The least positive integer n , for which (1+i)^ n (1-i)^ n-2 is positive, is

Step-by-step solution

aligned (1+i)^ n (1- i )^ n -2 &= ( 1+ i 1- i )^ n (1- i )² &= (1+ i ²-2 i ) ( 1+ i 1- i )^ n &=(1-1-2 i ) ( 1+ i 1- i )^ n &=(-2 i ) ( 1+ i 1- i )^ n aligned aligned &=(-2 i) (1+i)(1+i) (1-i)(1+i) ^ n &=(-2 i) 2 i 2 ^ n ( i ²=-1 ) &=(-2 i)(i)^ n &=(-2) i^ n+1 aligned Put n =1 , we get aligned &=(-2) i²=(-2)(-1) &=2 positive integer. aligned Hence, the least positive integer value of n=1 for which the given expression have positive value.