answer:1. Let's denote the operation in question as *. 2. We are given that for any integers X and Y, there exists a combination of X and Y such that X * Y results in any integer. 3. Also, we need to demonstrate that the operation * cannot simultaneously satisfy: a. A * B = - (B * A) b. (A * B) * C = A * (B * C) 4. Firstly, assuming properties a) and b) hold true, we analyze them step-by-step: 1. From property a: A * B = - (B * A), we have: [ A * B = - (B * A) ] This implies that for any integers A and B, swapping the operands and then applying * results in the negative of the previous result. 2. From property b: (A * B) * C = A * (B * C), we should hold for the associativity of the operation *. Let’s analyze both sides of this equality: [ (A * B) * C = A * (B * C) ] 5. Using property a) in property b), we proceed with the following steps: 1. Start with (A * B) * C: [ (A * B) * C = - (C * (A * B)) ] 2. Apply property a) to C * (A * B): [ - (C * (A * B)) = -(-((C * A) * B)) = (C * A) * B ] 3. Thus: [ (A * B) * C = (C * A) * B ] 6. Now, we need to use property b: [ (A * B) * C = A * (B * C) ] 7. According to property a), we rewrite A * (B * C): [ A * (B * C) = -((B * C) * A) = - (B * (C * A)) ] 8. Setting our two expressions for (A * B) * C equal to each other: [ (C * A) * B = -A * (B * C) ] 9. Given (C * A) * B = -(A * (B * C)), rewrite in a simplified form: [ (C * A) * B = -((A * B) * C) ] 10. Now, let’s set up (A * B) * C based on these associations: [ (A * B) * C = -((A * B) * C) ] 11. Hence, adding (A * B) * C to both sides: [ (A * B) * C + (A * B) * C = 0 ] 12. This implies: [ 2 * (A * B) * C = 0 ] 13. Consequently, we determine: [ (A * B) * C = 0 ] 14. Since A * B can take any integer value according to the problem condition: [ X * C = 0 ] 15. This leads to a contradiction because X can be any integer, and for any integer X there exists a C such that X * C neq 0. # Conclusion: Hence, the operation * cannot satisfy both properties simultaneously. [ blacksquare ]

answer:Let's consider the geometric progression with first term a = 1 and common ratio r, where 0 < r < 1. We're interested in finding the sum of this infinite progression, often denoted as S. We can express S as the sum of its terms: S = 1 + r + r^2 + r^3 + ... Now, let's multiply S by the common ratio r: rS = r + r^2 + r^3 + r^4 + ... Notice that rS is the same as S, but shifted one term forward. Subtracting rS from S allows us to eliminate all terms except the first: S - rS = 1 Factoring S out of the left side: S(1 - r) = 1 Now, we can solve for S by dividing both sides by (1 - r): S = 1 / (1 - r) This expression represents the sum of the infinite geometric progression when r < 1. This formula is valid only when |r| < 1, as the series diverges otherwise. In our case, since r is a positive real number less than 1, this condition is met, and the formula provides the desired closed expression for the sum.

answer:We recognize this series as a geometric progression with first term 1 and common ratio 2x. Since |x| < 1, we know that |2x| < 1, meeting the condition for convergence of an infinite geometric series. Using the formula we derived earlier for the sum of an infinite geometric progression, S = a / (1 - r), where a is the first term and r is the common ratio, we can find the sum of the given series. In this case, a = 1 and r = 2x. Substituting these values into the formula: S = 1 / (1 - 2x) This expression represents the sum of the infinite series. Note that the restriction |x| < 1 ensures that the series converges, and the formula provides a valid result.

answer:This series can be viewed as a geometric progression with first term 12 and common ratio 1/5. Since |1/5| < 1, the series converges. Using the formula for the sum of an infinite geometric progression, S = a / (1 - r), where a is the first term and r is the common ratio, we can find the sum of the given series. In this case, a = 12 and r = 1/5. Substituting these values into the formula: S = 12 / (1 - 1/5) = 12 / (4/5) = 12 * (5/4) = 15 So, the sum of the infinite series is 15.