Answer
$S_{50}=4,400$
Work Step by Step
RECALL:
(1) The sum of the first $n$ terms, $s_n$, of an arithmetic sequence can be found using the formula
$S_n = \frac{n}{2}(a_1+a_n)$
where
$a_1$=first term
$a_n$ = $n^{th}$ term
(2) The $n^{th}$ term, $a_n$, of an arithmetic sequence can be found using the formula
$a_n=a_1 + d(n-1)$
where
$d$=common difference
$a_1$ = first term
To find the sum of the first 50 terms of the sequence, we need to find the value of $a_{50}$. However, the value of $a_{50}$ can only be found if we know the value of $d$.
Solve for $d$ by subtracting the first term to the second term to obtain:
$d=-6-(-10)
\\d=-6+10
\\d=4$
The first term of the sequence is $-10$ so $a_1=-10$.
Substitute these values into the formula in (2) above to obtain;
$a_n= -10+4(n-1)$
Solve for the 50th term of the sequence to obtain:
$a_{50} = -10 + 4(50-1)
\\a_{50} = -10+ 4(49)
\\a_{50} = -10 + 196
\\a_{50} = 186$
Solve for the sum of the first 50 terms using the formula in (1) above to obtain:
$S_{50} = \frac{50}{2}(-10+186)
\\S_{50} = 25(176)
\\S_{50} = 4,400$
