College Algebra, Chapter 1, 1.3, Section 1.3, Problem 60

Solve $\displaystyle S = \frac{n(n + 1)}{2}$ for $n$.


$
\begin{equation}
\begin{aligned}

S =& \frac{n(n + 1)}{2}
&& \text{Given}
\\
\\
S =& \frac{n^2 + n}{2}
&& \text{Apply Distributive Property}
\\
\\
2S =& n^2 + n
&& \text{Multiply both sides by 2}
\\
\\
2S + \frac{1}{4} =& n^2 + n + \frac{1}{4}
&& \text{Complete the square: add } \left( \frac{1}{2} \right)^2 = \frac{1}{4}
\\
\\
2S + \frac{1}{4} =& \left( n + \frac{1}{2} \right)^2
&& \text{Perfect square}
\\
\\
n + \frac{1}{2} =& \pm \sqrt{\frac{8S + 1}{4}}
&& \text{Take the square root, and simplify the equation}
\\
\\
n =& \frac{-1}{2} \pm \frac{\sqrt{8S + 1}}{2}
&& \text{Subtract } \frac{1}{2}
\\
\\
n =& \frac{-1 + \sqrt{8S + 1}}{2} \text{ and } n = \frac{-1 - \sqrt{8S + 1}}{2}
&& \text{Solve for } n

\end{aligned}
\end{equation}
$