For Exercises 23 and 24, refer to the following. We have seen that the golden ratio is a positive root of the quadratic polynomial . We can generalize this polynomial to for and consider the positive roots of those polynomials as generalized means—the metallic means family, as they are sometimes known. In particular, for , and 5, we have, respectively, the silver, bronze, copper, and nickel means. It is surely surprising that these numbers arise both in connection with quasicrystals (investigated in Chapter 20) and in analyzing the behavior of some dynamical systems (investigated in Chapter 23) as the systems evolve into chaotic behavior.

Question 19.54

24. Just as the golden mean arises as the limiting ratio of consecutive terms of the Fibonacci sequence, each of the metallic means arises as the limiting ratio of consecutive terms of generalized Fibonacci sequences. A generalized Fibonacci sequence can be defined by

where and are positive integers. The Fibonacci sequence itself is the case .

  1. Try various small values of and and determine which means they lead to.
  2. image Divide the equation for by . Assume that and both tend toward the same number as gets large, replace those quantities by , and simplify the resulting equation. What must be the value of ?
  3. What happens to the sequence and to the mean if we allow one or both of and to be negative integers?