Botanists at UMass Boston recently discovered a new local flower species that they named the Boston powerflower. It has beautiful, blue blossoms, and each plant lives for only one summer. During that time, each plant produces 11 seeds. Five of these seeds will turn into plants in the following year, and the remaining six seeds will turn into flowers the year after. As the name powerflower suggests, these seeds always turn into flower plants; there is no failure ever. During the year of this discovery (let us call it year one), the scientists found two powerflowers on the UMass campus, and in the following year, there were already five of them. Required:
a. Devise a recurrence relation for the number of flower plants fn in year n, and specify the initial conditions. Use the above recurrence relation to predict the number of plants on the UMass campus in years 3, 4, 5, and 6.
b. Find an explicit formula for computing the number of plants in any given year without requiring iteration, i. e., repeated application of a formula. You should (but do not have to) check the correctness of your formulas using some of the results you obtained in (b). Remember that our initial conditions are for a1 and a2.