e^{\frac{i\pi}{7}}-e^{\frac{i2\pi}{7}}+e^{\frac{i3\pi}{7}}-...+(-1)^{n-1}e^{\frac{in\pi}{7}}=e^{\frac{i\pi}{7}}\cdot \frac{(-e^{\frac{i\pi}{7}})^n-1}{e^{\frac{i\pi}{7}}-1}=