Answer
The first five terms are:
$a_1 = 0
\\a_2 = \ln{5}
\\a_3 = 2\ln{5}
\\a_4=3\ln{5}
\\a_5=4\ln{5}$
The sequence is not geometric.
Work Step by Step
To find the first five terms, substitute 1, 2, 3, 4, and 5 to the given formula.
Use the rules
(1) $\ln{(a^n)} = n\cdot \ln{a}$
(2) $\ln{1} = 0$
$a_1 = \ln{(5^{1-1})}=\ln{(5^0)}=\ln{1}=0
\\a_2 = \ln{(5^{2-1})} = \ln{(5^1)} =\ln{5}
\\a_3 = \ln{(5^{3-1})} = \ln{(5^2)} =2\ln{5}
\\a_4=\ln{(5^{4-1})} = \ln{(5^3)} =3\ln{5}
\\a_5=\ln{(5^{5-1})} = \ln{(5^4)} =4\ln{5}$
RECALL:
A sequence is geometric if there is a common ratio among consecutive terms.
Note that consecutive terms do not have a common ratio.
Thus, the sequence is not geometric.