For n=0, we have
[tex]f(0)=3[/tex]
for n=1, we have
[tex]\begin{gathered} f(1)=2f(0) \\ f(1)=2\cdot3=6 \end{gathered}[/tex]
for n=2, we obtain
[tex]\begin{gathered} f(2)=2f(1) \\ f(2)=2\cdot6=12 \end{gathered}[/tex]
for n=3, we have
[tex]\begin{gathered} f(3)=2f(2) \\ f(3)=2\cdot12=24 \end{gathered}[/tex]
and so on. Then, the pattern is given by
[tex]\begin{gathered} f(n)=3\cdot2^n\text{ for n}\ge1 \\ \text{with f(0)=3} \end{gathered}[/tex]
Lets check our formula.
When n=1, we have
[tex]f(1)=3\cdot2=6[/tex]
when n=2, we have
[tex]f(2)=3\cdot2^2=3\cdot4=12[/tex]
when n=3, we have
[tex]f(3)=3\cdot2^3=3\cdot8=24[/tex]
