Question

Suppose that $2000 is invested at a rate of 2.8%, compounded quarterly. Assuming that no withdrawals are made, find the total amount after 10 years.
Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

To calculate the total amount after 10 years for an investment compounded quarterly, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal amount (the initial investment),
- \( r \) is the annual interest rate (in decimal),
- \( n \) is the number of times that interest is compounded per year,
- \( t \) is the number of years the money is invested or borrowed.

Given:
- \( P = 2000 \),
- \( r = 0.028 \) (2.8% expressed as a decimal),
- \( n = 4 \) (since the interest is compounded quarterly),
- \( t = 10 \).

Substituting these values into the formula, we have:

\[
A = 2000 \left(1 + \frac{0.028}{4}\right)^{4 \times 10}
\]

First, calculate \( \frac{r}{n} \):

\[
\frac{0.028}{4} = 0.007
\]

Now substitute to find \( A \):

\[
A = 2000 \left(1 + 0.007\right)^{40}
\]
\[
A = 2000 \left(1.007\right)^{40}
\]

Next, we calculate \( (1.007)^{40} \):

\[
(1.007)^{40} \approx 1.348850
\]

Now, substituting this back into our equation for \( A \):

\[
A = 2000 \times 1.348850
\]
\[
A \approx 2697.70
\]

Finally, rounding to the nearest cent, the total amount after 10 years is

\[
\boxed{2697.70}
\]