For Exercises 52–62, refer to the following. Call a sequence musical if repeatedly applying deflation to it eventually results in a single .
Question 20.91
61. (For students who have covered Chapter 19.)
Conclude from Exercise 56 that the sequence cannot be periodic or settle into a periodic repetition after a finite “burn-in” period. Thus, the sequence is nonperiodic. (Hint: φ is not a rational number; that is, it cannot be represented as a ratio / of whole numbers and .)
61.
If the sequence were periodic, the limiting ratio of s to s would be the same as the ratio in the repeating part, which would be a rational number, contrary to the result of Exercise 56.