jmc

Let $\omega =e^{2\pi i/13}.$ Then from the formula for a geometric sequence, $$\begin{array}{rl}{e}^{2\pi i/13}+e^{4\pi i/13}+e^{6\pi i/13}+\cdots +e^{24\pi i/13}& =\omega +{\omega }^2+{\omega }^3+\cdots +{\omega }^{12}\\ & =\omega (1+\omega +{\omega }^2+\cdots +{\omega }^{11})\\ & =\omega \cdot \frac{1-{\omega }^{12}}{1-\omega }\\ & =\frac{\omega -{\omega }^{13}}{1-\omega }.\end{array}$$Since ${\omega }^{13}=(e^{2\pi i/13}{)}^{13}=e^{2\pi i}=1,$ $$\frac{\omega -{\omega }^{13}}{1-\omega }=\frac{\omega -1}{1-\omega }=\boxed{-1}.$$