1-x-x-2-x-99-0-need-an-explanation-

Question Number 55992 by Mikael_Marshall last updated on 07/Mar/19

1 + x + x^{2} + . . . + x^{99} = 0

n e e d a n e x p l a n a t i o n .

Commented by mr W last updated on 07/Mar/19

i f x = - 1 :

1 + x + x^{2} + x^{3} + . . . + x^{98} + x^{99}

= (1 + x) + (x^{2} + x^{3}) + . . . + (x^{98} + x^{99})

= (1 - 1) + (1 - 1) + \dots + (1 - 1)

= 0

o r

i f x = - 1 :

1 + x + x^{2} + . . . + x^{99} = \frac{1 - x^{100}}{1 - x} = \frac{1 - 1}{2} = 0

Commented by maxmathsup by imad last updated on 07/Mar/19

i f x f r o m C (e) \Leftrightarrow \frac{1 - x^{100}}{1 - x} = 0 w i t h x \neq 1 \Rightarrow x^{100} = 1 = e^{i 2 k π} \Rightarrow

x_{k} = e^{\frac{i k π}{50}} a n d k \in [[1, 99]] a r e r o o t s o f t h i s e q u a t i o n .