A recursive rule for a geometric sequence:
[tex]a_1\\\\a_n=r\cdot a_{n-1}[/tex]
---------------------------------------------------
[tex]a_1=3\\\\a_n=\dfrac{1}{2}a_{n-1}\to \boxed{r=\dfrac{1}{2}}[/tex]
--------------------------------------------------
Exciplit rule:
[tex]a_n=a_1r^{n-1}[/tex]
Substitute:
[tex]a_n=3\left(\dfrac{1}{2}\right)^{n-1}=3\cdot\left(\dfrac{1}{2}\right)^n\cdot\left(\dfrac{1}{2}\right)^{-1}=3\cdot\left(\dfrac{1}{2}\right)^n\cdot2\\\\\boxed{a_n=6\cdot\left(\dfrac{1}{2}\right)^n}[/tex]
