Answer
The sequence is arithmetic.
$d=2$
$S_{50} = 2550$
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
$\bf\text{Identify the sequence as arithmetic or geometric.}$
Notice that the terms increase by $2$
This means that the sequence is arithmetic with $d=2$.
$\bf\text{Find the sum of the first 50 terms}:$
With $a_1=2$ and $d=2$, solve for the sum of the first 50 terms using the formula in (1) above to obtain:
$S_n = \dfrac{n}{2}\left(2a_1+(n-1)d)\right)
\\S_{50} = \dfrac{50}{2}\left(2\cdot 2 + 2(50-1)\right)
\\S_{50} = 25(4+2 \cdot 49)
\\S_{50} = 25(4+98)
\\S_{50}=25(102)
\\S_{50}=2550$
