Worst case analysis of $100n + 5$ for different $n$
Big-$O$ is the set of functions
$$O(f)=\{g\colon \exists C&gt;0, \exists N \in \mathbb{N}, \forall n &gt; N, g(n) \leqslant Cf(n)  \}$$
So we can write $100n+5 \leqslant  105\cdot n$, taking $C=105, N=1$, and we obtain
$$100n+5 \in O(n)$$