For n = 1, 2, 3. we can give a geometric argument in proving the formula
1 + 2 + 3 + .. + n = n(n+1)/2
1^2 + 2^2 + .. n^2 = n(n+1)(2n+1)/6
1^3 + 2^3 + + n^3 = (n(n+1)/2)^2
For n >3, there is a method using generating functions to obtain that formula.
We will go over the geometric and generating function arguments.

We will discuss basic properties of ultrametric spaces. Well-known examples of complete ultrametric spaces are p-adic numbers as well as p-adic integers. Also, it is known that the boundary at infinity of metric trees as well as more general Gromov 0-hyperbolic spaces is a complete bounded ultrametric space when equipped with a visual metric. We will discuss this result in details and show that the converse statement also holds. Namely, we show that every complete ultrametric space arises as the boundary at infinity of both a Gromov 0-hyperbolic space as well as a metric tree.