Question

Prove that i107 + ill2 +122 +i153 = 0​

Answer 1

Step-by-step explanation:

LHS = i^107 + i^ll2 + i^122 + i^153

we know that i = √-1 , i² = -1 , i³ = -i , i^4 = 1

i^107 = i^104 x i³ = (i^4)^26 x i³

= (1)^26 x i³ = 1 x ( -i ) = -i

i^ll2 = (i^4)^28 = (1)^28 = 1

i^122 = i^120 x i² = (i^4)^30 x i²

= (1)^30 x i² = 1 x ( -1 ) = -1

i^153 = i^152 x i = (i^4)^38 x i

= (1)^38 x i = 1 x ( i ) = i

LHS = i^107 + i^ll2 + i^122 + i^153 = -i + 1 + (-1) + i

= -i + 1 + -1 + i = 0 = RHS

Hence proved..