if n e N, then nˆ2 -n is even, prove it by applying proposition 4.2 to sum 1+2+3+···+(n-1)

To prove that \(n^2 - n\) is even using the sum \(1 + 2 + 3 + \dots + (n-1)\), we can interpret \(n^2 - n\) as the result of Proposition 4.2, which relates to sums of integers.

### Proof:

1. **Expression of \(n^2 - n\):**
   The expression \(n^2 - n\) can be factored as:
   \[
   n^2 - n = n(n - 1)
   \]
   This product is the multiplication of two consecutive integers \(n\) and \(n-1\). 

2. **Sum of first \(n-1\) integers:**
   The sum of the first \(n-1\) positive integers is:
   \[
   S = 1 + 2 + 3 + \dots + (n - 1)
   \]
   By the well-known formula for the sum of the first \(m\) integers, we have:
   \[
   S = \frac{(n - 1)n}{2}
   \]
   
3. **Relating \(n^2 - n\) to the sum:**
   We notice that the product \(n(n - 1)\) appears in the numerator of the sum formula. Therefore:
   \[
   n^2 - n = 2S
   \]
   This shows that \(n^2 - n\) is exactly twice the sum of the first \(n-1\) integers.

4. **Conclusion:**
   Since \(n^2 - n = 2S\), it is clear that \(n^2 - n\) is an even number, because it is twice the integer \(S\). Hence, \(n^2 - n\) is always even for any \(n \in \mathbb{N}\).

### Final Result:
\[
n^2 - n \text{ is even for any } n \in \mathbb{N}.
\]

Would you like any further details on this proof?

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Here are 5 related questions that build on this concept:

1. How do you prove that the sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\)?
2. Can you generalize the expression \(n^2 - n\) for non-integer values of \(n\)?
3. What is the significance of the sum of consecutive integers in other areas of mathematics?
4. How does the parity (even or odd nature) of an integer affect the expression \(n^2 - n\)?
5. Can you explore similar results for products of three consecutive integers?

**Tip:** When dealing with sums or products of integers, always check for common factorizations that can simplify the proof, such as factoring out constants like 2 to check for evenness or oddness.

This problem demonstrates how to prove that n^2 - n is even by applying Proposition 4.2 and the sum of the first n-1 integers. Explore the detailed algebraic steps involved in factoring and simplifying to show the evenness of the expression, suitable for Grades 10-12.

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