Is it ok to divide this formula by n-1?

Question

I was trying to find the general term of sequence 0,0,2,0,2,0,2......(for killing time). I found that the general term of sequence 0,0,4,0,8,0,12...... is $$(n-1)(1+(-1)^{n+1})$$ I thought If I added some term I could get the general term of sequence 0,0,2,0,2,0,2...... and it's n-1 $$\frac{(n-1)(1+(-1)^{n+1})}{n-1}$$ but I've heard that it is undefinable that dividing number by 0. then what is the value of the formula $$\frac{(n-1)(1+(-1)^{n+1})}{n-1}$$ when n=1? and why is it ok to divide some formula by n-1? It's embarassing that showing my lack of basic knowledges but I want to ask this question just because I'm curious.

Answer 1

I'm sure that soon enough you'll learn about functions and continuity, and every time you'll have a question like this, you'll reason by looking points "close to the discontinuity" and get a reasonable answer.

In this case, for every integer $n\geq 1$ the numerator and denominator cancel giving simply $(1+(-1)^n)$. You can define the only reasonable value for $n=0$ and that's perfectly OK. When you'll study continuity, you can think of the real function $(x-1)/(x-1)$ that of course should be 1 everywhere. You'll then define that its value is 1 also at $x=0$, basically because it's the reasonable thing to do. Beware though, expressions of the form 0/0 are indeterminate, that means they can take any value. As I said, normally we choose the reasonable value which depends on the context.