All of the sequences that we've seen so far have defined each term by it's place in the sequence, e.g. $f(n) = n^2$ or $f(n) = n+1$. Another kind, called a recursive sequence, defines each term based on the values of the terms that came before it. This kind of sequence is going to let us define something called a difference equation that can be used to model a variety of different practical situations where we're interested in (or just have information about) how a situation changes over time.
Video Lectures
All of the sequences we've seen so far have defined each term using nothing but the sequence number. Another method, called a recurrence relation, defines each term in terms of the values of previous terms. (lecture slides)
In this lecture, we're going to look at examples of two recursively defined sequences, factorials and the Fibonacci sequence, that illustrate some of their interesting properties. (lecture slides)
We've already talked about mathematical series, i.e. sums based on sequences, but our definition was a little vague. In this lecture, we're going to look at a more formal definition based on recursion. (lecture slides)
Difference equations are a kind of recurrence relation that we can use to analyze the behavior of systems based on the way they change over time. (lecture slides)
In this lecture, we're going to take the difference equation we developed in the previous one and think about how we can write this as an equivalent, explicitly defined sequence, i.e. one whose definition only depends on the value of n. (lecture slides)
Difference equations are a great tool but it isn't always possible (or practical) to find an equivalent explicit formula that corresponds to one. When this happens, an alternative approach is to come up with what's called a numeric solution. (lecture slides)
