we know that
the explicit formula in a geometric sequence is
[tex]a_n=a_1\cdot(r)^{(n-1)}[/tex]
In this problem
we have that
a3=2
a9=128
substitute in the expression above
[tex]\begin{gathered} a_3=a_1\cdot(r)^{(3-1)} \\ 2=a_1\cdot r^2 \end{gathered}[/tex]
and
[tex]\begin{gathered} a_9=a_1\cdot(r)^{(9-1)} \\ 128=a_1\cdot r^8 \end{gathered}[/tex]
Divide both expressions
128=a1*(r^8)
2=a1*(r^2)
----------------
128/2=r^8/r^2
64=r^6
r=2
Find out the value of a1
2=a1*(r^2)
2=a1*(2^2)------> 2=a1*4
a1=1/2
therefore
a1=1/2 and r=2
the explicit formula is equal to
[tex]a_n=\frac{1}{2}\cdot(2)^{(n-1)}[/tex]
the answer is option C
