Let a, b, c be the roots of x^3 - x - 1 = 0. Compute
(1 + a)^2/(1 - a) + (1 + b)^2/(1 - b) + (1 + c)^2/(1 - c).
About finding general an+bn+cn,an+bn+cn, there is an elementary aspect to this, although not necessarily what you want. You already know that a+b+c=0,a+b+c=0, and the other answer gives enough to find a2+b2+c2a2+b2+c2 and a3+b3+c3.a3+b3+c3. Call those x1,x2,x3,x1,x2,x3, then solve in perpetuity with
Two errors in the question,
So far, I get
Since the real root of x3−x−1x3−x−1 is slightly larger than 1,1, the numbers increase, are positive and so on. The two complex roots are smaller than 11 in modulus, so, for large n,n, we get xn≈Rnxn≈Rn where R≈1.3247R≈1.3247 is the real root. For example, R10≈16.643
For more in debt answer just ask (hopefully its debt enough through)