We often use subscript notation when the function is a recursive function, where we rely on knowing previous values in order to compute the next value, as in the example below. Example: Suppose \(a_{n+1} = 3a_{n} + 2\) and \(a_0 = 3.\)
One way to read this definition is: ``To get the next value of \(a\)--the value at \(n+1,\) we take 3 times the previous value of \(a\)--the value at \(n\)--and then add two. In this example, we know the value of \(a\) at \(n=0,\) because we are given that \(a_0 = 3.\) Then we can find the next value of \(a\)--the value at \(n = 1\)--as follows:
It looks like we actually need to know the value of \(a_3\text{...}\)and it turns out that in order to know the value of \(a_3\) we actually need to know the value of \(a_2\text{...}\)which requires that we know the value of \(a_1\)--which we already found. So here goes:
The bottom line with recursive functions is that if we know \(a_0,\) we can find \(a_1,\) and then we can find \(a_2,\) and so on. But if we want to know \(a_{17},\) for example, we need to know ALL of the previous \(a\)-values \(a_0, a_1, a_2, \ldots , a_{16}\text{.}\)
You may recall some of this from studying sequences and series in calculus. In calculus we sometimes looked for a pattern so we could avoid having to compute all previous values. That’s a great skill, and it might even work for the examples below, but we won’t generally be able to do that in differential equations.
