The given problem asks us to prove that the sum of the first $n$ natural numbers is a composite number if $n > 2$.

The sum of the first $n$ natural numbers is given by the formula: $S = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$

We need to show that $S$ is a composite number for $n > 2$.

Step-by-step proof:

1. Expression for the sum: The sum of the first $n$ natural numbers is $S = \frac{n(n+1)}{2}$.

2. Factor properties:

   • The product $n(n+1)$ is always even because one of $n$ or $n+1$ is even.
   • Dividing by 2 ensures that $S$ is an integer.

3. Analyze for small values of $n$: Let's check the expression for small values of $n$:

   • For $n = 1$: $S = \frac{1(1+1)}{2} = 1$ (not composite, but we are interested in $n > 2$).
   • For $n = 2$: $S = \frac{2(2+1)}{2} = 3$ (not composite).
   • For $n = 3$: $S = \frac{3(3+1)}{2} = 6$ (composite).
   • For $n = 4$: $S = \frac{4(4+1)}{2} = 10$ (composite).
   • For $n = 5$: $S = \frac{5(5+1)}{2} = 15$ (composite).

   For $n > 2$, it appears that $S$ is composite. We now generalize this observation.

4. Generalization: The sum $S = \frac{n(n+1)}{2}$ is the product of two integers, $n$ and $\frac{n+1}{2}$, when $n$ is odd, or $\frac{n}{2}$ and $n+1$, when $n$ is even. In either case, as long as $n > 2$, these integers will be greater than 1, which means $S$ is a product of two integers greater than 1, i.e., $S$ is composite.

Conclusion:

For $n > 2$, the sum $S = \frac{n(n+1)}{2}$ is always a composite number, as it is the product of two integers both greater than 1.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What happens if $n = 2$? Can we prove this special case differently?
2. How can we determine whether a number is composite or prime?
3. How does the sum formula change if we include only even or odd numbers?
4. What if we consider other patterns of natural numbers, like squares or cubes? Will their sums behave similarly?
5. Can this method be extended to sums of arithmetic sequences with different starting points and common differences?

Tip: When trying to prove that a number is composite, factorization is key. Try to express the number as a product of two integers greater than 1.