💡Okay, so I have this problem here: Given that ( a + 3 ) and ( 2a - 15 ) are the square roots of a certain positive number, the cube root of ( b ) is ( -2 ), and the arithmetic square root of ( c ) is itself, find the value of ( a + b - 2c ).Alright, let's break this down step by step. First, I need to understand what each part is saying.Starting with the first part: ( a + 3 ) and ( 2a - 15 ) are the square roots of a certain positive number. Hmm, so if both ( a + 3 ) and ( 2a - 15 ) are square roots of the same positive number, that means they must be equal because a positive number has only one positive square root and one negative square root. But wait, actually, no. For any positive number, there are two square roots: one positive and one negative. So, if ( a + 3 ) and ( 2a - 15 ) are both square roots of the same number, one of them must be the positive square root, and the other must be the negative square root. So, that means either ( a + 3 = 2a - 15 ) or ( a + 3 = -(2a - 15) ). Let me write that down:Case 1: ( a + 3 = 2a - 15 )Case 2: ( a + 3 = -(2a - 15) )Let me solve both cases.Starting with Case 1:( a + 3 = 2a - 15 )Subtract ( a ) from both sides:( 3 = a - 15 )Add 15 to both sides:( 18 = a )So, ( a = 18 ) in this case.Now, let's check if this makes sense. If ( a = 18 ), then ( a + 3 = 21 ) and ( 2a - 15 = 21 ). So both expressions are equal, which means they are both the positive square roots of the same number. But wait, the problem says they are the square roots, which could include both positive and negative. So, if both are positive, that's fine, but I need to make sure that the number they are square roots of is positive, which it is because 21 squared is 441, which is positive. So, that seems okay.Now, let's look at Case 2:( a + 3 = -(2a - 15) )Simplify the right side:( a + 3 = -2a + 15 )Add ( 2a ) to both sides:( 3a + 3 = 15 )Subtract 3 from both sides:( 3a = 12 )Divide by 3:( a = 4 )Okay, so ( a = 4 ) in this case. Let's check this. If ( a = 4 ), then ( a + 3 = 7 ) and ( 2a - 15 = 8 - 15 = -7 ). So, one is 7 and the other is -7, which are indeed the two square roots of 49. That makes sense because 7 squared is 49 and (-7) squared is also 49. So, this is also a valid solution.So, from the first part, we have two possible values for ( a ): 18 and 4. But wait, I need to make sure if both are valid or if one of them is extraneous.Let me think. If ( a = 18 ), then both ( a + 3 ) and ( 2a - 15 ) are 21, which is a positive number. So, they are both positive square roots of 441. That's fine.If ( a = 4 ), then ( a + 3 = 7 ) and ( 2a - 15 = -7 ). So, one is positive and the other is negative, which are the two square roots of 49. That's also fine.So, both solutions are valid. Therefore, ( a ) can be either 18 or 4.Wait, but the problem says "a certain positive number." So, both 441 and 49 are positive numbers, so both are acceptable. So, we have two possible values for ( a ).Now, moving on to the next part: the cube root of ( b ) is ( -2 ). So, that means ( sqrt[3]{b} = -2 ). To find ( b ), I need to cube both sides. So, ( b = (-2)^3 = -8 ). So, ( b = -8 ). That's straightforward.Next, the arithmetic square root of ( c ) is itself. Hmm, arithmetic square root usually refers to the principal (non-negative) square root. So, if the arithmetic square root of ( c ) is itself, that means ( sqrt{c} = c ). Let's solve this equation.( sqrt{c} = c )Square both sides:( c = c^2 )Bring all terms to one side:( c^2 - c = 0 )Factor:( c(c - 1) = 0 )So, ( c = 0 ) or ( c = 1 ).Therefore, ( c ) can be either 0 or 1.Now, we need to find ( a + b - 2c ). So, let's compute this expression for each possible value of ( a ), ( b ), and ( c ).First, let's list all possible combinations:1. ( a = 18 ), ( b = -8 ), ( c = 0 )2. ( a = 18 ), ( b = -8 ), ( c = 1 )3. ( a = 4 ), ( b = -8 ), ( c = 0 )4. ( a = 4 ), ( b = -8 ), ( c = 1 )So, we have four possible scenarios. Let's compute ( a + b - 2c ) for each.1. ( a = 18 ), ( b = -8 ), ( c = 0 ):( 18 + (-8) - 2(0) = 18 - 8 - 0 = 10 )2. ( a = 18 ), ( b = -8 ), ( c = 1 ):( 18 + (-8) - 2(1) = 18 - 8 - 2 = 8 )3. ( a = 4 ), ( b = -8 ), ( c = 0 ):( 4 + (-8) - 2(0) = 4 - 8 - 0 = -4 )4. ( a = 4 ), ( b = -8 ), ( c = 1 ):( 4 + (-8) - 2(1) = 4 - 8 - 2 = -6 )So, the possible values of ( a + b - 2c ) are 10, 8, -4, and -6.But wait, the problem says "find the value of ( a + b - 2c )." So, does that mean there are multiple possible answers, or is there something I missed?Let me go back to the first part. The problem says ( a + 3 ) and ( 2a - 15 ) are the square roots of a certain positive number. So, if both are square roots, they can be either both positive or one positive and one negative. But in the case of ( a = 18 ), both are positive, which is fine because they are both square roots of 441. Similarly, ( a = 4 ) gives one positive and one negative, which are square roots of 49.So, both are valid. Therefore, both ( a = 18 ) and ( a = 4 ) are acceptable. So, the expression ( a + b - 2c ) can have multiple values depending on ( a ) and ( c ).But the problem asks for "the value," implying a single answer. Maybe I need to consider which value of ( a ) is correct. Wait, perhaps I made a mistake earlier.Let me double-check the first part. If ( a + 3 ) and ( 2a - 15 ) are square roots of the same number, then they must satisfy ( (a + 3)^2 = (2a - 15)^2 ). Let me write that equation:( (a + 3)^2 = (2a - 15)^2 )Expanding both sides:Left side: ( a^2 + 6a + 9 )Right side: ( 4a^2 - 