Reading a worked example in my notes on recurrence relations, I am told that the following recurrence relation R1
can be rewritten in the form below, R2:
This is done by replacing $n$ with ($n - 1$).
I feel I may be slightly confused on why there is a need to replace $n$ with ($n - 1$) within the fraction.
Rearranging R1, we get
Since any two consecutive terms in the sequence should defer by the same amount/multiple/ratio, when we re-write R1, why is there a need to change the fraction at all?. I.e. shouldn't
Thus, R2 should be re-written as
rather than what was shown in my text?
I feel like I'm confusing this concept with the idea of a "common ratio" for geometric sequences, which this is not since common ratio is a constant, but I still can't link this to why $n$ must be replaced with ($n - 1$) when re-writing R1 into R2.
Would appreciate explanations of what I am getting wrong and why my notes are right.
$ U_{n} = \dfrac{n+1}{n-1} U_{n-1} $ is true for $n = 2, 3, 4 \cdots$. Your equation is true for $n = 1, 2, 3 \cdots$ since you didn't substitute $n$ for anything. You have to substitute $n$ for $(n-1)$ in order to find an equation for $n = 2, 3, 4 \cdots$.