Answer
The sequence is geometric.
The length of the fourth swing is $25.6$ inches.
Refer to the step-by-step part below for the explanation and solution.
Work Step by Step
RECALL:
A geometric sequence is a sequence that has a common ratio.
The next term of a geometric sequence can be found by multiplying the common ratio to the previous term.
Since the length of each successive arc is $80\%$ of the previous one, then the given sequence involves a common ratio of $80\%$ or $0.8$.
Thus, this sequence is geometric.
The given geometric sequence has $a_1 = 50$ inches and a common ratio $r=80\%=0.8$.
The $n^{th}$ term ($a_n$) of a geometric sequence is given by the formula $a_n=a_1 \cdot r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio.
Thus, using the formula above gives the 4th term:
$a_4 = a_1 \cdot r^{4-1}
\\a_4=50 \cdot 80\%^{3}
\\a_4=50 \cdot 0.8^{3}
\\a_4= 25.6$
Therefore. the length of the fourth swing is $25.6$ inches.