60a + 225 )Set them equal:( a^2 + 6a + 9 = 4a^2 - 60a + 225 )Bring all terms to one side:( 0 = 3a^2 - 66a + 216 )Simplify by dividing both sides by 3:( 0 = a^2 - 22a + 72 )Now, solve the quadratic equation:( a^2 - 22a + 72 = 0 )Using the quadratic formula:( a = frac{22 pm sqrt{(-22)^2 - 4(1)(72)}}{2(1)} )Calculate discriminant:( 484 - 288 = 196 )So,( a = frac{22 pm 14}{2} )Thus,( a = frac{22 + 14}{2} = frac{36}{2} = 18 )or( a = frac{22 - 14}{2} = frac{8}{2} = 4 )So, indeed, ( a = 18 ) or ( a = 4 ). So, both are valid solutions.Therefore, the expression ( a + b - 2c ) can have multiple values depending on the values of ( a ) and ( c ).But the problem says "find the value of ( a + b - 2c )." Maybe I need to consider all possible values and present them. Alternatively, perhaps there's a constraint I missed.Wait, let's think about the square roots. If ( a + 3 ) and ( 2a - 15 ) are square roots, they can be either both positive or one positive and one negative. But in the case of ( a = 18 ), both are positive, which is fine. In the case of ( a = 4 ), one is positive and one is negative, which is also fine.So, both solutions are acceptable. Therefore, the expression ( a + b - 2c ) can have multiple values.But the problem asks for "the value," which is singular. Maybe I need to consider that the arithmetic square root of ( c ) is itself, which means ( c ) is non-negative. So, ( c = 0 ) or ( c = 1 ). So, both are acceptable.Therefore, the possible values are 10, 8, -4, and -6. But perhaps the problem expects all possible values. Alternatively, maybe I need to consider that ( a + 3 ) and ( 2a - 15 ) are the two square roots, meaning one is positive and one is negative. So, perhaps ( a = 4 ) is the correct solution because ( a = 18 ) gives both positive, which would mean they are the same square root, not two distinct square roots.Wait, that's a good point. If ( a = 18 ), then both ( a + 3 ) and ( 2a - 15 ) are 21, which is the same positive square root. So, they are not distinct square roots. Therefore, perhaps ( a = 4 ) is the correct solution because it gives one positive and one negative square root, which are distinct.So, maybe ( a = 4 ) is the only valid solution. Let me think about that.If ( a = 18 ), then both expressions are equal, so they are not two distinct square roots, just the same square root. So, perhaps the problem implies that they are two distinct square roots, meaning one positive and one negative. Therefore, ( a = 4 ) is the correct solution.So, with ( a = 4 ), ( b = -8 ), and ( c = 0 ) or ( c = 1 ), let's compute ( a + b - 2c ).First, ( c = 0 ):( 4 + (-8) - 2(0) = 4 - 8 - 0 = -4 )Second, ( c = 1 ):( 4 + (-8) - 2(1) = 4 - 8 - 2 = -6 )So, the possible values are -4 and -6.But the problem says "the value," so maybe both are acceptable. Alternatively, perhaps the arithmetic square root being itself implies ( c ) is non-negative, so both 0 and 1 are acceptable.Therefore, the possible values of ( a + b - 2c ) are -4 and -6.But let me check again. If ( a = 4 ), then ( a + 3 = 7 ) and ( 2a - 15 = -7 ), which are indeed the two square roots of 49. So, that's correct.If ( a = 18 ), then both are 21, which is just the positive square root, so they are not two distinct square roots. Therefore, ( a = 4 ) is the correct solution.Therefore, ( a = 4 ), ( b = -8 ), and ( c = 0 ) or ( c = 1 ).So, computing ( a + b - 2c ):For ( c = 0 ):( 4 - 8 - 0 = -4 )For ( c = 1 ):( 4 - 8 - 2 = -6 )Therefore, the possible values are -4 and -6.But the problem asks for "the value," so perhaps both are acceptable. Alternatively, maybe the problem expects a single answer, and I need to consider which one is correct.Wait, let's think about the arithmetic square root. The arithmetic square root is the principal (non-negative) square root. So, if the arithmetic square root of ( c ) is itself, that means ( c ) must be non-negative, which it is in both cases (0 and 1). So, both are acceptable.Therefore, the expression ( a + b - 2c ) can be either -4 or -6.But the problem says "find the value," which is singular. Maybe I need to consider that ( c ) can be 0 or 1, so both answers are possible.Alternatively, perhaps the problem expects both answers. Let me check the original problem again."Given that ( a + 3 ) and ( 2a - 15 ) are the square roots of a certain positive number, the cube root of ( b ) is ( -2 ), and the arithmetic square root of ( c ) is itself, find the value of ( a + b - 2c )."So, it doesn't specify to find all possible values, just "the value." But since there are multiple possible values, perhaps both -4 and -6 are acceptable.Alternatively, maybe I made a mistake earlier. Let me think again.If ( a = 4 ), then ( a + 3 = 7 ) and ( 2a - 15 = -7 ). So, they are the two square roots of 49. So, that's correct.If ( a = 18 ), then both are 21, which is just the positive square root. So, they are not two distinct square roots. Therefore, ( a = 4 ) is the correct solution.Therefore, ( a = 4 ), ( b = -8 ), and ( c = 0 ) or ( c = 1 ).So, computing ( a + b - 2c ):For ( c = 0 ):( 4 - 8 - 0 = -4 )For ( c = 1 ):( 4 - 8 - 2 = -6 )Therefore, the possible values are -4 and -6.But since the problem asks for "the value," perhaps both are acceptable. Alternatively, maybe the problem expects a single answer, and I need to consider that ( c ) is 1 because 0 is trivial. But I'm not sure.Alternatively, perhaps the problem expects both answers, so I should present both.Therefore, the values of ( a + b - 2c ) are -4 and -6.

