jmc

First, we can take out a factor of ${\omega }^{16}$: $${\omega }^{16}+{\omega }^{18}+{\omega }^{20}+\cdots +{\omega }^{54}={\omega }^{16}(1+{\omega }^2+{\omega }^4+\cdots +{\omega }^{38}).$$By the formula for a geometric series, $${\omega }^{16}(1+{\omega }^2+{\omega }^4+\cdots +{\omega }^{38})={\omega }^{16}\cdot \frac{1-{\omega }^{40}}{1-{\omega }^2}.$$(Note that this expression is valid, because $\omega \ne 1$ and $\omega \ne -1.$)

Since ${\omega }^7=1,$ $${\omega }^{16}\cdot \frac{1-{\omega }^{40}}{1-{\omega }^2}={\omega }^2\cdot \frac{1-{\omega }^5}{1-{\omega }^2}=\frac{{\omega }^2-{\omega }^7}{1-{\omega }^2}=\frac{{\omega }^2-1}{1-{\omega }^2}=\boxed{-1}.$$