Question 19.51

21. Shari Lynn Levine, as a high school student, published an article in The Fibonacci Quarterly that investigated the “Beta-nacci” sequence that results if, instead of bearing one pair of baby rabbits per month, mature rabbits bear two pairs every month, starting when they reach two months of age. Here, we ask you to rediscover some of Shari’s results.

  1. How many rabbits will there be each month for the first 12 months?
  2. What is the recursive rule for the nth Beta-nacci number ?
  3. For the terms of the sequence in part (a), calculate the ratios of successive terms. (Motivating hint: It’s not the golden ratio this time.)
  4. image Suppose that the ratio of successive terms approaches a number . We show how to find exactly. For very large , we have . Inserting these values into the recursive rule for the sequence and dividing by gives the equation . Solve this equation for (you can use the quadratic formula). Make a table of values of versus . From the evidence, can you suggest a formula for ?

21.

(a) 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1,365

(b)

(c) 1, 3, 1.667, 2.2, 1.909, 2.048, 1.977, 2.012, 1.994, 2.003, 1.999

(d) ; we discard the root.