2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Students should already know that a sequence is an ordered list of numbers with each term defined by its position in the list. Like how every year on your birthday you're a year older. Well, until you're 29 and then you never get older.

You can say that in year 1, you are year old. In year 2, you're 2 years old, and so on. At any point, you can ask the question, "After x years, how old will I be?" You can answer this question pretty easily since the term in the sequence is its position. In other words, if f(x) is your age and x is the number of years that have passed since your birth, f(x) = x. A pretty simple function.

...Or is it? Students should know that they can write explicit functions recursively, too. For instance, with every year that passes, your age increases by 1. We can interpret this as constantly adding 1 to the age you were before. In other words, we could just as easily write your age as f(x) = f(x – 1) + 1 starting with f(1) = 1. Make sure that your students know that it's important to have a basis for recursive functions. Otherwise, we'll end up chasing pavements.

Of course, students should know how to handle sequences (and therefore functions) that are a bit more complicated than that. Arithmetic functions that take the explicit form A(n) = A(1) + (n – 1)d have the recursive form A(n) = A(n – 1) + d and geometric functions with the form G(n) = G(1) × r^{n – 1} have the recursive form G(n) = G(n – 1) × r.

Once again, students should be sure to define A(1) and G(1). After students can convert between explicit and recursive functions, they should be able to construct either (or both) from a given context.