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Workshop: Session 5 — Summary

Before we started talking about sequences, we looked another folding problem involving medial triangles.

We started talking about some fun sequences, and we found the next terms for most of them.  Hopefully no one lost too much sleep over the tricky ones!

We talked a bit about the method of finite differences for finding an n-th term formula for a given sequences.  For example, by looking at the difference of consecutive terms

we see that it takes two iterations to get to a constant difference.  This means that nth term of the original sequence is quadratic!  Naturally, we recognize the original sequence

as triangle numbers, and the formula for the nth triangle number is well known to be \frac {n(n+1)}{2}, a quadratic function.

We also looked at some ways to use the method of finite differences backwards, that is, to find the sum of sequence, i.e. a series.  For instance, if we wanted to find the sum of the triangle numbers, we could work up like this

and get the tetrahedral numbers (the sequence in red), whose formula is well-known to be cubic.

In the second half, we looked at several activities that use Excel to explore sequences and series.  In particular, two of the activities I showed you were taken directly from these videos on Excel and Pascal’s Triangle and Excel and Fibonacci Numbers.

Last, we talked a little about the Purple Comet contest, which is going on now!  Get a team together, sign them up, and have fun!

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