If a + 1/a = -2, then a17 + 1/(a^102) equals to:

a + \(\frac{1}{a}\) = -2           [If a + \(\frac{1}{a}\) = -2, then a = -1 ⇒ a + \(\frac{1}{a}\) = 2 ⇒ a = 1 (always)]

So, (-1)¹⁷ + \(\frac{1}{(-1)^{102}}\) = -1 + 1 = 0