A linear algebra follow up to our probability project

My older son is learning linear algebra right in Strang’s Linear Algebra book. Over the weekend we did a fun project on problem number #1 in Frederick Mosteller’s probability challenge book and I wanted to show him a neat follow up involving linear algebra today.

The original project is here:

Going through problem #1 from Frederick Mosteller’s probability challenge book with kids

Today what I wanted to show my son is how you solve the recurrence relation for the terms we found in the original project. It is a fun linear algebra example.

We started by reminding ourselves what the terms were and then looking at the recurrence relation we found on the Internet Sequence Database:

By the end of the last video we had a cubic equation written down and in this video my son worked to find the roots of that cubic:

Now we looked at linear combinations of the powers of the roots that we found in the last video to see if we could find the general solution. The general solution involved a matrix equation:

We did not solve the matrix equation by hand, but rather went to Mathematica to save time.

Once we solved the equation we saw that we could find any term in the sequence that we wanted!

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