Zoom International Math League: Andy, Are we there yet? (Weekly Brain Potion Discussion)

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Andy, Are we there yet? (Weekly Brain Potion Discussion)

To attend the CAG Conference this weekend, some staff from Areteem are headed to Palm Springs, California from Orange County, California. Being very organized they start listing out some of the cities they will need to drive through to get to Palm Springs. They notice that:

Riverside is roughly halfway there.
Roughly halfway from Riverside to Palm Springs is Beaumont.
Then roughly halfway from Beaumont to Palm Springs is Cabazon.

At this point they start to get worried. This process can go on forever! There will always be another distance that is halfway from any location to Palm Springs. How will they ever get to Palm Springs?

This of course is a version of the very famous Zeno's paradox. Try to explain why, in your own words, it is possible for Areteem staff to get to Palm Springs, even though their trip can be divided into a never-ending (infinite) set of steps. What type of sequence does this relate to?

For a greater challenge, consider another common riddle involving motion.

A snail starts at the top of a 1 foot tall fern. Being a very slow snail, each night it crawls 1 inch down the fern. Unfortunately, each day the fern grows 1 inch taller. Assuming the fern grows uniformly, will the snail ever reach the bottom? (For example, if the snail stops moving 40% down the fern, then the snail will be 40% down the fern if it grows uniformly.)

Please share any thoughts or questions you have below. We'll monitor the responses and give our thoughts as well! Have your own request, idea, or feedback for the Brain Potion series? Share with us in our Request and Idea Thread available here.