💡Alright, so I've got this sequence problem here. Let me try to wrap my head around it. The sequence is defined by ( a_1 = frac{1}{2} ) and ( a_{n+1} = frac{a_n^2}{a_n^2 - a_n + 1} ) for any natural number ( n ). There are two parts to this problem: first, I need to prove that each term is smaller than the previous one, meaning ( a_{n+1} < a_n ). Second, I have to show that the sum of the first ( n ) terms, denoted ( S_n ), is less than 1.Starting with part (I). I need to show that the sequence is decreasing. That means each subsequent term is smaller than the one before it. Given the recursive formula, I can try to subtract ( a_{n+1} ) from ( a_n ) and see if the result is positive.So, let's compute ( a_n - a_{n+1} ):[a_n - a_{n+1} = a_n - frac{a_n^2}{a_n^2 - a_n + 1}]To combine these terms, I'll get a common denominator:[= frac{a_n(a_n^2 - a_n + 1) - a_n^2}{a_n^2 - a_n + 1}]Expanding the numerator:[= frac{a_n^3 - a_n^2 + a_n - a_n^2}{a_n^2 - a_n + 1}]Combine like terms:[= frac{a_n^3 - 2a_n^2 + a_n}{a_n^2 - a_n + 1}]Factor the numerator:[= frac{a_n(a_n^2 - 2a_n + 1)}{a_n^2 - a_n + 1}]Notice that ( a_n^2 - 2a_n + 1 ) is a perfect square:[= frac{a_n(a_n - 1)^2}{a_n^2 - a_n + 1}]Now, let's analyze the denominator ( a_n^2 - a_n + 1 ). Completing the square:[a_n^2 - a_n + 1 = left(a_n - frac{1}{2}right)^2 + frac{3}{4}]Since squares are always non-negative, this denominator is always positive. Also, ( a_n ) is given as ( frac{1}{2} ) initially, and from the recursive formula, it seems all terms are positive because the denominator is always positive and the numerator is a square, hence positive. So, ( a_n > 0 ) for all ( n ).Looking back at the expression for ( a_n - a_{n+1} ), the numerator is ( a_n(a_n - 1)^2 ). Since ( a_n > 0 ) and ( (a_n - 1)^2 ) is always non-negative, the entire numerator is non-negative. However, since ( a_n ) is less than 1 (as ( a_1 = frac{1}{2} ) and the sequence is decreasing), ( (a_n - 1)^2 ) is positive. Therefore, the numerator is positive, and the denominator is positive, so the entire expression is positive. Hence, ( a_n - a_{n+1} > 0 ), which means ( a_{n+1} < a_n ). So, part (I) is proven.Moving on to part (II). I need to show that the sum ( S_n = a_1 + a_2 + dots + a_n < 1 ). Hmm, this seems a bit trickier. Maybe I can find a telescoping series or some pattern when summing the terms.Let me consider the expression for ( a_{n+1} ):[a_{n+1} = frac{a_n^2}{a_n^2 - a_n + 1}]I can rewrite this as:[a_{n+1} = frac{a_n^2}{(a_n^2 - a_n + 1)}]Let me try to manipulate this equation to find a relationship that can help me sum the terms. Maybe express ( 1 - a_{n+1} ) in terms of ( a_n ):[1 - a_{n+1} = 1 - frac{a_n^2}{a_n^2 - a_n + 1}]Combine the terms:[= frac{(a_n^2 - a_n + 1) - a_n^2}{a_n^2 - a_n + 1}]Simplify the numerator:[= frac{-a_n + 1}{a_n^2 - a_n + 1}]So,[1 - a_{n+1} = frac{1 - a_n}{a_n^2 - a_n + 1}]Interesting. Let's take reciprocals on both sides:[frac{1}{1 - a_{n+1}} = frac{a_n^2 - a_n + 1}{1 - a_n}]Let me simplify the right-hand side:[= frac{a_n^2 - a_n + 1}{1 - a_n} = frac{a_n^2 - a_n + 1}{-(a_n - 1)} = -frac{a_n^2 - a_n + 1}{a_n - 1}]Wait, maybe another approach. Let me perform polynomial division or see if I can split the fraction:Let me write ( a_n^2 - a_n + 1 ) as ( (a_n^2 - a_n) + 1 ). Then,[frac{a_n^2 - a_n + 1}{1 - a_n} = frac{(a_n^2 - a_n) + 1}{1 - a_n} = frac{a_n(a_n - 1) + 1}{1 - a_n}]Which is:[= frac{-a_n(1 - a_n) + 1}{1 - a_n} = -a_n + frac{1}{1 - a_n}]So,[frac{1}{1 - a_{n+1}} = -a_n + frac{1}{1 - a_n}]Rearranging terms:[frac{1}{1 - a_{n+1}} + a_n = frac{1}{1 - a_n}]Therefore,[a_n = frac{1}{1 - a_n} - frac{1}{1 - a_{n+1}}]Wow, that's a nice telescoping relationship! So each term ( a_n ) can be expressed as the difference between two reciprocals. That should help when summing up the terms.Let me write this out for each term from ( n = 1 ) to ( n = k ):For ( n = 1 ):[a_1 = frac{1}{1 - a_1} - frac{1}{1 - a_2}]For ( n = 2 ):[a_2 = frac{1}{1 - a_2} - frac{1}{1 - a_3}]...For ( n = k ):[a_k = frac{1}{1 - a_k} - frac{1}{1 - a_{k+1}}]Now, if I sum all these equations from ( n = 1 ) to ( n = k ), the left-hand side will be ( S_k = a_1 + a_2 + dots + a_k ), and the right-hand side will telescope:[S_k = left( frac{1}{1 - a_1} - frac{1}{1 - a_2} right) + left( frac{1}{1 - a_2} - frac{1}{1 - a_3} right) + dots + left( frac{1}{1 - a_k} - frac{1}{1 - a_{k+1}} right)]All the intermediate terms cancel out, leaving:[S_k = frac{1}{1 - a_1} - frac{1}{1 - a_{k+1}}]We know that ( a_1 = frac{1}{2} ), so:[frac{1}{1 - a_1} = frac{1}{1 - frac{1}{2}} = frac{1}{frac{1}{2}} = 2]Therefore,[S_k = 2 - frac{1}{1 - a_{k+1}}]Now, since from part (I) we know that the sequence ( {a_n} ) is decreasing and all terms are positive, ( a_{k+1} ) is positive and less than ( a_k ), which is less than ( a_1 = frac{1}{2} ). Therefore, ( 1 - a_{k+1} ) is less than ( 1 - 0 = 1 ), but actually, since ( a_{k+1} ) is positive, ( 1 - a_{k+1} ) is less than 1 but greater than 0.Therefore, ( frac{1}{1 - a_{k+1}} ) is greater than 1. So,[S_k = 2 - frac{1}{1 - a_{k+1}} < 2 - 1 = 1]Hence, ( S_k < 1 ) for all ( k ). Therefore, the sum of the first ( n ) terms is always less than 1.Let me just double-check my steps. I derived that each ( a_n ) can be written as the difference of reciprocals, which telescopes when summed. That seems solid. Then, substituting ( a_1 = frac{1}{2} ) gives 2, and since ( frac{1}{1 - a_{k+1}} > 1 ), subtracting it from 2 gives something less than 1. That makes sense.I think this approach is correct. So, both parts (I) and (II) are proven.**Final Answer**Ⅰ. boxed{a_{n+1} < a_n}Ⅱ. boxed{S_n < 1}

💡Okay, so I have this problem where I need to find the ratio A/B, where A and B are two series defined in a specific way. Let me try to understand what each series is and how they relate to each other.First, let me write down the series A and B again to make sure I have them correctly:A is defined as:[ A = frac{1}{1^3} + frac{1}{5^3} - frac{1}{7^3} - frac{1}{11^3} + frac{1}{13^3} + frac{1}{17^3} - dotsb ]This series omits all terms where the denominator is an odd multiple of 3. So, terms like 1/(3^3), 1/(9^3), etc., are not included in A.On the other hand, B is defined as:[ B = frac{1}{3^3} - frac{1}{9^3} + frac{1}{15^3} - frac{1}{21^3} + frac{1}{27^3} - frac{1}{33^3} + dotsb ]This series includes only terms where the denominator is an odd multiple of 3. So, it's all the terms that A omits.The goal is to find the ratio A/B.Hmm, okay. So, A and B are related because A includes all the terms except those that are multiples of 3, and B includes only those multiples of 3. Maybe I can express A in terms of B or vice versa.Let me think about how these series are structured. Both A and B have terms with denominators that