Answer
$\frac{8}{7}$ cubic units
Work Step by Step
Volume of I stair = 1 cubic unit
Volume of II stair = $\left(\dfrac{1}{2}\right)^3$ cubic units
Volume of III stair = $\left(\dfrac{1}{4}\right)^3$ cubic units
Volume of IV stair = $\left(\dfrac{1}{8}\right)^3$ cubic units
and so on.
Assuming that the staircase has an infinite number of stairs, the total volume in cubic units is:
\begin{align*}
\sum_{n=0}^\infty \left(\frac{1}{8}\right)^n &= 1 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{2^2}\right)^3 + \left(\frac{1}{2^3}\right)^3 + \dots \\
&= 1 + \left(\frac{1}{8}\right)^1 + \left(\frac{1}{8}\right)^2 + \left(\frac{1}{8}\right)^3 + \dots \\
&= \sum_{n=0}^\infty \left(\frac{1}{8}\right)^n
\end{align*}
The series $\sum_{n=0}^\infty \left(\frac{1}{8}\right)^n$ is a geometric series $\sum_{n=0}^\infty ar^n$ with $a=1$ and $r=\frac{1}{8}$. Its value is:
\begin{align*}
\sum_{n=0}^\infty \left(\frac{1}{8}\right)^n &= \frac{1}{1-\frac{1}{8}} = \frac{8}{7}
\end{align*}
Thus, the total volume of the staircase is $\frac{8}{7}$ cubic units.
Result
The total volume of the staircase is $\frac{8}{7}$ cubic units.
