Question

Calculati: $1+(1+i)+(1+i)^2+...+(1+i) ^{2n-1}$, n∈N, i∈C.
Mersi

Answer 1

fie 1+i=x
$1+x+ x^{2} + x^{3} +.............+ x^{2n-1} =S/x \\ x+ x^{2} + x^{3} +.................... x^{2n} =Sx \\ \\ x^{2n} -1=S(x-1) \\ S= \frac{ x^{2n} -1}{x-1} = \f \frac{( 1+i)^{2n}-1 }{i}$