Complex numbers in algebraic and polar form

Given complex numbers \( z_1 = -2 + \mathrm{i} \), \( z_2 = 1 - 4\mathrm{i} \) and \( z_3 = -3\mathrm{i} \), find \(\frac{z_1\cdot z_2}{3\cdot z_3}\).

Evaluate \( \mathrm{i}^5 - \mathrm{i}^{10} + \mathrm{i}^{15} - \mathrm{i}^{20} + \mathrm{i}^{25} - \mathrm{i}^{30} + \mathrm{i}^{35} - \mathrm{i}^{40} \).

Evaluate \( \mathrm{i}^{100}+\mathrm{i}^{99}+\mathrm{i}^{98}+\mathrm{i}^{97}+\dots+\mathrm{i}^4+\mathrm{i}^3+\mathrm{i}^2+\mathrm{i}\).

Given \(z_{1} = 4\left (\cos \frac{5} {3}\pi + \mathrm{i}\sin \frac{5} {3}\pi \right )\) and \(z_{2} = 2\left (\cos \frac{1} {6}\pi + \mathrm{i}\sin \frac{1} {6}\pi \right )\), evaluate \(\frac{z_{1}} {z_{2}} \).