To develop the exponential equation that models the account's balance \( A \) after \( t \) years, we can start from the recursive formula provided:
\[ f(x) \approx f(x−1) \cdot 1.375 \text{ where } f(0) = 3500 \]
From the recursive relationship, we can deduce how the value of the account changes annually. This means that for each year \( t \), the value can be written as:
\[ f(t) = f(t-1) \cdot 1.375 \]
After \( t \) years, we can unfold this to:
\[ f(t) = 3500 \cdot (1.375)^t \]
This is the standard form of exponential growth where
- the initial amount (or principal) \( P = 3500 \)
- the growth factor per year \( r = 1.375 \)
Thus, the equation that models the account's balance after \( t \) years is:
\[ A = 3500 \cdot (1.375)^t \]
So, the correct equation is:
\[ A = 3,500 (1.375)^t \]
Where \( P = 3500 \) (rounded to two decimal places) and \( r = 1.375 \) (which is already rounded to three decimal places).
So the final answer is:
A = 3,500(1.375)^{t}