Question

$1,000 are deposited into an account with a 5% interest rate, compounded monthly (12 times per year). find the accumulated amount after 20 years. hint: $a = p(1 + \frac{r}{k})^{kt}$ round your answer to the nearest cent (hundredth).

Explanation:

Step1: Identify the values

We know that the principal amount \( P = 1000 \), the annual interest rate \( r = 5\%=0.05 \), the number of times compounded per year \( k = 12 \), and the number of years \( t = 20 \).

Step2: Substitute into the formula

Using the compound - interest formula \( A=P(1 +\frac{r}{k})^{kt} \), we substitute the values:
\( A = 1000\times(1+\frac{0.05}{12})^{12\times20} \)

First, calculate the exponent: \( 12\times20 = 240 \)
Then, calculate the value inside the parentheses: \( 1+\frac{0.05}{12}=\frac{12 + 0.05}{12}=\frac{12.05}{12}\approx1.004167 \)

Now, we need to calculate \( (1.004167)^{240} \). Using a calculator, \( (1.004167)^{240}\approx2.71286 \)

Then, multiply by the principal: \( A=1000\times2.71286 = 2712.86 \)

Answer:

\( \$2712.86 \)