The students may need some support to imagine what the recurrence relation is telling them. Many may not realise that the sequence of a's is in some sense already there, and the recurrence relation tells them how to calculate one term from others no matter what part of the sequence they are looking at.
My experience is that pictures often help students feel that mathematical things are real and so allows them to do new things with them. It also gives them something they can "hang their understanding on" - something that they can go back to until they understand in a symbolic way.
Here is a suggestion based on that idea that may help. (I'd appreciate if you try it telling me if it does help!)
Draw a picture like this to represent the sequence itself:

You can also draw a general part of the sequence centred at a$_n$:

The recurrence relation tells you how to calculate a$_{n+1}$ from a$_n$, and you can represent this as an arrow. I put a curl in the arrow to indicate that something possibly complex happens in between:

The key point students need to realise is that this applies no matter what $n$ is. To start off with, you can calculate a$_2$ using a$_1$, you can calculate a$_3$ using a$_2$, and so on. Slowly add the arrows from left to right to indicate how this happens:

Then you can say that it works anywhere in the sequence, even if you don't know what n is. You calculate a$_{n+1}$ using a$_n$, you can calculate a$_n$ using a$_{n-1}$ and so on. So now you can add extra arrows to the left and right of the picture with general n:

Now you can circle the bit you want, which is the bit that shows how to go from a$_{n-1}$ to a$_n$, to highlight that this is what you are looking for.