Answer
geometric, sum=$\displaystyle \frac{7}{4}(625\sqrt{5}-1)\approx 2443.95$
Work Step by Step
A geometric sequence has a common ratio
$a_{n}=ar^{n-1}$, with nth partial sum:
$S_{n}=a\displaystyle \cdot\frac{1-r^{n}}{1-r}$
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The sum has $9$ terms (for n=0,1,2,...$8$).
We can rewrite it as
$\displaystyle \sum_{k=1}^{9}7(5)^{(k-1)/2}=\sum_{k=1}^{9}7(5^{1/2})^{k-1}$
The terms make up a geometric sequence,
with $a=7,\ r=5^{1/2}$, and $n=9$
$a_{n}=7(5^{1/2})^{n-1}$
The sum equals
$S_{9}=a\displaystyle \cdot\frac{1-r^{n}}{1-r}=7\cdot\frac{1-5^{9/2}}{1-5}$
$=\displaystyle \frac{-7}{-4}(5^{9/2}-1)$
$=\displaystyle \frac{7}{4}(625\sqrt{5}-1)$
