Chapter 7.5, Problem 8MRPS

Solution

To state: The number of trees eventually exists on the farm after an extended period of time if initially it has 8000 trees. Each year 10 percent trees are harvested and 500 seedlings are planted.

After a long time there will be 5000 trees only.

Given information:

Initially a farm has 8000 trees. Each year 10 percent trees are harvested and 500 seedlings are planted.

Explanation:

Initially a farm has 8000 trees. That means the first term $a_1=8000$ .

Each year 10 percent trees are harvested and 500 seedlings are planted.

The recursive relation is:

  $a_1=8000,a_n=0.9a_{n-1}+500$

Find the second term:

  $\begin{array}{c}{a}_2=0.9a_{2-1}+500\\ =0.9a_1+500\\ =0.9\left(8000\right)+500\\ =7200+500\\ =7700\end{array}$

Find the third term:

  $\begin{array}{c}{a}_3=0.9a_{3-1}+500\\ =0.9a_2+500\\ =0.9\left(7700\right)+500\\ =6930+500\\ =7430\end{array}$

Find the fourth term:

  $\begin{array}{c}{a}_4=0.9a_{4-1}+500\\ =0.9\left(7430\right)+500\\ =6687+500\\ =7187\end{array}$

If it is continued like this then after long time the number of trees will become 5000.