are odd numbers, but A excludes multiples of 3, while B includes only multiples of 3.Perhaps I can consider the entire series of reciprocals of cubes of odd numbers and see how A and B fit into that.Let me denote the full series as C:[ C = frac{1}{1^3} + frac{1}{3^3} + frac{1}{5^3} + frac{1}{7^3} + frac{1}{9^3} + frac{1}{11^3} + dotsb ]So, C is the sum of reciprocals of cubes of all odd numbers.From the definitions, A is C without the terms that are multiples of 3, and B is exactly those terms that are multiples of 3. Therefore, we can write:[ C = A + B ]But wait, in the original definitions, A and B have alternating signs. Let me check:Looking back at A:[ A = frac{1}{1^3} + frac{1}{5^3} - frac{1}{7^3} - frac{1}{11^3} + frac{1}{13^3} + frac{1}{17^3} - dotsb ]It seems that the signs are alternating in pairs: two positive, two negative, etc.Similarly, B:[ B = frac{1}{3^3} - frac{1}{9^3} + frac{1}{15^3} - frac{1}{21^3} + frac{1}{27^3} - frac{1}{33^3} + dotsb ]Here, the signs alternate every term: positive, negative, positive, etc.So, both A and B have alternating signs, but A alternates every two terms, while B alternates every term.This complicates things a bit because it's not a straightforward separation of terms. I need to figure out how the signs affect the relationship between A and B.Let me try to express A in terms of C and B, considering the signs.Wait, in the original problem, the user mentioned that in B, each term is 1/(3n)^3 where n is an odd number, but with alternating signs. Let me see:B is:[ B = frac{1}{3^3} - frac{1}{9^3} + frac{1}{15^3} - frac{1}{21^3} + dotsb ]So, each term is 1/(3*(2k-1))^3 with alternating signs starting with positive.Similarly, A is:[ A = frac{1}{1^3} + frac{1}{5^3} - frac{1}{7^3} - frac{1}{11^3} + frac{1}{13^3} + frac{1}{17^3} - dotsb ]Here, the denominators are 1, 5, 7, 11, 13, 17, etc., which are odd numbers not divisible by 3. The signs are two positive, two negative, etc.So, perhaps I can factor out 1/3^3 from B to see if it relates to A.Let me try that.Factor out 1/3^3 from B:[ B = frac{1}{3^3} left(1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb right) ]Wait, that's interesting. The expression inside the parentheses is similar to A but with alternating signs every term instead of every two terms.But in A, the signs are two positive, two negative, etc. So, perhaps there's a relationship between the two.Let me denote the series inside the parentheses as D:[ D = 1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb ]So, D is an alternating series of reciprocals of cubes of odd numbers.But A is:[ A = 1 + frac{1}{5^3} - frac{1}{7^3} - frac{1}{11^3} + frac{1}{13^3} + frac{1}{17^3} - dotsb ]Which can be written as:[ A = (1 + frac{1}{5^3}) - (frac{1}{7^3} + frac{1}{11^3}) + (frac{1}{13^3} + frac{1}{17^3}) - dotsb ]So, A groups terms in pairs with the same sign.Comparing D and A, D is an alternating series, while A is grouped into pairs with alternating signs.Is there a way to relate D and A?Alternatively, maybe I can express A in terms of D.Wait, let's see:If I take D:[ D = 1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb ]And then group the terms in pairs:[ D = left(1 + frac{1}{5^3}right) - left(frac{1}{3^3} + frac{1}{7^3}right) + left(frac{1}{9^3} + frac{1}{13^3}right) - dotsb ]Hmm, that's similar to A but with different groupings.Wait, in A, the first two terms are positive, then the next two are negative, etc. In D, the grouping is different: it's alternating every term.So, perhaps A can be expressed as D plus some correction terms.Alternatively, maybe I can write A in terms of D and B.Wait, since A is C without the multiples of 3, and C is the full series, but A has a different sign pattern.Alternatively, maybe I can consider generating functions or use known series expansions to relate A and B.I recall that the sum of reciprocals of cubes of odd numbers can be related to the Riemann zeta function. Specifically, the sum over n odd of 1/n^3 is (1 - 1/2^3)^{-1} ζ(3), but I'm not sure if that's directly applicable here because of the alternating signs.Wait, actually, the sum over all odd n of 1/n^3 is (1 - 1/2^3) ζ(3) = (7/8) ζ(3). But in our case, A and B have alternating signs, so it's more complicated.Alternatively, perhaps I can express A and B in terms of the Dirichlet eta function or other alternating series.The Dirichlet eta function is defined as:[ eta(s) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} ]For s=3, it's:[ eta(3) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^3} ]But our series A and B are over odd numbers only, with specific sign patterns.Wait, let me think differently. Maybe I can factor out 1/3^3 from B and see if the remaining series relates to A.From the original problem, the user factored out 1/27 from B and noticed that the remaining series includes all terms of A but with some sign changes.Wait, let me try that again.Given:[ B = frac{1}{3^3} - frac{1}{9^3} + frac{1}{15^3} - frac{1}{21^3} + dotsb ]Factor out 1/3^3:[ B = frac{1}{27} left(1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb right) ]So, the series inside the parentheses is:[ D = 1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb ]Which is an alternating series over all odd reciprocals cubed.Now, comparing D with A:A is:[ A = 1 + frac{1}{5^3} - frac{1}{7^3} - frac{1}{11^3} + frac{1}{13^3} + frac{1}{17^3} - dotsb ]So, A has two positive terms, then two negative, etc.If I look at D, it's:[ D = 1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb ]Which alternates every term.So, perhaps D can be expressed in terms of A and B.Wait, let me see:If I take D:[ D = 1 - frac{1}{3^3} + frac{1}{5^3} - frac{1}{7^3} + frac{1}{9^3} - dotsb ]And notice that A is:[ A = 1 + frac{1}{5^3} - frac{1}{7^3} - frac{1}{11^3} + frac{1}{13^3} + frac{1}{17^3} - dotsb ]So, if I subtract the terms involving multiples of 3 from D, I get A.Wait, but D includes the terms that are multiples of 3 with alternating signs, while A excludes them.Wait, actually, if I take D and subtract the terms that are multiples of 3, I get A.But in D, the multiples of 3 are:[ -frac{1}{3^3} + frac{1}{9^3} - frac{1}{15^3} + frac{1}{21^3} - dotsb ]Which is exactly -B.Because B is:[ B = frac{1}{3^3} - frac{1}{9^3} + frac{1}{15^3} - frac{1}{21^3} + dotsb ]So, the multiples of 3 in D are -B.Therefore, D can be written as:[ D = A - B ]Because D is the full alternating series, which is A (without multiples of 3) plus the multiples of 3 terms, which are -B.But wait, from earlier, we have:[ B = frac{1}{27} D ]So, substituting D = A - B into this:[ B = frac{1}{27} (A - B) ]Multiply both sides by 27:[ 27B = A - B ]Bring B to the left:[ 27B + B = A ][ 28B = A ]Therefore, the ratio A/B is 28.So, the answer is 28.

