To model the given scenario of a tree's height using an arithmetic sequence, we need to set up the sequence based on the provided information.

Given:

• The tree's current height is 8 feet (initial height).
• The tree grows 3 feet per year (common difference in the arithmetic sequence).

The general form of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \cdot d$

Where:

• $a_n$ is the height of the tree after $n$ years.
• $a_1$ is the initial height of the tree (8 feet).
• $d$ is the common difference, which is the growth per year (3 feet).
• $n$ is the number of years.

Explicit Formula:

$a_n = 8 + (n - 1) \cdot 3$

Simplifying this:

$a_n = 8 + 3n - 3 = 3n + 5$

Thus, the explicit formula for the height of the tree after $n$ years is:

$a_n = 3n + 5$

This equation gives the tree's height after $n$ years, where $n$ starts from 1 (the current year).

Would you like further details or clarification? Here are a few related questions:

1. How would the formula change if the growth rate was different?
2. How can we calculate the height of the tree after 10 years using this model?
3. What is the significance of the common difference in an arithmetic sequence?
4. How does the arithmetic sequence differ from a geometric sequence in modeling growth?
5. How can we extend this model to account for different growth patterns over time?

Tip: Arithmetic sequences are useful for modeling linear growth, where the change between successive terms remains constant!