A nice series problem for kids from Five Triangles

Back in 2013 we did a neat problem on Numberphile’s “Pebbling the Chessboard” video:

That video also reminded me of a neat “proof without words” that Patrick Honner had written about:

Our project is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

and Patrick Honner’s blog post is here:

Proof Without Words: Two Dimensional Geometric Series

Tonight I saw a neat tweet from Five Triangles that reminded me of the prior project:

I thought it would be a fun one to try out with my older son, though I didn’t quite know how to introduce the problem. I started with a slightly easier series as a trial: 1/2 + 2/4 + 3/8 + 4 / 16 + . . .

Since things seemed to go pretty well with the first problem I decided to go ahead and try out the series posted by Five Triangles:

So, a neat problem for kids building off of a the “simple” infinite series 1 + 1/2 + 1/4 + . . . . As our project from 2013 shows, the more complicated versions can have interesting geometric interpretations, but I’ll leave those for another time. Tonight it was just fun to see some neat arithmetic with infinite series.

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