In the sequence a_1=2017 and . . .

Hey M7MBA,

When you have a function that you can't seem to simplify, and you're asked to find a crazy high value, the usual trick is to find a pattern.

Here, we have a_1, so let's find a_2:

$$a_2 = \frac{2017\ -\ 1}{2017} = \frac{2016}{2017}$$

No pattern yet, so we'll keep going:

$$a_3 = \frac{\frac{2016}{2017}\ -\ 1}{\frac{2016}{2017}} = \frac{\frac{-1}{2016}}{\frac{2016}{2017}} = \frac{-1}{2016}$$

And one more:

$$a_4 = \frac{\frac{-1}{2016}\ -\ 1}{\frac{-1}{2016}} = \frac{\frac{-2017}{2016}}{\frac{-1}{2016}} = 2017$$

So we see that a_4 = a_1. This means that the pattern will repeat again until a_7 = 2017, then a_10, and so on. So every time we add one to a multiple of three, we get 2017. Following that pattern on up, we see that a_2017=2017.