Solve the problem.

Refer to Exercise 97. If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and growth will slow. According to Verhulst’s model, the number of bacteria N_j at time 40(j - 1) in minutes can be determined by the sequence

where K is a constant and j ≥ 1.

(a) If N₁ = 230 and K = 5000, make a table of N_j for j = 1, 2, 3, . . . , 20. Round values in the table to the nearest integer.

(b) Graph the sequence N_j for j = 1, 2, 3, . . . , 20. Use the window [0, 20] by [0, 6000].

(c) Describe the growth of these bacteria when there are limited nutrients.

(d) Make a conjecture about why K is called the saturation constant. Test the conjecture by changing the value of K in the given formula.

Exercise 97.

(a) Write N_j+1 in terms of N_j for j ≥ 1.

(b) Determine the number of bacteria after 2 hr if N₁ = 230.

(c) Graph the sequence N_j for j = 1, 2, 3, . . . , 7, where N₁ = 230. Use the window [0, 10] by [0, 15,000].

(d) Describe the growth of these bacteria when there are unlimited nutrients.

N, Nj+1 K