Example 2
Completion requirements
Javier stocks shelves at a grocery store. His manager has asked him to
make a triangular prism display of pop boxes at the entrance. The top
row has two boxes, and each subsequent row has one additional box, as
shown in the diagram. There are \(15\) rows of boxes. How many boxes are
used to make the display?
Because the number of boxes in the last row, \(t_n \), is unknown, use the first version of the arithmetic series formula.
There are \(135\) boxes in the display.
\(\begin{align}
t_1 &= 2\\
n &= 15\\
d &= 1\\
S_n &= ?\\
\end{align}\)
t_1 &= 2\\
n &= 15\\
d &= 1\\
S_n &= ?\\
\end{align}\)
\[\begin{align}
S_n &= \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right] \\
S_{15} &= \frac{{15}}{2}\left[ {2\left( 2 \right) + \left( {15 - 1} \right)1} \right] \\
S_{15} &= 7.5\left[ {4 + 14} \right] \\
S_{15} &= 135 \\
\end{align}\]
S_n &= \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right] \\
S_{15} &= \frac{{15}}{2}\left[ {2\left( 2 \right) + \left( {15 - 1} \right)1} \right] \\
S_{15} &= 7.5\left[ {4 + 14} \right] \\
S_{15} &= 135 \\
\end{align}\]