Teach 180: A Sequence of Logs (Day 117)

Rarely do I have time to do something extra in my classes and that is especially the case this year.  Even though we are about 2 weeks behind where we were last year, my class of PreCalculus students is pretty hardworking and inquisitive.  They are ahead of the other PreCalculus classes and it looks like I'll end up doing something fun after spring break to allow the other PreCalculus classes time catch up.

Today I was starting to purge my files at school and I ran across this gem of a question.  I have some thoughts about how I will teach it.  My students aren't familiar with the concepts of geometric sequences and arithmetic sequences, but they do understand exponential and linear functions.  I'll have plenty of time between now and March 19th (the day back from spring break) to design a lesson around this question.  One of my goals will be for the language of sequences (terms, initial term, geometric, arithmetic, common ratio and common difference) to develop naturally as a consequence of playing with this problem.  Any comments or thoughts on this welcomed!