## Proof of the Sum of the First n Positive Integers is Even ### Overview The proof involves decomposing the problem into propositions and formalizing them using a programming language, specifically Dafny. The goal is to verify that for any even integer \( n \), the sum of the first \( n \) positive integers is an even number. ### Components/Axes - **Proposition 1**: Define what it means for a number to be even. - **Proposition 2**: State that the sum of the first \( n \) positive integers follows a specific formula. - **Proposition 3**: Prove that if \( n \) is even, then the sum of the first \( n \) positive integers is even. ### Detailed Analysis or ### Content Details 1. **Proposition 1**: A number is even if it can be divided by 2 without leaving a remainder. Mathematically, this is expressed as \( n \mod 2 = 0 \). 2. **Proposition 2**: The sum of the first \( n \) positive integers can be calculated using the formula \( \frac{n(n+1)}{2} \). 3. **Proposition 3**: To prove that if \( n \) is even, then the sum of the first \( n \) positive integers is even, we can use the formula from Proposition 2. If \( n \) is even, then \( n = 2k \) for some integer \( k \). Substituting \( n = 2k \) into the formula gives \( \frac{2k(2k+1)}{2} = k(2k+1) \). Since \( k \) and \( 2k+1 \) are both integers, their product is also an integer. Moreover, since \( 2k+1 \) is odd, the product \( k(2k+1) \) is even. ### Key Observations - The proof relies on the properties of even and odd numbers. - The formula for the sum of the first \( n \) positive integers is a well-known result in mathematics. - The proof demonstrates the logical coherence of the propositions and the correctness of the encoded propositions. ### Interpretation The proof demonstrates that the sum of the first \( n \) positive integers is even if and only if \( n \) is even. This result is significant in number theory and has applications in various areas of mathematics and computer science. The proof uses logical reasoning and mathematical formulas to establish the correctness of the proposition. The integration of Dafny with theorem provers allows for the verification of the correctness of the propositions, ensuring that the proof is logically sound and free of errors.