Answer:
[tex]a_{n} = 4*3^{n-1}[/tex]
Step-by-step explanation:
Step-by-step explanation:
Geometric sequence concepts:
The nth term of a geometric sequence is given by the following equation.
[tex]a_{n+1} = ra_{n}[/tex]
In which r is the common ratio.
This can be expanded for the nth term in the following way:
[tex]a_{n} = a_{1}r^{n-1}[/tex]
Or also
[tex]a_{n} = a_{m}r^{n-m}[/tex]
In which [tex]a_{1}[/tex] is the first term
In this question:
[tex]a_{2} = 12, a_{5} = 324[/tex]
Then
Finding the common ratio:
[tex]a_{n} = a_{m}r^{n-m}[/tex]
[tex]a_{5} = a_{2}r^{5-2}[/tex]
[tex]12r^{3} = 324[/tex]
[tex]r^{3} = \frac{324}{12}[/tex]
[tex]r = \sqrt[3]{\frac{324}{12}}[/tex]
[tex]r = 3[/tex]
Finding the first term:
[tex]a_{n} = a_{1}r^{n-1}[/tex]
[tex]a_{2} = a_{1}*r[/tex]
[tex]a_{1} = \frac{a_{2}}{r} = \frac{12}{3} = 4[/tex]
Explicit rule:
[tex]a_{n} = a_{1}r^{n-1}[/tex]
[tex]a_{n} = 4*3^{n-1}[/tex]
