The value of i¹⁰⁰⁴+i¹⁰⁰⁶+i¹⁰⁰⁸+i¹⁰¹⁰+i¹⁰¹² i⁵¹⁰+i⁵⁰⁸+i⁵⁰⁶+i⁵⁰⁴+i⁵⁰² is

Step-by-step solution

The powers of i repeat in a cycle of 4 : i¹ = i , i² = -1 , i³ = -i , i⁴ = 1 . For the numerator, we have i¹⁰⁰⁴ + i¹⁰⁰⁶ + i¹⁰⁰⁸ + i¹⁰¹⁰ + i¹⁰¹² . Since 1004, 1008, 1012 are multiples of 4 , i¹⁰⁰⁴ = i¹⁰⁰⁸ = i¹⁰¹² = 1 . Since 1006 = 1004 + 2 , i¹⁰⁰⁶ = i² = -1 . Since 1010 = 1008 + 2 , i¹⁰¹⁰ = i² = -1 . Numerator = 1 + (-1) + 1 + (-1) + 1 = 1 . For the denominator, we have i⁵¹⁰ + i⁵⁰⁸ + i⁵⁰⁶ + i⁵⁰⁴ + i⁵⁰² . Since 508, 504 are multiples of 4 , i⁵⁰⁸ = i⁵⁰⁴ = 1 . Since 510 = 508 + 2 , i⁵¹⁰ = i² = -1 . Since 506 = 504 + 2