Question

1. lacey is standing on top of one of the tallest buildings in her town. she stands at the edge of the roof and drops a bouncy ball from a height of 120 feet. each bounce is 20% shorter than the previous bounce. which explicit formula describes this situation?

\\(a_n = 120(0.20)^{n-1}\\)
\\(a_n = 20(0.80)^{n-1}\\)
\\(a_n = 120(0.80)^{n-1}\\)
\\(a_n = 120(80)^{n-1}\\)

Explanation:

⚡ Using what you learned: Geometric Sequences · Recursive and Explicit Formulas

Step 1: Identify the initial value

The first term \( a_1 \) represents the height of the first bounce. Looking at the options, the initial value is:

$$ a_1 = 120 $$

Step 2: Determine the common ratio

Each bounce is \( 20\% \) shorter than the previous bounce. This means each bounce retains \( 100\% - 20\% = 80\% \) of the previous height.

Convert the percentage to a decimal to find the common ratio \( r \):

$$ r = 0.80 $$

Step 3: Write the explicit formula

The explicit formula for a geometric sequence is:

$$ a_n = a_1(r)^{n-1} $$

Substitute \( a_1 = 120 \) and \( r = 0.80 \):

$$ a_n = 120(0.80)^{n-1} $$

Answer:

\( a_n = 120(0.80)^{n-1} \)