Answer
$r=5^c$
$a_5=5^{4c+1}$
The $n^{th}$ term of the geometric sequence is: $a_n=5 \cdot \left(5^c\right)^{n-1}$
Work Step by Step
RECALL:
(1) The common ratio of a geometric sequence is equal to the quotient of any two consecutive terms:
$r =\dfrac{a_n}{a_{n-1}}$
(2) The $n^{th}$ term of a geometric sequence is given by the formula:
$a_n = a\cdot r^{n-1}$
where
$a$ = first term
$r$ = common ratio
The sequence is said to be geometric.
Thus, we can proceed to solving for the common ratio:
$\require{cancel}
r=\dfrac{5^{c+1}}{5}$
Use the rule $\dfrac{a^m}{a^n} = a^{m-n}$ to obtain:
$r=5^{c+1-1}
\\r=5^c$
The fifth term can be found by multiplying the common ratio to the fourth term.
The fourth term is $5^{3c+1}$.
Thus, the fifth term is:
$a_5=5^{3c+1} \cdot 5^c$
Use the rule $a^m \cdot a^n = a^{m+n}$ to obtain:
$a_5=5^{3c+1+c}
\\a_5=5^{4c+1}$
With a first term of $5$ and a common ratio of $r=5^c$, the $n^{th}$ term of the geometric sequence is:
$a_n=a \cdot r^{n-1}
\\a_n=5 \cdot \left(5^c\right)^{n-1}$
