College Algebra, Chapter 9, 9.2, Section 9.2, Problem 50

Find the partial sum of an arithmetic sequence $\displaystyle -3 + \left( \frac{-3}{2} \right) + 0 + \frac{3}{2} + 3 +.......+30$

Using both formulas to solve for $n$, since $\displaystyle d = \frac{3}{2}$.


$
\begin{equation}
\begin{aligned}

\frac{n}{2} \left[ 2a + (n - 1) d \right] =& n \left( \frac{a + a_n}{2} \right)
&&
\\
\\
2a + (n - 1)d =& a+ a_n
&& \text{Multiply both sides by } \frac{2}{n}
\\
\\
(n - 1)d =& a_n - a
&& \text{Combine like terms}
\\
\\
n - 1 =& \frac{a_n - a}{d}
&& \text{Divide by } d
\\
\\
n =& \frac{a_n - a}{d} + 1
&& \text{Add } 1
\\
\\
n =& \frac{30 - (-3)}{\displaystyle \frac{3}{2}} + 1
&&
\\
\\
n =& 23
&&

\end{aligned}
\end{equation}
$


Now we solve the partial sum,


$
\begin{equation}
\begin{aligned}

S_{23} =& 23 \left( \frac{-3 + 30}{2} \right)
\\
\\
S_{23} =& \frac{621}{2}

\end{aligned}
\end{equation}
$