Answer
$S_{10}=1735$
Work Step by Step
RECALL:
(1) The sum of the first $n$ terms of an arithmetic sequence is given by the formula:
$S_n=\dfrac{n}{2}(a+a_n)$
where
$a$ = first term
$d$ = common difference
$a_n$ = $n^{th}$ term
(2) The $n^{th}$ term $a_n$ of an arithmetic sequence is given by the formula:
$a_n = a + (n-1)d$
where
$a$ = first term
$d$ = common difference
The given arithmetic sequence has:
$a=250
\\a_n = 97
\\d=233-250=-17$
The formula for the partial sum requires the values of $a$, $a_n$ and $n$.
However, only $a$ and $a_n$ are known at the moment.
Solve for $n$ using the formula for $a_n$ to obtain:
$\require{cancel}
a_n = a + (n-1)d
\\97 = 250+(n-1)(-17)
\\97-250 = (n-1)(-17)
\\-153=(n-1)(-17)
\dfrac{-153}{-17}=\dfrac{(n-1)(-17)}{-17}
\\9=n-1
\\9+1=n-1+1
\\10=n$
Now that it is known that $n=10$, the sum of the first 10 terms can be computed using the formula above.
$S_{10} = \dfrac{10}{2}(250+97)
\\S_{10}=5(347)
\\S_{10}=1735$
