a) Every vampire bites one human every night starting the first night after the initiation. Write down a recurrent and an explicit formula for $v(n)$.

b) Every vampire bites one human every night starting the second night after the initiation. Write down a recurrent formula for $v(n)$. Prove that

\[v\left ( n \right )=\frac{\left ( \frac{1+\sqrt{5}}{2} \right )^{n+2}-\left ( \frac{1-\sqrt{5}}{2} \right )^{n+2}}{\sqrt{5}}\]

c) Every vampire bites $m$ humans on the $n^\text{th}$ night after the initiation. Write down a recurrent and an explicit formula for $v(n)$.