# Another poorly written word problem (Part 9)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

While the ball is on the 20-yard line, a defensive end is suddenly cursed so that he commits a penalty every down that causes the following:

a. The ball is moved half the distance to the goal line, and
b. The down is replayed.

Show that the ball will eventually travel the entire 20-yard distance to the goal.

Sigh. The textbook expects students to use the formula for an infinite geometric series

$\displaystyle \sum_{n=0}^\infty ar^n = \displaystyle \frac{a}{1-r}$

with $a = 10$ and $r = 0.5$. However, this series only works if there are an infinite number of terms, so that any finite partial sum will be less than 20. Therefore, saying that the ball “will eventually travel” all 20 yards is misleading, as this implies that this happens after a finite amount of time.

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