Explicit to recursive formula for arithmetic series

Students will be introduced to discrete unary recursive functions. They'll start with function like the following:

$$\begin{align} & f(0) = 0 \\ & f(x + 1) = f(x) + 2 \end{align}$$

This is a funny way of writing \(f(x) = 2x.\) Students should also learn these functions can be expressed as piecewise functions. For example, here's another way of expressing \(f:\) $$f(x) = \begin{cases} 0 & \text{if }x = 0 \\ f(x - 1) + 2 & \text{if }x \gt 0 \end{cases}$$

Then students will learn how to express linear functions. Finally, students will learn how to convert between the explicit formula for an arithmetic series:

$$a_n = a_1 + (n - 1)d$$

and the recursive formula:

$$a_n = \begin{cases} a_1 & \text{if }n = 1 \\ a_{n - 1} + d & \text{if }n \gt 1 \end{cases}$$