punctuation - Should there be a period after an equation?

I don't know if LaTeX is considered a definitive source for mathematics writing style (although it was developed for typesetting math equations), but this link and this one seem to indicate that, yes, a period would be inserted after the equation in the example

We used the equation

x + y = z.

This is the next sentence.

The Wikipedia Manual of Style (Mathematics), which cites several published mathematics style guides, offers this wisdom (emphasis added by me):

Just as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula. This equally applies to displayed formulae (that is, formulae that take up a line by themselves). Similarly, if the conventional punctuation rules would require a question mark, comma, semicolon, or other punctuation at that place, the formula must have that punctuation at the end.

The reference for this section is Higham, Nicholas J. (1998), Handbook of Writing for the Mathematical Sciences (second ed.), SIAM, ISBN 0-89871-420-6. The notation states that this is the style adopted by "many mathematics journals," so it is probably a safe choice.

Alternatively, you could use

We used Equation 1.

Equation 1.

x + y = z

This is the next sentence.