I need to construct a formula for the $n$th term of the sequence $$\{2, 2/3, 3/4, 4/5,\cdots \}$$
It's easy to come up with someting for every term after $2$, something like $\cfrac{n+1}{n+2}$ with lower limit $n=1$.
But after countless attempts I have not been able to come up with a formula that includes the very first term.
If you really need to, you can come up with some artificial way to express this. For instance:
$$u_n=\frac{n+1}{n+2}+\frac32\left\lfloor\frac{1}{n+1}\right\rfloor$$ where $\lfloor\cdot\rfloor$ is the floor function (so that $\lfloor\frac{1}{n+1}\rfloor$ is zero unless $n=0$).
But that is pointless, and just serves to obfuscate the fact that $n=0$ is a special case. Better is simply $$ u_n = \begin{cases} 2 & \text{if $n=0$} \\ \frac{n+1}{n+2} & \text{if $n\ne 0$} \end{cases} $$