a Calculate the second and third term, then find a recursive pattern.
B
b Use a spreadsheet.
A
aRecursive Rule: a_1=9000, a_n=0.9a_(n-1)+800 for n>1
B
b The number stabilizes at 8000 trees.
Practice makes perfect
a Let a_n denote the number of trees in the farm at the beginning of the n^(th) year. The farm initially has 9000 trees. This tells us that a_1=9000. Next, we will find a recursive equation for the sequence a_n.
Recursive Equation
At the beginning of the (n-1)^(th) year the number of trees is a_(n-1). During this year, 10 % of trees are harvested. Therefore, 90 %=0.9 of the old trees remain. Furthermore, 800 new trees are planted.
ccccc
c Trees at the start of the $n^(th)$ year
&
=0.9 *
&
c Trees from previous year
&
+
&
New trees
⇓ & & ⇓ & & ⇓
a_n & =0.9 * & a_(n-1) & + & 800
Therefore, we can write a recursive rule: a_1=9000, a_n=0.9a_(n-1)+800 for n>1.
b We are asked to find what happens to the number of trees after an extended period of time. To do this we will use the spreadsheet. Let's enter 9000 in cell A1 and write =Round((0.9)*(A1)+800,2) in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cells A3 through A100, and paste. Note that the numbers stabilize around cell A94.
After cell A93 we will keep getting the same value, 8000.05. This tells us that the number of trees stabilizes around 8000 trees.
