# Complex Arithmetic/Examples/Sum of Powers of i from 0 to 7

## Example of Complex Arithmetic

$1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7 = 0$

## Proof

 $\ds$  $\ds 1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7$ $\ds$ $=$ $\ds \dfrac {1 - i^8} {1 - i}$ Sum of Geometric Sequence $\ds$ $=$ $\ds \dfrac {1 - \paren {i^4}^2} {1 - i}$ simplifying $\ds$ $=$ $\ds \dfrac {1 - 1^2} {1 - i}$ Powers of Imaginary Unit $\ds$ $=$ $\ds 0$

$\blacksquare$