Solve each problem. Refer to Exercise $85 .$ If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and the growth will slow. According to Verhulst's model, the number of bacteria $N_{j}$ at time $40(j-1)$ minutes can be determined by the sequence $$N_{j+1}=\left[\frac{2}{1+\left(N_{j} / K\right)}\right] N_{j}$$, where $K$ is a constant and $j \geq 1$. (a) If $N_{1}=230$ and $K=5000,$ make a table of $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Round values in the table to the nearest integer.
(b) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Use the window $[0,20]$ by $[0,6000]$.
(c) Describe the growth of these bacteria when there are limited nutrients.
(d) Make a conjecture as to why $K$ is called the saturation constant. Test your conjecture by changing the value of $K$ in the given formula.