Choose $r \in R$. $f(0)= a = 1$ and $f(n+1)= a + r* f(n)$ for all $n \ge 0$.

Draw a mapping diagram and graph of $f$ yourself, or consider the GeoGebra figures and table below.

Martin Flashman, Dec. 26, 2013, Created with GeoGebra

Notice how the points on the graph are paired with the points on the mapping diagram.Given a point / number, $n$, on the source line, there is a unique arrow meeting the target line at the point / number, $f(n)$. which corresponds to the function's value for $n$.

Click on the point marked $n$ in the mapping diagram and use the up and down cursor controls on the keyboard to change the value of $n$ by $1$.

Click on a cell in column B of the spreadsheet to see how the number in that cell is calculated by the recursion formula.

You can use the slider for $r$ or the input box in the mapping diagram frame to change the value of $r$ between $-1$ and $1$.

You can enter a different value for $a$ in the appropriate input box in the mapping diagram frame.

Check the box to show more points and arrows to match the data in the spreadsheet.