 Hi, and welcome back to a second video dealing with linear homogeneous recurrence relations. In this video, we're going to introduce the concepts of the characteristic equation and characteristic roots for linear homogeneous recurrence relations of orders one or two. So let's think back to an example we saw in class where the recurrence relation was given by a sub n is equal to two times a sub n minus one. This is a linear homogeneous first order recurrence relation. We saw that a closed form solution to this recurrence relation was a of n equals two to the nth power. Similar reasoning should convince you that if I have any similar recurrence relation where r here is just a constant, just a single number, then a sub, then a of n equals r to the nth power will be a solution to that one. So building from this, what happens when we have a linear second order homogeneous recurrence relation like this one? You should first check to see that just peeling off the numbers and making exponential functions out of them, which worked for the starter example, does not work here. These two things are not closed formula solutions to the recurrence relation. However, maybe there is some number out there, let's call it r, such that a of n equals r to the nth is a solution. So we're going to make this guess and then see what would have to follow. Well, if r to the nth power is a solution, then if I plug that formula in for the terms of the recurrence relation, I should get this formula. Now, all those formulas have a lot of factors of r, so if we divide off both sides of this equation by r to the n minus 2, that will give me this formula, which now no longer has any n's in it. If I get all the terms onto one side of the equation, I see an equation that would not be out of place in a high school algebra class. This equation which we get from inserting the initial guess of r to the nth power into the recurrence relation and then simplifying is called the characteristic equation for the recurrence relation. Here are a couple more examples. The first one is a simple first order recurrence relation. We first substitute the formula in for the recurrence relation, simplify, and then get this equal to zero. So the simple linear equation r minus 3 equals zero is the characteristic equation for this recurrence relation. The second example here is a linear homogeneous second order recurrence relation. Substitute the r to the nth power formula in for the recurrence relation, then simplify, and then get this equal to zero. The quadratic you see here is the characteristic equation for the second order recurrence relation. One last concept for this video is the characteristic roots for a recurrence relation. These are just the algebraic roots of the characteristic equation. That is what you would get by actually solving the characteristic equation once you have set it up. In the first example from earlier, the characteristic equation was this, and so when we factor it, we see that the characteristic roots are one and two. In that simple first order example from the last slide, we have this characteristic equation, and so obviously the characteristic root, there's only one of them this time is r equals 3. Again, there's only one of these because this was a first order recurrence relation. Generally speaking, a second order recurrence relation should have two roots, although in some cases those roots may be a single root that's repeated or two imaginary roots. The last example here gave us this characteristic equation, and by factoring the characteristic equation, we see that the roots are three and two. Finally, note that not all quadratic equations can be solved easily by factoring as we know, and so sometimes we may have to use the quadratic formula to get at these characteristic roots. Thanks for watching.