Progression and series doubt

Each term ai·ai+1 is either +1 or −1, so the sum is just n such terms. For the total to be 0, the number of +1 and −1 terms must be equal, so n must be even. Now if you multiply all terms, everything cancels nicely and you get a1²·a2²·…·an² = 1, which means the number of −1 terms must be even. Since the number of −1 terms is n/2, that must be even, so n is a multiple of 4.