Answer
Arithmetic sequence
$S=5050\sqrt 5$
Work Step by Step
if it is arithmetic sequence then
$a_{n+1}+a_{n-1}=2a_{n}$
if it is geometric sequence then
$a_{n+1}\times a_{n-1}=a^{2}_{n}$
$a_{1}=\sqrt {5};a_{2}=2\sqrt {5};a_{3}=3\sqrt {5}\Rightarrow a_{1}+a_{3}=4\sqrt {5}=2\times \left( 2\sqrt {5}\right) =2a_{2}$
So this sequence is arithmetic and the sum:
$d=a_{2}-a_{1}=2\sqrt {5}-\sqrt {5}=\sqrt {5}$
$S_{n}=\dfrac {n}{2}\left( 2a+\left( n-1\right) d\right) =\dfrac {100}{2}\left( 2\times \sqrt {5}+\left( 100-1\right) \times \sqrt {5}\right) =101\sqrt {5}\times 50=5050\sqrt {5} $
