Suppose initially there are 1200 trees in the forest. Let the number of trees at the start of each spring be
\[y_n\]
then the number in year \[n+1\]
be \[y_{n+1}\]
, then \[y_{n+1}=100+0.9y_n\]
- 100 saplings are planted ND 10% of trees are harvested so 90% are left.\[y_0=1200\]
\[y_1=100+0.9 y_0=100+0.9 \times 1200=1180\]
\[y_2=100+0.9 y_1=100+0.9 \times 1180=1162\]
\[y_3=100+0.9 y_2=100+0.9 \times 1162=1145.8\]
and so on. If the number of trees is to reach a limit
\[l\]
then \[y_{n+1} \rightarrow y_n \rightarrow l\]
as \[n \rightarrow \infty\]
.\[l=100+0.9l \rightarrow l-0.9l=100 \rightarrow 0.1l=100 \rightarrow l=\frac{100}{0.1}=1000\]
.In the long term the forest will contain 1000 trees and 100 trees will be harvested every year.