Question

The number of individuals in a population is modeled by the following discrete-difference equation and initial condition: xn+1=0.90xn+20,x0=100 In this model, xn represents the population at year n. Find a solution to the problem.

Answer 1

To find the population of individuals at year n given the discrete-difference equation xn+1=0.90xn+20 with initial condition x0=100, one must iterate the equation starting from x0. This results in a sequence that describes the population for each year.

Step-by-step explanation:

The student is provided with a discrete-difference equation to model the population of individuals over time given by xn+1=0.90xn+20, with the initial condition x0=100. To find a solution to the problem means to determine the population at year n. This can be done by iterating the equation starting with x0.

At year 1 (n=1), the population would be x1 = 0.90(100) + 20 = 90 + 20 = 110. For year 2 (n=2), we substitute x1 into the equation: x2 = 0.90(110) + 20, and so on. The population for each subsequent year is calculated by taking 90% of the previous year's population and then adding 20 individuals.

The sequence generated from this equation can be observed to approach a limiting value as n gets larger, which can be found by setting xn+1 = xn and solving for x, indicating the long-term behavior of the population size.