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Prove that: (i) 1+i^(2)+i^(4)+i^(6)=0 ...

Prove that:
(i) `1+i^(2)+i^(4)+i^(6)=0`
(ii) `1+i^(10)+i^(100)+i^(1000)=2`
(iii) `i^(104)+i^(109)+i^(114)+i^(119)=0`
(iv) `6i^(54)+5i^(37)-2i^(11)+6i^(68)=7i`
(v)` (i^(592)+i^(590)+i^(588)+i^(586)+i^(584))/(i^(582)+i^(580)+i^(578)+i^(576)+i^(574))=-1`

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1+i^(10)+i^(110)+i^(1000)

the value of (i^(592)+i^(590)+i^(588)+i^(586)+i^(584))/(i^(582)+i^(580)+i^(578)+i^(576)+i^(574))+2

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Prove that: (i) 1+i^(10)+i^(100)-i^(1000)=0 (ii) i^(107)+i^(112)+i^(117)+i^(122)=0 (iii) (1+i^(14)+i^(18)+i^(22)) is real number.

Write the following in the form x+iy: (i) i+i^(2)+i^(3)+i^(4) (ii) i^(4)+i^(8)+i^(12)+i^(16) (iii) i+i^(5)+i^(9)+i^(13) (iv) i^(9)+i^(10)+i^(11)+i^(12) .

1+i^(2)+i^(4)+i^(6)+i^(8)++i^(20)