Answer
$a_1=1, a_n=(2)^{-\frac{n-1}{2}}$
Work Step by Step
Step 1. Identify the given quantities: initial height $8ft$, bounce back to 1/2 of the previous height, the first complete bounce takes $1s$, and each subsequent complete bounce requires $\frac{1}{\sqrt 2}=(2)^{-\frac{1}{2}}$ as long as the preceding complete bounce.
Step 2. We can set up a geometric sequence $a_n$ to represent the time interval from the instant the ball touches the ground as: $a_1=1, a_2=(2)^{-\frac{1}{2}}, a_3=(2)^{-1}, ... a_n=(2)^{-\frac{n-1}{2}}$
