[tex]a_1=2,\ a_2=6,\ a_3=18,\ a_4=54,\ ...[/tex]
The Geometric Sequence Formula
Explicit formula: [tex]a_n=a_1r^{n-1}[/tex]
Calculate the common ratio (r):
[tex]r=\dfrac{a_{n+1}}{a_n}\to r=\dfrac{6}{2}=3[/tex]
Substitute:
[tex]a_n=2(3)^{n-1}=2\cdot3^n\cdot3^{-1}=2\cdot3^n\cdot\dfrac{1}{3}=\dfrac{2}{3}(3)^n[/tex]
Recursive formula:
[tex] \left\{\begin{array}{ccc}a_1=2\\a_{n+1}=3a_n\end{array}\right [/tex]
