To calculate the total amount after 4 years when $2000 is invested at a rate of 3.1% compounded quarterly, we will use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the annual interest rate (decimal),
- \( n \) is the number of times that interest is compounded per year,
- \( t \) is the number of years the money is invested.
Given:
- \( P = 2000 \)
- \( r = 3.1% = 0.031 \)
- \( n = 4 \) (since the interest is compounded quarterly)
- \( t = 4 \)
Now, let's substitute these values into the formula:
\[ A = 2000 \left(1 + \frac{0.031}{4}\right)^{4 \times 4} \]
Calculating \( \frac{r}{n} \):
\[ \frac{0.031}{4} = 0.00775 \]
Now, compute \( 1 + \frac{r}{n} \):
\[ 1 + 0.00775 = 1.00775 \]
Next, calculate \( nt \):
\[ nt = 4 \times 4 = 16 \]
Now substitute back into the formula:
\[ A = 2000 \left(1.00775\right)^{16} \]
Calculating \( (1.00775)^{16} \):
\[ (1.00775)^{16} \approx 1.130325186 \]
Now, we will find \( A \):
\[ A \approx 2000 \times 1.130325186 \approx 2260.650372 \]
Finally, rounding to the nearest cent:
\[ A \approx 2260.65 \]
Thus, the total amount after 4 years is
\[ \boxed{2260.65} \]
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