💡Okay, so I have this problem here with four positive integers a, b, c, and d. The problem gives me some greatest common divisors (gcds) between pairs of these numbers and then tells me that the gcd of d and a is between 90 and 120. I need to figure out which of the given options must be a divisor of a.Let me write down the given information:1. gcd(a, b) = 182. gcd(b, c) = 453. gcd(c, d) = 604. 90 < gcd(d, a) < 120And the options are 5, 7, 11, 13, and 17. I need to determine which one must divide a.First, I think it's helpful to factorize the given gcds into their prime factors because that might help me understand the relationships between a, b, c, and d.So, let's factorize each gcd:- 18 = 2 × 3²- 45 = 3² × 5- 60 = 2² × 3 × 5Okay, so from gcd(a, b) = 18, I know that both a and b must have at least the prime factors 2 and 3². Similarly, gcd(b, c) = 45 tells me that both b and c must have at least 3² and 5. Then, gcd(c, d) = 60 means c and d must have at least 2², 3, and 5.Now, I need to figure out what this implies about a, b, c, and d individually.Let me try to express each variable in terms of their prime factors, considering the gcds.Starting with a and b:Since gcd(a, b) = 18 = 2 × 3², this means that both a and b have at least these factors. So, I can write:a = 2 × 3² × wb = 2 × 3² × xWhere w and x are integers that don't share any common factors with each other (since 18 is the gcd, which is the product of the minimum exponents of the shared primes).Next, looking at gcd(b, c) = 45 = 3² × 5. Since b already has 2 × 3², and the gcd with c is 3² × 5, this means that c must have at least 3² × 5, but it can't have the factor 2 because otherwise, the gcd(b, c) would include 2 as well. So, c must be:c = 3² × 5 × yWhere y is an integer that doesn't share any common factors with b beyond 3² × 5.Similarly, gcd(c, d) = 60 = 2² × 3 × 5. Since c has 3² × 5, and the gcd with d is 2² × 3 × 5, this tells me that d must have at least 2² × 3 × 5. So, d can be written as:d = 2² × 3 × 5 × zWhere z is an integer that doesn't share any common factors with c beyond 3 × 5.Now, we need to find gcd(d, a). Let's express both d and a in terms of their prime factors:a = 2 × 3² × wd = 2² × 3 × 5 × zSo, gcd(a, d) will be the product of the smallest powers of the common prime factors. Let's see:- For prime 2: the minimum exponent between a (which has 2¹) and d (which has 2²) is 1.- For prime 3: the minimum exponent between a (which has 3²) and d (which has 3¹) is 1.- For prime 5: a doesn't have 5, and d does, so the minimum exponent is 0.- Any other primes: since w and z are co-prime with each other and with the other terms, they don't contribute to the gcd.Therefore, gcd(a, d) = 2¹ × 3¹ = 6. Wait, that can't be right because the problem states that 90 < gcd(d, a) < 120. 6 is way too small. So, I must have made a mistake in my reasoning.Let me go back. Maybe I didn't consider that w and z could have common factors. Because in my initial breakdown, I assumed that w and z are co-prime, but actually, they might share some common factors, which would increase the gcd(a, d).So, let me correct that. The gcd(a, d) is actually:gcd(a, d) = gcd(2 × 3² × w, 2² × 3 × 5 × z) = 2^min(1,2) × 3^min(2,1) × 5^min(0,1) × gcd(w, z)But since a doesn't have 5, the exponent for 5 is 0, and d has 5^1. So, 5 doesn't contribute. Similarly, for other primes, only the common ones contribute, which would be the gcd(w, z).So, gcd(a, d) = 2^1 × 3^1 × gcd(w, z) = 6 × gcd(w, z)Given that 90 < gcd(d, a) < 120, so 90 < 6 × gcd(w, z) < 120Divide all parts by 6:15 < gcd(w, z) < 20So, gcd(w, z) must be an integer between 15 and 20, not including 15 and 20. So, possible values are 16, 17, 18, 19.But wait, gcd(w, z) must be a divisor of both w and z. Since w and z are parts of a and d respectively, and a and d have certain prime factors, we need to see what primes can divide both w and z.Looking back:- a = 2 × 3² × w- d = 2² × 3 × 5 × zSo, w is part of a, which already has 2 and 3². z is part of d, which has 2², 3, and 5. So, w and z can have other prime factors, but they can't have 2, 3, or 5 because otherwise, the gcd(a, b) or gcd(c, d) would have higher exponents or include those primes.Wait, actually, let me think again. Since w is part of a, which already has 2 and 3², w could have other primes, but if w had 2 or 3, it would increase the gcd(a, b) beyond 18. Similarly, z is part of d, which has 2², 3, and 5, so z can't have 2, 3, or 5 because otherwise, the gcd(c, d) would be higher.Therefore, w and z must be composed of primes other than 2, 3, and 5. So, the gcd(w, z) must be a product of primes greater than 5.But the possible gcd(w, z) values are 16, 17, 18, 19. Let's factorize these:- 16 = 2⁴- 17 is prime- 18 = 2 × 3²- 19 is primeBut since w and z can't have 2, 3, or 5, the only possible gcd(w, z) values are 17 and 19 because 16 and 18 include primes that w and z can't have.So, gcd(w, z) must be either 17 or 19.Therefore, gcd(a, d) = 6 × 17 = 102 or 6 × 19 = 114.Both 102 and 114 are between 90 and 120, so that fits.Now, since gcd(w, z) is either 17 or 19, which are primes, this means that both w and z must be multiples of 17 or 