Answer
The sequence is geometric.
$r=\dfrac{5}{4}$
$S_{50} \approx 350,319.62$
Work Step by Step
$\bf\text{RECALL:}$
$\bf\text{(1) Arithmetic Sequence }$
A sequence is arithmetic if there exists a common difference $d$ among consecutive terms.
$d=a_n-a_{n-1}$
The sum of the first $n$ terms of an arithmetic sequence is given by the formulas:
$S_n=\frac{n}{2}(a_1 +a_n)$
or
$S_n=\frac{n}{2}\left(2 a_1 + (n-1)d\right)$
$\bf\text{(2) Geometric Sequence }$
A sequence is geometric if there exists a common ratio $r$ among consecutive terms.
$r=\dfrac{a_n}{a_{n-1}}$
The sum of the first $n$ terms of a geometric sequence is given by the formula:
$S_{n}=a_1 \cdot \dfrac{1-r^n}{1-r}$
In the formulas listed above,
$d$ = common difference
$r$ = common ratio
$a_1$ = first term
$a_n$ = nth term
$n$ = number of terms
$\bf\text{List the first few terms of the sequence.}$
$\bf\text{Identify the sequence as arithmetic or geometric.}$
Substitute $1, 2, 3$ for $n$ to list the first three terms:
$a_1 =(\frac{5}{4})^1=\frac{5}{4}$
$a_2 = (\frac{5}{4})^2=\frac{25}{4}$
$a_3 = (\frac{5}{4})^3=\frac{125}{64}$
There is no common difference, so the sequence is not arithmetic.
Solve for the ratio of pairs of consecutive terms to obtain:
$\require{cancel}\dfrac{a_2}{a_1} = \dfrac{\frac{25}{16}}{\frac{5}{4}}=\dfrac{25}{16} \cdot \dfrac{5}{4} = \dfrac{\cancel{25}5}{\cancel{16}4} \cdot \dfrac{\cancel{5}}{\cancel{4}} = \dfrac{5}{4}$
$\require{cancel}\dfrac{a_3}{a_2} = \dfrac{\frac{125}{64}}{\frac{25}{16}}=\dfrac{125}{64} \cdot \dfrac{16}{25} = \dfrac{\cancel{125}5}{\cancel{64}4} \cdot \dfrac{\cancel{16}}{\cancel{25}}=\dfrac{5}{4}$
The sequence has a common ratio of $\dfrac{5}{4}$.
Thus, the sequence is geometric with $r=\frac{5}{4}$.
$\bf\text{Find the sum of the first 50 terms}:$
With $a_1=\dfrac{5}{4}$ and $r=\frac{5}{4}$, solve for the sum of the first 50 terms using the formula in (2) above to obtain:
$S_n = a_1 \cdot \dfrac{1-r^n}{1-r}
\\S_{50} = \dfrac{5}{4} \cdot \left(\dfrac{1-\cdot(\frac{5}{4})^{50}}{1-\frac{5}{4}}\right)
\\S_{50} \approx 350,319.62$
