Find a formula to calculate the sum of the angles of a convex polygon with $n$ sides $(n \geqslant 3)$.
Let's prove by induction on $n \geqslant 3$ the property $P(n):$ The sum of the angles of an $n$-gon is $(n-2) \cdot 180^{\circ}$:

— Initialization: $P(3)$ is true: the sum of the angles of a triangle is indeed $180^{\circ}$.

- Heredity: Suppose that $P(j)$ is true and let's show $P(j+1)$. Consider three consecutive vertices of a polygon with $j+1$ sides and draw the diagonal that connects the two extreme vertices. We have separated the original polygon into a triangle and a polygon with $j$ sides, and the sum of the angles of the original polygon is therefore $180^{\circ}+(j-2) \cdot 180^{\circ} = ((j+1)-2) \cdot 180^{\circ}$.
- Conclusion: By induction, $P(n)$ is therefore true for all $n$ greater than or equal to 3, and the statement is proved.