Răspuns:
a) i·i²·i³·.........·i²⁰¹⁹ = - 1
b) 1+i+i²+i³+ ......+i²⁰¹⁹ =0
Explicație pas cu pas:
a) i·i²·i³·.........·i²⁰¹⁹ = ?
i·i²·i³·.........·i²⁰¹⁹ = i¹⁺²⁺³⁺⁻⁻⁻⁻⁻⁺²⁰¹⁹ = i⁽¹⁺²⁰¹⁹⁾ˣ²⁰¹⁹⁽² =
= i²⁰¹⁹ˣ¹⁰¹⁰ = i²⁰³⁹¹⁹⁰
i = i ; i² = -1 ; i³ = -i ; i⁴ = 1
2039190 = 509797 x 4 + 2 =>
i²⁰³⁹¹⁹⁰ = i² = -1 =>
i·i²·i³·.........·i²⁰¹⁹ = - 1
b) 1+i+i²+i³+ ......+i²⁰¹⁹ = ?
i = i ; i² = -1 ; i³ = -i ; i⁴ = 1
Numarul de termeni ai sumei = 2019+1 = 2020
2020 = 505 x 4
1+i+i²+i³+ ......+i²⁰¹⁹ = 1+i+(-1)+(-i) +1+i+(-1)+(-i)+.......+1+i+(-1)+(-i) =
= (1+i-1+-i )·505 = 0·505 = 0
