We have the following sequence: 3,6,12, ...
We can note that the common ratio between consecutive numbers is
[tex]\frac{6}{3}=\frac{12}{6}=2[/tex]This implies that the given sequence is a geometric one. Generally, to check whether a given sequence is geometric, we simply checks whether successive entries in the sequence all have the same ratio.
The formula for the n-th term of the sequence is given by
[tex]a_n=a\cdot r^{n-1}[/tex]where a is the first term of the sequence and r is the common ratio. From our last results, we have that
[tex]\begin{gathered} a=3 \\ r=2 \end{gathered}[/tex]By substituting these result into the above formula, the last term is given as
[tex]a_n=3\cdot2^{n-1}[/tex]Therefore the last term a_n, is given by:
[tex]a_n=3\cdot2^{n-1}\text{ for }n=1,2,3,\ldots[/tex]