Substitution in recurrence relations

Question

Let's say we study a reccurence relation such as $$a_{n+1}=a_{n}+n, n \geq 1$$ I find many students are having difficulties when you ask them to find $$a_{n}$$ in terms of $$a_{n-1}$$ where they just have to replace $n$ by $n-1$ in the above relation, and get $$a_{n}=a_{n-1}+n-1, n \geq 2$$. They don't get that $n$ is a just a variable, some placeholder for a value, and could be substituted with whatever we want, even with something also depending on $n$. They get it if we replace $n$ by some specific number such as 2 or 3 but they don't see replacing $n$ with another expression which also contains $n$ such as $n-1$ legit. What could be done to make them understand this better? Maybe use another letter, say replace $n$ by $k-1$? And then work only with $k$?

I'm interested in another ways of making them understand this substitution better.

Answer 1

One idea: Try using function notation instead $$a(n+1)=a(n)+n.$$ It's more familiar than subscripts. Then ask for $a(m)$ possibly with the hint that $m=n+1$.

Answer 2

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.