19. Therefore, a, which is 2 × 3² × w, must have 17 or 19 as a factor.Looking back at the options, 17 is one of them (option E), and 19 is not listed. So, 17 must be a divisor of a.Wait, but hold on. The problem says "which of the following must be a divisor of a". So, if gcd(w, z) could be either 17 or 19, does that mean a must have both 17 and 19? Or is it that a must have at least one of them?No, because depending on whether gcd(w, z) is 17 or 19, a would have either 17 or 19 as a factor, but not necessarily both. So, a must have at least one of them, but since only 17 is an option, that must be the answer.But wait, let me double-check. The problem says "must be a divisor of a". So, if in some cases, a has 17, and in others, it has 19, but since 19 isn't an option, then 17 is the only one that must be a divisor in all cases? Hmm, no, actually, if gcd(w, z) could be 17 or 19, then a could have either 17 or 19, but not necessarily both. So, the only prime that must divide a is 5, because looking back, a is 2 × 3² × w, and w could have 17 or 19, but 5 is not necessarily a factor of a.Wait, hold on, let me go back to the beginning. The gcd(a, d) is 6 × gcd(w, z). But in the problem, the gcd(d, a) is between 90 and 120, so 6 × gcd(w, z) is between 90 and 120, which means gcd(w, z) is between 15 and 20. But as we saw, gcd(w, z) can only be 17 or 19 because 16 and 18 include primes that w and z can't have.But wait, let me think about the prime factors again. a is 2 × 3² × w, and d is 2² × 3 × 5 × z. So, the gcd(a, d) is 2 × 3 × gcd(w, z). So, 2 × 3 is 6, and then multiplied by gcd(w, z). So, 6 × gcd(w, z) is between 90 and 120. So, gcd(w, z) is between 15 and 20.But since w and z can't have 2, 3, or 5, the only possible gcd(w, z) is 17 or 19 because 16 is 2⁴, which is not allowed, and 18 is 2 × 3², which is also not allowed.Therefore, gcd(w, z) must be 17 or 19, so a must have either 17 or 19 as a factor. Since 17 is an option, that must be the answer.Wait, but hold on again. The problem is asking which of the options must be a divisor of a. So, if a could have either 17 or 19, but not necessarily both, then 17 is a possible divisor, but not necessarily the only one. However, since 17 is an option and 19 isn't, 17 must be the answer.But wait, let me think about the initial factorizations again. a is 2 × 3² × w, and d is 2² × 3 × 5 × z. So, the gcd(a, d) is 2 × 3 × gcd(w, z). So, 6 × gcd(w, z). So, if gcd(w, z) is 17, then gcd(a, d) is 102, and if it's 19, it's 114. Both are between 90 and 120, so both are valid.Therefore, a must have either 17 or 19 as a factor, but since only 17 is an option, that must be the answer.Wait, but hold on. Let me check if 5 is a divisor of a. From the initial factorizations, a is 2 × 3² × w. There's no 5 in a unless w has it. But from the gcd(a, d), which is 6 × gcd(w, z), and since d has 5, but a doesn't necessarily have 5 unless w has it. However, the gcd(a, d) doesn't include 5 because a doesn't have 5. So, 5 is not necessarily a divisor of a.Wait, but in the initial step, I thought that since gcd(c, d) is 60, which includes 5, and c is 3² × 5 × y, so c has 5, and d has 5. But a is 2 × 3² × w, which doesn't necessarily have 5. So, 5 isn't necessarily a divisor of a.Therefore, the only prime that must divide a is either 17 or 19, but since 19 isn't an option, 17 is the answer.Wait, but hold on again. Let me think about the chain of gcds:gcd(a, b) = 18, so a has 2 × 3².gcd(b, c) = 45, so b has 3² × 5, and c has 3² × 5.gcd(c, d) = 60, so c has 2² × 3 × 5, and d has 2² × 3 × 5.Wait, hold on, I think I made a mistake earlier. Let me re-examine the factorizations.From gcd(a, b) = 18 = 2 × 3², so a and b have at least 2 × 3².From gcd(b, c) = 45 = 3² × 5, so b and c have at least 3² × 5. Therefore, b must have 2 × 3² × 5, because it already has 2 from gcd(a, b) and 3² × 5 from gcd(b, c). So, b = 2 × 3² × 5 × x, where x is an integer co-prime with a and c.Similarly, c must have 3² × 5, but from gcd(c, d) = 60 = 2² × 3 × 5, so c must have 2² × 3 × 5. Wait, but earlier I thought c was 3² × 5 × y, but now it seems c must have 2² × 3 × 5. So, c is 2² × 3 × 5 × y, where y is co-prime with b.Wait, but hold on, if c has 2² × 3 × 5, but from gcd(b, c) = 45 = 3² × 5, that would mean that c has at least 3² × 5, but from gcd(c, d) = 60 = 2² × 3 × 5, c has 2² × 3 × 5. So, c must have the maximum of these, which is 2² × 3² × 5. So, c = 2² × 3² × 5 × y, where y is co-prime with b and d.Therefore, c is 2² × 3² × 5 × y.Similarly, d must have 2² × 3 × 5, so d = 2² × 3 × 5 × z, where z is co-prime with c.Now, let's re-express a, b, c, d:a = 2 × 3² × wb = 2 × 3² × 5 × xc = 2² × 3² × 5 × yd = 2² × 3 × 5 × zNow, let's compute gcd(a, d):a = 2 × 3² × wd = 2² × 3 × 5 × zSo, gcd(a, d) is the product of the minimum exponents of the shared primes:- For 2: min(1, 2) = 1- For 3: min(2, 1) = 1- For 5: min(0, 1) = 0- Any other primes: gcd(w, z)So, gcd(a, d) = 2 × 3 × gcd(w, z) = 6 × gcd(w, z)Given that 90 < 6 × gcd(w, z) < 120, so dividing by 6:15 < gcd(w, z) < 20So, gcd(w, z) must be 16, 17, 18, or 19.But since w is part of a, which is 2 × 3² × w, and z is part of d, which is 2² × 3 × 5 × z, w and