f(1) = -8, simply is a way of saying, the 1st value in the arithmetic sequence is -8.
f(n) = f(n-1) - 3, is another way of saying, following values are just, the previous minus -3.
so in short, -3 is the common difference, since that is what we're "adding" to get the new value in the sequence.
[tex]\bf n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\[-0.5em] \hrulefill\\ d = -3\\ a_1 = -8 \end{cases} \\\\\\ a_n=-8+(n-1)(-3)\implies a_n=-8-3n+3\implies \stackrel{f(n)}{a_n}=\stackrel{\stackrel{explicit}{\downarrow }}{-3n-5}[/tex]
