Chapter 8.1, Problem 130E

a.

To determine

To write: a recursive sequence that gives the population $p_n$ of trout in the lake in terms of the year n, with n = 0 corresponding to 2020.

Answer to Problem 130E

The recursive sequence that gives the population $p_n$ of trout in the lake in terms of the year n is: $P_n=0.75P_{n-1}+500.$

Explanation of Solution

Given information: A landlocked lake has been selected to be stocked in the year 2020

with 5500 trout and to be restocked each year thereafter with 500 trout. Each year the fish population declines 25% due to harvesting as well as natural causes.

Calculation:

The recursive sequence that gives the population $p_n$ of trout in the lake in terms of the year n is:

  $\begin{array}{l}{P}_0=5500.\\ P_n=0.75P_{n-1}+500.\end{array}$

b.

To determine

To find: the numbers of trout in the lake for n = 1, 2, 3 and 4 and interpret these values in the context of the situation.

Answer to Problem 130E

In 2021, there will be $P_1=4625\text{fish}.$

In 2022, there will be $P_2=3969\text{fish}.$

In 2023, there will be $P_3=3477\text{fish}.$

In 2024, there will be $P_4=3107\text{fish}.$

Explanation of Solution

Given information: $P_n=0.75P_{n-1}+500.$

Calculation:

In 2021, there will be $P_1=0.75P_0+500=0.75\times 5500+500=4625\text{fish}.$

In 2022, there will be $P_2=0.75P_1+500=0.75\times 4625+500=3969\text{fish}.$

In 2023, there will be $P_3=0.75P_2+500=0.75\times 3969+500=3477\text{fish}.$

In 2024, there will be $P_4=0.75P_3+500=0.75\times 3477+500=3107\text{fish}.$

c.

To determine

To find: the number of trout as time passes infinitely using a graphing utility and explains.

Answer to Problem 130E

2000 trout.

Explanation of Solution

Given information: $P_n=0.75P_{n-1}+500.$

Calculation:

  ${\displaystyle \sum _{p=0}^{\infty }0.75P_{n-1}+500=2000\text{trout}.}$ As time passes, the population of trout decreases at a decreasing rate. Because the population is growing smaller and still declines 25 %, each time 25% is taken from a smaller number there is a smaller decline in the number of trout.