I would like to confirm that my reasoning for the answer to the following Summation is sound/sufficient:
Let $F_n$ be the Fibonacci number at $n$
$$\sum_{n=1}^{\infty}[\frac{1}{F_nF_{n+1}}-\frac{1}{F_{n+1}F_{n+2}}]$$ $$= \frac{1}{1\times1}-\frac{1}{1\times2}+\frac{1}{1\times2}-\frac{1}{2\times3}+\frac{1}{2\times3}-\frac{1}{3\times5}...$$ $$= 1$$
Is thi correct? Is my reasoning sufficient to reach the answer I did? Or am I missing something?
That's convincing, though some well placed parenthesis may help. You might more formally define the sequence of partial sums
$$ S_n = \sum_{j=1}^n \frac{1}{F_j F_{j+1}} - \frac{1}{F_{j+1} F_{j+2}}$$
and then show inductively that
$$ S_n = 1 - \frac{1}{F_{n+1} F_{n+2}}$$
and then note that $F_n \to \infty$ and conclude that $S_n \to 1$.
If you needed to be even more formal, you'd have to use a theorem to justify that last step. It all depends on your audience and their expectations, and how comfortable you feel with the topic at hand. For example, I feel that I have a respectable amount of experience with analysis so I'd accept your proof of that claim at face value. But I'm just starting my study of logic so a similarly difficult statement in logic, I might demand full rigor to accept.