While solving some questions based on factorials, I noticed a pattern and sat down to prove what I had been observing, for all $n$.
I solved some questions like this one:
Solve for $x$ - $$\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$$
On computing, I found out that the value of $x$ came out to be $8^2 = 64$.
On solving some other similar questions, I noticed that the value of $x$ came out to be the square of the denominator of $x$ itself. In other words, I can express it as follows -
Solve for $x$ - $$\frac{1}{n!} + \frac{1}{(n+1)!} = \frac{x}{(n+2)!}$$
So, the value of $x$ in such situations came out to $(n+2)^2$. Here is how I proved it -
$(n+1)! = (n+1).n!$
$(n+2)! = (n+2).(n+1).n!$
So, LHS becomes, $\frac{n+2}{(n+1).n!} = \frac{x}{(n+2).(n+1).n!}$
$\implies x = (n+2)^2$
Is my intuition correct? The pattern which I noticed led me to this result. Is this result valid?
Yes, this is correct.
Technically, I wouldn't call that intuition, I'd call it a proof.
Essentially, you are seeing $m(m-1) + m=m^2$, where $m=n+2$.