Introduction to geometric series

Earlier in the week we looked at a problem that involved the series 1 + 2 + 2^2 + 2^3 + \ldots + 2^n. Each kid had a different and interesting approach to summing that series. For today’s project I wanted to review each of their approaches and then explore a few other simple geometric series to see if we could use the same ideas to add them up, too.

We started with a review of my older son’s approach to the original series. His approach was essentially mathematical induction, which was pretty cool since we’ve never talked about mathematical induction. Oddly, though, telling him that his approach had a name seems to have confused him a little bit, and it takes a little while for him to remember what he did. That little bit of confusion made me happy that we decided to revisit this topic today:


Next I had my younger son explain his approach to summing the series. His approach came from noticing a connection to binary numbers when he first saw the series. That connection to binary is a really clever way to think about this sum:


With that introduction and review of their prior ideas, we decided to see if we could apply those ideas to a new series: 1 + 3 + 3^2 + \ldots + 3^n. My older son tries his approach first – the connection isn’t obvious, but then we compare a few of the sums to the powers of 3 and make some progress.


Now my younger son takes a shot a the new series with his “trinary” ideas.


Finally, we wrap up the project by looking at the series 1 + 9 + 9^2 + \ldots + 9^n At this point they’ve seen each of the two prior sums in two different ways and they are able to see how the prior ideas apply to the new series. They even speculate about what a general formula would be!


So, a fun little exploration. I’m happy that the boys had a chance to review their prior solutions, especially since some of the ideas weren’t quite in the front of their mind anymore. Hopefully that review was helpful even if it wasn’t intended to be the main point of the project. By the end of the discussion today they seemed much more comfortable applying their ideas to a few new (though obviously similar) series. It is fun to show them how two ideas that seem pretty different help you work through these problems.


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