z can't have 2, 3, or 5 as factors because otherwise, the gcds would be higher.Therefore, w and z must be composed of primes greater than 5. So, the possible gcd(w, z) values are 17 or 19 because 16 is 2⁴ and 18 is 2 × 3², which are not allowed.Thus, gcd(w, z) is either 17 or 19, so gcd(a, d) is either 102 or 114.Therefore, a must have either 17 or 19 as a factor. Since 17 is an option, that must be the answer.Wait, but hold on. Let me check if 5 is a divisor of a. From the initial factorizations, a is 2 × 3² × w. There's no 5 in a unless w has it. But from the gcd(a, d), which is 6 × gcd(w, z), and since d has 5, but a doesn't necessarily have 5 unless w has it. However, the gcd(a, d) doesn't include 5 because a doesn't have 5. So, 5 isn't necessarily a divisor of a.Therefore, the only prime that must divide a is either 17 or 19, but since 19 isn't an option, 17 is the answer.Wait, but in the initial problem, the options are 5, 7, 11, 13, 17. So, 17 is an option, so that must be the answer.But wait, let me think again. If gcd(w, z) is 17, then a has 17, and if it's 19, a has 19. Since 19 isn't an option, 17 must be the answer.But hold on, the problem says "must be a divisor of a". So, if a could have either 17 or 19, but not necessarily both, then 17 is a possible divisor, but not necessarily the only one. However, since 17 is an option and 19 isn't, 17 is the answer.Wait, but I'm a bit confused because earlier I thought 5 might be a divisor, but now I'm convinced it's 17. Let me check the initial factorizations again.a = 2 × 3² × wb = 2 × 3² × 5 × xc = 2² × 3² × 5 × yd = 2² × 3 × 5 × zSo, a doesn't have 5 unless w has it. But from the gcd(a, d), which is 6 × gcd(w, z), and since d has 5, but a doesn't necessarily have 5 unless w has it. However, the gcd(a, d) doesn't include 5 because a doesn't have 5. So, 5 isn't necessarily a divisor of a.Therefore, the only prime that must divide a is either 17 or 19, but since 19 isn't an option, 17 is the answer.Wait, but in the initial problem, the options are 5, 7, 11, 13, 17. So, 17 is an option, so that must be the answer.But hold on, let me think about the chain of gcds again. Since gcd(b, c) = 45, which includes 5, and gcd(c, d) = 60, which also includes 5, does that mean that 5 must be a factor of a?Wait, no, because a is only connected to b, which has 5, but a's gcd with b is 18, which doesn't include 5. So, a doesn't necessarily have 5.Therefore, 5 isn't necessarily a divisor of a, so the answer must be 17.Wait, but in the initial problem, the answer given was 5, but now I'm thinking it's 17. I must have made a mistake earlier.Wait, let me go back to the initial problem. The user provided the problem and then a solution that concluded 5 is the answer. But in my analysis, I think it's 17. So, maybe I need to figure out where I went wrong.Wait, in the initial solution, the user wrote:"Thus, gcd(a, d) = 2 × 3 × 5 × gcd(w, z). Neither 2 nor 5 can divide gcd(w, z), as that would make gcd(a, d) exceed the upper limit of 120. Thus, gcd(w, z) must be divisible by a prime number greater than 5. Among the possibilities, gcd(a, d) = 90 = 2 × 3 × 5 × 3. This value is in the range (90, 120) and implies 3 divides w and z."Wait, but in my analysis, I found that gcd(a, d) is 6 × gcd(w, z), not 30 × gcd(w, z). So, perhaps the initial solution was incorrect.Wait, let me recast the problem.Given:gcd(a, b) = 18 = 2 × 3²gcd(b, c) = 45 = 3² × 5gcd(c, d) = 60 = 2² × 3 × 5We need to find gcd(d, a) between 90 and 120.Expressing each variable:a = 2 × 3² × wb = 2 × 3² × 5 × xc = 2² × 3² × 5 × yd = 2² × 3 × 5 × zNow, gcd(a, d) = gcd(2 × 3² × w, 2² × 3 × 5 × z) = 2^1 × 3^1 × gcd(w, z) = 6 × gcd(w, z)Given 90 < 6 × gcd(w, z) < 120So, 15 < gcd(w, z) < 20Possible gcd(w, z): 16, 17, 18, 19But w and z can't have 2, 3, or 5, so gcd(w, z) must be 17 or 19Thus, gcd(a, d) is 102 or 114Therefore, a must have either 17 or 19 as a factorSince 17 is an option, that's the answerWait, but in the initial solution, the user concluded that 5 must be a divisor of a, but from my analysis, that's not necessarily the case. So, perhaps the initial solution was incorrect.Therefore, the correct answer should be 17, which is option E.But wait, the initial problem's solution concluded 5, but I think that's wrong. Let me check again.Wait, in the initial solution, the user wrote:"Thus, gcd(a, d) = 2 × 3 × 5 × gcd(w, z). Neither 2 nor 5 can divide gcd(w, z), as that would make gcd(a, d) exceed the upper limit of 120. Thus, gcd(w, z) must be divisible by a prime number greater than 5. Among the possibilities, gcd(a, d) = 90 = 2 × 3 × 5 × 3. This value is in the range (90, 120) and implies 3 divides w and z."Wait, but in reality, gcd(a, d) is 6 × gcd(w, z), not 30 × gcd(w, z). So, the initial solution incorrectly included 5 in the gcd(a, d), which isn't the case because a doesn't have 5.Therefore, the initial solution was wrong, and the correct answer is 17.But wait, the problem says "90 < gcd(d, a) < 120". So, if gcd(a, d) is 102 or 114, both of which are multiples of 6 and 17 or 19, then a must have 17 or 19.Since 17 is an option, that's the answer.Therefore, the correct answer is E) 17.

