An arithmetic sequence with common difference $r$ is a sequence of numbers $s_{0}, s_{1}, s_{2}, \ldots$ that satisfies for all indices $k, s_{k+1}=s_{k}+r$. Determine an expression for $s_{n}$ in terms of $s_{0}, r$, and $n$.
Let's look at the first terms: $s_{0}, s_{0}+r,\left(s_{0}+r\right)+r=s_{0}+2 r,\left(s_{0}+2 r\right)+r=$ $s_{0}+3 r$ and so on.

Let's prove the conjectured formula (i.e., $s_{n}=s_{0}+n r$) by induction.

$\triangleright$ The formula is verified for $n=0$.

$\triangleright$ Let $n$ be an integer such that $s_{n}=s_{0}+n r$. Then,

$$
s_{n+1}=s_{n}+r=\left(s_{0}+n r\right)+r=s_{0}+(n+1) r
$$

hence the result by induction.