Answer
$S_{23}=310.5$
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:
$a=-3
\\a_n = 30
\\d=\frac{3}{2}-0=\frac{3}{2}$
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:
$\require{cancel}
a_n = a + (n-1)d
\\30 = -3+(n-1)\frac{3}{2}
\\30+3 = (n-1)(\frac{3}{2})
\\33=(n-1)(\frac{3}{2})
\\\frac{2}{3}(33)=(n-1)(\frac{3}{2}) \cdot \frac{2}{3}
\\\frac{2}{\cancel{3}}(\cancel{3}(11))=(n-1)(\frac{\cancel{3}}{\cancel{2}}) \cdot \frac{\cancel{2}}{\cancel{3}}\\
\\22=n-1
\\22+1=n-1+1
\\23=n$
Now that it is known that $n=23$, the sum of the first 23 terms can be computed using the formula above.
$S_{23} = \dfrac{23}{2}(-3+30)
\\S_{23}=\dfrac{23}{2}(27)
\\S_{23}=310.5$
