To create an exponential equation from the given recursive formula, we can start by analyzing the equation you've provided:
The recursive equation is:
\[ f(x) \approx f(x-1) \cdot 1.025 \]
This indicates that the balance grows by a factor of \(1.025\) each year. The initial balance given is \(f(0) = 2310\).
We can express \(f(x)\) in terms of an exponential function. The recursive relationship can be interpreted as:
\[ f(x) = f(0) \cdot (1.025)^x \]
Substituting \(f(0) = 2310\), we get:
\[ f(x) = 2310 \cdot (1.025)^x \]
Now to express this in terms of \(A(t)\) (the amount of money in the account after \(t\) years), we can use the notation:
\[ A(t) = 2310 \cdot (1.025)^t \]
Now we can calculate the amount of money in the account after \(t = 2\) years:
\[ A(2) = 2310 \cdot (1.025)^2 \]
Calculating \( (1.025)^2 \):
\[ (1.025)^2 \approx 1.050625 \]
Now substituting this back into the equation for \(A(2)\):
\[ A(2) = 2310 \cdot 1.050625 \approx 2431.44 \]
Rounding to the nearest cent:
\[ A(2) \approx 2431.44 \]
Therefore, the amount of money in the account after 2 years is approximately $2431.44.