Question

why there's no minus sign in front of the Rydberg equation?

Answer 1

There would be if it was
\[-R(1/r _{2}^{2}-1/r _{1}^{2})\]
but they like to get rid of the "-" in front of R giving\[R(1/r_{1}^{2}-1/r _{2}^{2})\]
(note the switched subscripts on r)