The function \(F(n)\) is defined for all positive integers as follows; \(F(1)=0\), and for all \(n \geq 2\),
\[\begin{align*}
&F(n) = F(n-1) + 2 \quad\quad\quad\quad \text{if $2$ divides $n$ but $3$ does not divide $n$};\\
&F(n) = F(n-1) + 3 \quad\quad\quad\quad \text{if $3$ divides $n$ but $2$ does not divide $n$};\\
&F(n) = F(n-1) + 4 \quad\quad\quad\quad \text{if $2$ and $3$ both divide $n$};\\
&F(n) = F(n-1) \quad\quad\quad\quad\quad\quad \text{if neither $2$ nor $3$ divides $n$}.
\end{align*}\]
The value of \(F(6000)\) equals
\[(a)\quad 9827,\quad(b)\quad 10121,\quad(c)\quad 11000,\quad(d)\quad 12300,\quad(e)\quad 12352.\]
Can we use the rules to evaluate \(F(n)\) for small \(n\)?
Can we find any pattern in the sequence given by the amounts we add on each time? Might this sequence be periodic?
