Certainly! Let's analyze the problem and derive the formula for the total value of the investment over [tex]$t$[/tex] years.
### Step-by-Step Solution:
1. Initial Investment and Interest Rate:
- Initial Investment (P): \[tex]$2000
- Annual Interest Rate (r): 2%, which can be written as 0.02 in decimal form.
2. Compound Interest Formula:
The formula for compound interest, where interest is compounded annually, can be expressed as:
\[
V(t) = P \left(1 + r\right)^t
\]
In this case:
- \( P = 2000 \)
- \( r = 0.02 \)
- \( t \) is the number of years
3. Substitute the Known Values:
\[
V(t) = 2000 \left(1 + 0.02\right)^t
\]
4. Simplify the Expression:
\[
V(t) = 2000 \left(1.02\right)^t
\]
5. Conclusion:
The function that represents the total value of the investment $[/tex]V(t)[tex]$, in dollars, after $[/tex]t[tex]$ years is:
\[
V(t) = 2000 \left(1.02\right)^t
\]
So, the final formula that gives the value of Shota's investment $[/tex]t$ years from now is:
[tex]\[
\boxed{V(t) = 2000 \left(1.02\right)^t}
\][/tex]
