a−bca+1b+1c−1a−1b−1c+1+(−1)na+1a−1ab+1b−1−bc−1c+1c=a−bca+1b+1c−1a−1b−1c+1+(−1)na+1b+1c−1a−1b−1c+1a−bc=a−bca+1b+1c−1a−1b−1c+1+(−1)n+1a+1b+1c−1a−bca−1b−1c+1=a−bca+1b+1c−1a−1b−1c+1+(−1)n+2a−bca+1b+1c−1a−1b−1c+1 This is equal to zero only if n+2 is odd i.e. n is odd integer.