Answer
The length of the fifth swing is 16.4 inches (round to the nearest tenth of an inch).
The total length swung during the first five swings is 134.5 inches (round to the nearest tenth of an inch).
Work Step by Step
For the pendulum, since each successive swing is $\frac{4}{5}$ of the preceding swing, the situation can be modeled by a geometric sequence with $a_{1} = 40$ and $r = \frac{4}{5}$, and the general term to be $a_{n} = 40(\frac{4}{5})^{n-1}$.
For the fifth swing, the length is $a_{5}$ = $40(\frac{4}{5})^{5-1}$ = 16.4 inches (round to the nearest tenth of an inch).
And, the total length swung during the first five swings will be
$\sum\limits_{n=1}^5 a_{n}$ = $\sum\limits_{n=1}^5 40(\frac{4}{5})^{n-1}$ = 40 + 32 + 25.6 + 20.48 + 16.384 = 134.5 inches (round to the nearest tenth of an inch).
