1. Discrete mathematics is a branch of mathematics dealing with discrete elements that uses algebra and arithmetic. 2. Common topics include logic, set theory, combinatorics, graph theory, and algorithms. 3. Let's start with a fundamental problem in discrete math: proving that the sum of the first $n$ natural numbers is given by $$\frac{n(n+1)}{2}.$$ 4. We use induction to prove this. 5. Base case: for $n=1$, sum is $1$. The formula gives $$\frac{1\cdot(1+1)}{2} = 1,$$ which holds. 6. Inductive step: assume it holds for $n=k$, i.e., $$1 + 2 + \cdots + k = \frac{k(k+1)}{2}.$$ 7. For $n=k+1$, sum is $$1 + 2 + \cdots + k + (k+1) = \frac{k(k+1)}{2} + (k+1)$$ by the inductive hypothesis. 8. Simplify: $$\frac{k(k+1)}{2} + (k+1) = (k+1)\left(\frac{k}{2} + 1\right) = (k+1)\left(\frac{k+2}{2}\right) = \frac{(k+1)(k+2)}{2}.$$ 9. This matches the formula for $n=k+1$, completing the induction. 10. Therefore, the sum of the first $n$ natural numbers is $$\boxed{\frac{n(n+1)}{2}}.$$