Mimi, I am that student.

Yesterday I opened class with this question, taken from the University of Maryland High School Mathematics Competition (2005):

What is the value of    log (2/1) + log(3/2) + log(4/3) + … + log(99/98) + log (100/99) ?

Students who remembered their logarithm properties (from the last time we had class…) saw almost immediately that they could multiply the arguments together and, ta-da, have the numerator cancel with the denominator until it becomes just log (100) = 2.   But there were a few groups who didn’t remember this property.  Well, one group knew they could expand the arguments into log 2 – log 1 + log 3 – log 2… but didn’t follow this logic far enough to solve the problem.  That would’ve been cool, and it’s exactly why I ask students to share ideas that didn’t work– sometimes it’s a bad idea, other times it’s bad execution.

But anyway, I write because one group was kind of sitting and starting at the problem.  I heard one of them say, “You know the first one is going to be the biggest, and the others get smaller… and they are getting closer to 0.”   And I thought: cool!  They just turned this log properties problem into a sequences problem, even looking at what the terms are approaching!   I missed this structure of the terms because I had so quickly calculated the answer.   I had acted exactly like the students Mimi was talking about in her post (A critical mass problem)– I missed subtlety in my haste to get the answer.   It was a little unfortunate that this group couldn’t actually solve the problem with their realization.  It’s always fun when a cool insight actually helps you to solve the problem.  Maybe next time.

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