Let A = 1 + i 1 - i 0 where i = - 1 . Then, the number of elements in the set n ∈ 1 , 2 , … . , 100 : A n = A is

Step-by-step solution

We have A = 1 + i 1 - i 0 A 2 = A · A = 1 + i 1 - i 0 1 + i 1 - i 0 = i 1 + i - i + 1 - i A 3 = A 2 · A = i 1 + i 1 - i - i 1 + i 1 - i 0 = 0 i 1 1 - i A 4 = A 3 · A = 0 i 1 1 - i 1 + i 1 - i 0 1 0 0 1 = I ∴       A 4 = I So, A 5 = A 4 · A = I · A = A A 6 = A 4 · A 2 = I · A 2 = A 2 and so on ∵ A 1 = A 5 = A 9 = . . . . . = A 97 = A Hence, possible values of n , such that A n = A = 1 , 5 , 9 , … , 97 Clearly, above sequence is in A.P. where a = 1 ,