Question

a toy engineer is testing the bouncing of a ball that is made of a new type of material. she drops the ball from an initial height of 6,250 millimeters and records how far the ball falls before each bounce. the geometric series below represents the total distance the ball has fallen before its sixth bounce. which expression represents the total distance, in millimeters, that the ball has fallen before its sixth bounce? 6,250 + 5,000 + 4,000 + ... + 2,048

Explanation:

Step1: Identify the first - term and common ratio of the geometric series

The first - term $a = 6250$. To find the common ratio $r$, we note that $5000\div6250 = 0.8$, $4000\div5000 = 0.8$. So, $r = 0.8$.

Step2: Use the sum formula for a geometric series

The sum formula for a geometric series is $S_n=\frac{a(1 - r^n)}{1 - r}$, where $n$ is the number of terms. Here, we want to find the sum of the series before the sixth bounce, so $n = 5$ (because the first - term is the initial drop and then there are 4 more terms before the sixth bounce).
The sum $S_5=\frac{6250(1 - 0.8^5)}{1 - 0.8}$.

Answer:

$\frac{6250(1 - 0.8^5)}{1 - 0.8}$