In English, ordinals are usually created by appending a “-th” or “-eth” sound to the end. In mathematics, this extends to variables, hence the word “$n$th”. But the numbers 1, 2, 3, and 5 are irregular—first, second, third, and fifth instead of *oneth, *twoth, *threeth, and *fiveth, however those would be pronounced. 5 is the odd one out here because the difference is only apparent if you spell it out (“fifth”) or pronounce it aloud (/fɪfθ/) rather than using the digit (“5th”) as is more common in mathematics. This applies not only to these irregular numbers themselves, but to all numbers that end in those same digits: twenty-first, twenty-second, hundred-and-first, etc.
What does this mean for mathematical writing when you have an expression that ends one of these numbers? What's the ordinal for $n+1$? $n+1$st? $n+1$th? $(n+1)$th? $n$th$+1$? $n$th$+1$st? In informal contexts I have seen all of these examples.
Alterantively, are these all proscribed, and if so, what alternative is there? Do you introduce a new variable on the spot (e.g. “$m$th where $m=n+1$”)?
(Not sure how to tag this question, by the way; edit me if I’m wrong there.)
The standard is to append the -th ending to all integer-denoting expressions other than those ending in the digits $1$, $2$, or $3$. If the expression contains typographic spacing (as e.g. surrounds the plus sign in $n+1$), then the expression must be enclosed in parentheses or brackets, to prevent the -th from visually attaching specifically to the elements of the expression following the last space. Thus we might have the $[m+\frac12n(n+1)]$th term in a sequence, but not the $m+\frac12n(n+1)$th term. The only use of -st, -nd, or -rd will then occur in such “words” as $4371$st, $22$nd, or $50$,$003$rd.
An exceptional case arises if we use a thin space rather than an unspaced comma to parse the groups of three digits in a large number. For example, $29\,403\,002$nd would be more natural than $(29\,403\,002)$th.