Question

Rezolvați ecuația : 1+x^1+x^2+...+x^99=0

Answer 1

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

$\frac{x^{100}- 1}{x-1}=0~<=>~x^{100}-1=0$

$x^{100}-1=0~=>x^{100}=1~=>x=+-1$

Daca x diferit de 1, deci x=-1

Daca x=-1
1+x^1+x^2+...+x^99
=1+(-1)^1+(-1)^2+...+(-1)^99
=1+(-1)+1+(-1)+...+(-1)=
(se reduc toti 1 si -1)
=0