Let z = 1 + a i , be a complex number, a > 0 , such that z 3 is a real number. Then, the sum 1 + z + z 2 + … . + z 11 is equal to :

Question

Let z = 1 + a i , be a complex number, a &#62; 0 , such that z 3 is a real number. Then, the sum 1 + z + z 2 + &#8230; . + z 11 is equal to :

Aicharya · Accepted Answer

z = 1 + a i Cubing both sides, z 3 = 1 - a 2 + 2 a i + 1 - a 2 a i - 2 a 2 z 3 = 1 - 3 a 2 + ( 3 a - a 3 ) i &#8757; z 3 is a real number. a 3 - a 2 = 0 &#160; &#8658; &#160; &#160; a = &#160; 3 a &#62; 0 1 + z + z 2 &#8230; . . z 11 = z 12 - 1 z - 1 = 1 + &#160; 3 i 12 - 1 1 + &#160; 3 i - 1 = 1 + &#160; 3 i 12 - 1 3 i 1 + &#160; 3 i 12 = 2 12 1 2 + 3 2 i 12 = 2 12 cos &#8289; &#960; 3 + i sin &#8289; &#960; 3 12 = 2 12 &#160; cos &#8289; 4 &#960; + i sin &#8289; 4 &#960; = 2 12 = 2 12 - 1 3 i = - 1365 3 i

Let z = 1 + a i , be a complex number, a > 0 , such that z 3 is a real number. Then, the sum 1 + z + z 2…

Solution

Let z = 1 + a i , be a complex number, a &#62; 0 , such that z 3 is a real number. Then, the sum 1 + z + z 2…

Solution

Let z = 1 + a i , be a complex number, a > 0 , such that z 3 is a real number. Then, the sum 1 + z + z 2…