1 = 0.So, the sequence is 0, 1, 2, 3, ..., and S_{2n} = sum from k=0 to 2n-1 of k = (2n - 1)(2n)/2.Wait, that's (2n)(2n - 1)/2 = n(2n - 1).Which is certainly greater than or equal to n for n ≥ 1.If d is between 0 and 1, say d = 0.5, then a₁ = 1 - 0.5 = 0.5.The sequence is 0.5, 1, 1.5, 2, ..., and S_{2n} = sum from k=0 to 2n-1 of (0.5 + 0.5k).Which is 2n * 0.5 + 0.5 * sum from k=0 to 2n-1 of k = n + 0.5 * (2n - 1)(2n)/2 = n + 0.5 * n(2n - 1).Which simplifies to n + n(2n - 1)/2 = n + n² - n/2 = n² + n/2.Which is definitely greater than or equal to n.If d is negative, say d = -0.5, then a₁ = 1 - (-0.5) = 1.5.The sequence is 1.5, 1, 0.5, 0, -0.5, ..., and S_{2n} would be the sum of these terms.But for d negative, the terms eventually become negative, and the sum might not necessarily be greater than or equal to n for all n.Wait, let's test n = 2.For d = -0.5, a₁ = 1.5.So, the first four terms are 1.5, 1, 0.5, 0.Sum S₄ = 1.5 + 1 + 0.5 + 0 = 3.Which is equal to n = 2? Wait, S_{2n} where n=2 is S₄.But S₄ = 3, which is equal to n=2? Wait, no, n=2, S_{2n}=S₄=3, which is greater than or equal to n=2.But for n=3, S₆ would be sum of first 6 terms:1.5, 1, 0.5, 0, -0.5, -1.Sum = 1.5 + 1 + 0.5 + 0 + (-0.5) + (-1) = 1.5 + 1 + 0.5 - 0.5 -1 = 1.5 + 1 -1 = 1.5.Which is less than n=3.So, S₆ = 1.5 < 3, which violates the condition S_{2n} ≥ n.Therefore, d cannot be negative.Hence, d must be greater than or equal to 0.But earlier, for n=1, d ≤ 1.So, combining both, d must be between 0 and 1, inclusive.Therefore, the range of d is [0, 1].I think that's the answer for part 2.**Final Answer**(1) boxed{4}(2) boxed{[0, 1]}

💡First, determine the number of pages remaining after losing one page.Subtract the lost page from the total number of pages:12 pages - 1 page = 11 pagesNext, calculate the total number of stickers by multiplying the number of remaining pages by the number of stickers per page.11 pages × 20 stickers per page = 220 stickersTherefore, the total number of stickers after losing one page is 220.

💡First, I need to find the first term ( a_1 ) of the arithmetic sequence. I know that the sum of the first 5 terms is 35, and the fifth term ( a_5 ) is 11. Using the formula for the sum of the first ( n ) terms of an arithmetic sequence:[S_n = frac{n}{2} times (a_1 + a_n)]Substituting the known values:[35 = frac{5}{2} times (a_1 + 11)]Solving for ( a_1 ):[35 = frac{5}{2} times (a_1 + 11) 35 times 2 = 5 times (a_1 + 11) 70 = 5a_1 + 55 5a_1 = 70 - 55 5a_1 = 15 a_1 = 3]Next, I'll find the common difference ( d ) using the formula for the ( n )-th term of an arithmetic sequence:[a_n = a_1 + (n - 1)d]Substituting ( a_5 = 11 ) and ( a_1 = 3 ):[11 = 3 + 4d 4d = 11 - 3 4d = 8 d = 2]Finally, I'll calculate the fourth term ( a_4 ):[a_4 = a_1 + 3d a_4 = 3 + 3 times 2 a_4 = 3 + 6 a_4 = 9]