Question

to find the term \\(f(4)\\), repeatedly multiply the common ratio \\(r = 0.8\\).

the ball bounces about ______ meters on the ______ bounce.

Explanation:

Identify the sequence parameters

We are given a geometric sequence representing a bouncing ball.
The first term is \(f(1) = 500\) (as seen in the partially visible fraction \(r = \frac{\text{something}}{500}\)).
The common ratio is Common Ratio \(r = 0.8\).

Calculate the fourth term

To find the term \(f(4)\), we repeatedly multiply by the common ratio \(r\).
Using Geometric Sequence Terms:

$$ f(2) = 500 \times 0.8 = 400 $$

$$ f(3) = 400 \times 0.8 = 320 $$

$$ f(4) = 320 \times 0.8 = 256 $$

Interpret the result

The value \(f(4) = 256\) represents the height of the ball on the 4th bounce.
Thus, the ball bounces about \(256\) meters on the \(4\)-th bounce.
This completes the Geometric Sequences application.

Answer:

The ball bounces about <blank>256</blank> meters on the <blank>4</blank> bounce.