20)
[tex]z = 1+i+i^2+...+i^n\\ \\ z = 1\cdot \dfrac{i^{n+1}-1}{i-1} \Rightarrow z = \dfrac{i^{n+1}-1}{i-1}\\ \\a)\quad n = 2010 \Rightarrow z = \dfrac{i^{2011}-1}{i-1} =\dfrac{(i^2)^{1005}\cdot i-1}{i-1} =\dfrac{(-1)^{1005}\cdot i-1}{i-1} = \\ \\=\dfrac{-i-1}{i-1} \Rightarrow |z| =\bigg|\dfrac{-i-1}{i-1}\bigg| =\\ \\=\dfrac{|-i-1|}{|i-1|} = \dfrac{\sqrt{(-1)^2+(-1)^2}}{\sqrt{1^2+(-1)^2}} = \dfrac{\sqrt 2}{\sqrt 2} = 1[/tex]
[tex]b)\quad z = 1+i+i^2+...+i^n \\ \\ 1+i+i^2+i^3= 1+i-1-i = 0\in \mathbb{R} \Rightarrow n \in \{3,7,11,15,...\}\\ \\ 1+i+i^2+i^3+i^4 = 1+i-1-i+1 = 1\in \mathbb{R}\Rightarrow n \in \{0,4,8,12,...\} \\ \\ \Rightarrow n \in \Big\{4k,\, 4k+3\Big\},\quad k\in \mathbb{N}[/tex]
