Question

Suppose that $2000 is invested at a rate of 3.1%, compounded quarterly. Assuming that no withdrawals are made, find the total amount after 4 years.
Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

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

\[
\boxed{2260.65}
\]