Answer
Arithmetic;
$5050\sqrt 5$
Work Step by Step
Let's note by $\{a_n\}$ the given sequence. The sum of its first $k$ terms is:
$$S_k=\sqrt 5+2\sqrt 5+3\sqrt 5+\dots+100\sqrt 5.$$
The terms of the sequence are:
$$\begin{align*}
a_1&=\sqrt 5\\
a_2&=2\sqrt 5\\
a_3&=3\sqrt 5\\
&\dots\\
a_k&=100\sqrt 5.
\end{align*}$$
We notice that the common difference of consecutive terms is $2\sqrt 5-\sqrt 5=3\sqrt 5-2\sqrt 5=\sqrt 5$, therefore constant, so the sequence is arithmetic with the elements:
$$\begin{cases}
a_1=\sqrt 5\\
d=\sqrt 5.
\end{cases}$$
Before calculating $S_k$ we must find $k$ (the number of terms in the sum):
$$\begin{align*}
a_k&=a_1+(k-1)d\\
100\sqrt 5&=\sqrt 5+(k-1)\sqrt 5\\
100&=1+k-1\\
k&=100.
\end{align*}$$
Calculate the partial sum $S_{100}$ using the formula:
$$S_n=\dfrac{n(a_1+a_n)}{2}.$$
For $n=100$ we have:
$$S_{100}=\dfrac{100(\sqrt 5+100\sqrt 5)}{2}=5050\sqrt 5.$$