💡Okay, so I have this problem where I need to find the minimum value of the expression (frac{1}{x + y} + frac{1}{x + z} + frac{1}{y + z}) given that (x), (y), and (z) are positive real numbers and their sum is 2. Hmm, let me think about how to approach this.First, I remember that when dealing with expressions that involve reciprocals and sums, inequalities like the Cauchy-Schwarz inequality or the AM-HM inequality might be useful. Maybe I can apply one of those here.Let me write down what I know:1. (x + y + z = 2)2. The expression to minimize is (frac{1}{x + y} + frac{1}{x + z} + frac{1}{y + z})I notice that each term in the expression is the reciprocal of the sum of two variables. Since (x + y + z = 2), each pair sum (x + y), (x + z), and (y + z) will be equal to (2 - z), (2 - y), and (2 - x) respectively. So, the expression can also be written as:[frac{1}{2 - z} + frac{1}{2 - y} + frac{1}{2 - x}]But I'm not sure if that helps directly. Maybe I should think about symmetry here. Since the problem is symmetric in (x), (y), and (z), it's likely that the minimum occurs when all three variables are equal. Let me test that.If (x = y = z), then each variable is (frac{2}{3}) because (3x = 2) implies (x = frac{2}{3}). Plugging this into the expression:[frac{1}{frac{2}{3} + frac{2}{3}} + frac{1}{frac{2}{3} + frac{2}{3}} + frac{1}{frac{2}{3} + frac{2}{3}} = 3 times frac{1}{frac{4}{3}} = 3 times frac{3}{4} = frac{9}{4}]So, the value is (frac{9}{4}) when all variables are equal. But is this the minimum? I need to confirm.Maybe I can use the Cauchy-Schwarz inequality. The Cauchy-Schwarz inequality states that for positive real numbers (a_i) and (b_i):[left( sum a_i b_i right)^2 leq left( sum a_i^2 right) left( sum b_i^2 right)]But I'm not sure how to apply it directly here. Alternatively, the AM-HM inequality says that for positive real numbers (a_1, a_2, ..., a_n):[frac{a_1 + a_2 + ... + a_n}{n} geq frac{n}{frac{1}{a_1} + frac{1}{a_2} + ... + frac{1}{a_n}}]Which can be rearranged to:[frac{1}{a_1} + frac{1}{a_2} + ... + frac{1}{a_n} geq frac{n^2}{a_1 + a_2 + ... + a_n}]Maybe I can use this. Let me consider the terms (x + y), (x + z), and (y + z). Let me denote them as (a = x + y), (b = x + z), and (c = y + z). Then, the expression to minimize is (frac{1}{a} + frac{1}{b} + frac{1}{c}).From the problem, (x + y + z = 2). Also, adding the three equations (a = x + y), (b = x + z), (c = y + z), we get:[a + b + c = 2x + 2y + 2z = 2(x + y + z) = 4]So, (a + b + c = 4). Now, applying the AM-HM inequality on (a), (b), and (c):[frac{a + b + c}{3} geq frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}]Plugging in (a + b + c = 4):[frac{4}{3} geq frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}]Taking reciprocals (and remembering to reverse the inequality):[frac{3}{4} leq frac{frac{1}{a} + frac{1}{b} + frac{1}{c}}{3}]Wait, that doesn't seem right. Let me double-check.Actually, the AM-HM inequality is:[frac{a + b + c}{3} geq frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}]So, substituting (a + b + c = 4):[frac{4}{3} geq frac{3}{frac{1}{a} + frac{1}{b} + frac{1}{c}}]Multiplying both sides by (frac{1}{a} + frac{1}{b} + frac{1}{c}) and then by (frac{3}{4}):[frac{1}{a} + frac{1}{b} + frac{1}{c} geq frac{9}{4}]Ah, that's better. So, the expression (frac{1}{a} + frac{1}{b} + frac{1}{c}) is at least (frac{9}{4}). And equality holds when (a = b = c), which would mean (x + y = x + z = y + z). Solving these equations, we get (y = z), (x = z), and (x = y), so all variables are equal. Therefore, equality occurs when (x = y = z = frac{2}{3}), which matches our earlier calculation.So, the minimum value is indeed (frac{9}{4}).But just to be thorough, let me consider if there's any possibility that the expression could be smaller when the variables aren't equal. Suppose one variable is very small, approaching zero. Let's say (z) approaches 0. Then, (x + y) approaches 2, and the expression becomes:[frac{1}{x + y} + frac{1}{x} + frac{1}{y}]Since (x + y = 2 - z approx 2), so (frac{1}{x + y} approx frac{1}{2}). Then, we have (frac{1}{x} + frac{1}{y}). Since (x + y approx 2), by AM-HM inequality again:[frac{x + y}{2} geq frac{2}{frac{1}{x} + frac{1}{y}} implies frac{2}{2} geq frac{2}{frac{1}{x} + frac{1}{y}} implies 1 geq frac{2}{frac{1}{x} + frac{1}{y}} implies frac{1}{x} + frac{1}{y} geq 2]So, the expression becomes approximately (frac{1}{2} + 2 = frac{5}{2}), which is 2.5, which is larger than (frac{9}{4} = 2.25). So, making one variable very small doesn't give a smaller value.Similarly, if two variables are equal and the third is different, let's say (x = y) and (z = 2 - 2x). Then, the expression becomes:[frac{1}{2x} + frac{1}{x + (2 - 2x)} + frac{1}{x + (2 - 2x)} = frac{1}{2x} + frac{2}{2 - x}]Let me set (x = y), so (z = 2 - 2x). Then, the expression is:[frac{1}{2x} + 2 times frac{1}{2 - x}]Let me denote this as (f(x) = frac{1}{2x} + frac{2}{2 - x}). To find the minimum, take the derivative with respect to (x):[f'(x) = -frac{1}{2x^2} + frac{2}{(2 - x)^2}]Set (f'(x) = 0):[-frac{1}{2x^2} + frac{2}{(2 - x)^2} = 0 implies frac{2}{(2 - x)^2} = frac{1}{2x^2}]Cross-multiplying:[4x^2 = (2 - x)^2]Expanding the right side:[4x^2 = 4 - 4x + x^2]Bring all terms to one side:[4x^2 - x^2 + 4x - 4 = 0 implies 3x^2 + 4x - 4 = 0]Solving this quadratic equation:[x = frac{-4 pm sqrt{16 + 48}}{6} = frac{-4 pm sqrt{64}}{6} = frac{-4 pm 8}{6}]So, (x = frac{4}{6} = frac{2}{3}) or (x = frac{-12}{6} = -2). Since (x) must be positive and less than 1 (since (z = 2 - 2x > 0 implies x < 1)), we have (x = frac{2}{3}). Thus, (z = 2 - 2 times frac{2}{3} = 2 - frac{4}{3} = frac{2}{3}). So, all variables are equal again, leading to the same minimum value of (frac{9}{4}).Therefore, regardless of whether I assume symmetry or try to vary one variable, the minimum occurs when all variables are equal. Hence, the minimum value is (frac{9}{4}).

💡First, I need to determine the total cost of the ping pong balls without any discount. Each ping pong ball costs 0.10, and Johnny is purchasing 10,000 balls. So, the total cost without the discount would be:10,000 balls * 0.10/ball = 1,000Johnny paid 700 after receiving the discount. To find the discount amount, I subtract the amount paid from the original total cost:1,000 (original cost) - 700 (amount paid) = 300 (discount amount)Next, to calculate the percentage of the discount, I divide the discount amount by the original total cost and then multiply by 100:(300 discount / 1,000 original cost) * 100 = 30%Therefore, Johnny received a 30% discount on his purchase.