Question

A tree farm initially has 9000 trees. Each year, 10% of the trees are harvested and 800 seedlings are planted. a. Write a recursive rule for the number of trees on the tree farm at the beginning of the \(n\)th year. b. What happens to the number of trees after an extended period of time? cant copy image

Short answer

a) The recursive rule for the number of trees is given by \(T_n = 0.9*T_{n-1} + 800\). b) Over the long term, the number of trees on the farm stabilizes at 8000.

Step-by-step solution

Formulate the Recursive Rule

The recursive rule will start with the initial number of trees, which is 9000. Each year, the tree farm loses 10% of its trees and adds 800 new ones. So, for the nth year, the number of trees, denoted \(T_n\), can be estimated as \(T_n = 0.9*T_{n-1} + 800\), where \(T_{n-1}\) denotes the number of trees in the previous year.

Apply the Rule for the First Few Years

To see how the rule works, you can apply it for the first few years. For example, for the first year (\(n=1\)), the number of trees is \(T_1 = 0.9*9000 + 800 = 8900\). Similarly, for the second year (\(n=2\)), it is \(T_2 = 0.9*8900 + 800 = 8810\), and so on. You should observe that the number of trees decreases every year.

Determine Long-term Behavior

Over the long term, since 10% of the trees are harvested each year and 800 new trees are planted regardless of the current total, the farm will tend towards a steady state where the number of trees harvested equals the number planted each year. As the total reduces, the 10% harvested becomes less, and will eventually nearly equal to 800 trees. Solving the equation \(0.1*T_n = 800\) gives the steady state value as \(T_n = 8000\), which is the long-term number of trees at the farm.

Key concepts

Recursive Rule

When dealing with recursive sequences in algebra, a recursive rule is a mathematical formula that allows you to determine the next term in the sequence based on the previous term(s). This rule is particularly useful in modeling processes that involve repeated actions over time. In our tree farm example, the number of trees at the beginning of each year is determined by the changes from the previous year.

To formulate the recursive rule, we start with the initial condition that the tree farm has 9000 trees. Each year, 10% are harvested, so only 90% of the trees remain. Additionally, 800 new seedlings are planted. The recursive formula for the number of trees, \( T_n \), at the beginning of the \( n \)th year is:

• \( T_0 = 9000 \), the initial number of trees.
• \( T_n = 0.9 \times T_{n-1} + 800 \) for \( n \geq 1 \).

This formula makes it straightforward to calculate the number of trees for any future year by using the result from the previous year.

Long-term Behavior in Sequences

Understanding the long-term behavior of sequences can reveal stable patterns or steady-state values. A sequence that stabilizes or converges to a constant value can be highly informative about the dynamics of the system it represents.

In the context of the tree farm, as time goes on, the recursive formula's outputs reveal an interesting pattern: the number of trees approaches a steady state. Although the number of trees decreases initially due to the 10% harvest rate, the consistent planting of 800 seedlings each year counteracts this loss.

Eventually, the sequence reaches a point where the rate of tree harvest and planting balance each other, leading the number of trees to converge to a stable value. By solving the equation \( 0.1 \times T_n = 800 \), we discover that this steady-state value is 8000 trees. Thus, in the long run, the farm maintains a stable tree count of 8000.

Algebraic Modeling

Algebraic modeling involves creating equations and formulas that represent real-world processes and systems in a mathematical language. This modeling simplifies the understanding of complex scenarios and helps predict future behavior.

In practice, algebraic modeling helps answer fundamental questions about systems—like how changes in variables might affect outcomes. For the tree farm example, the recursive model allows farmers to predict how different harvesting or planting strategies would impact the overall number of trees over time. By adjusting variables within the model, such as the percentage harvested or the number of seedlings planted, they can simulate different scenarios and make informed decisions.

Algebraic modeling doesn’t just stop at prediction, it also assists in optimization. It allows decision-makers to establish parameters that lead to desired outcomes, such as maintaining a minimum tree count or maximizing yield, by examining the relationships between different elements of the system.