Consider the following function:

A(P,r,n,t)=P(1+rn)nt

where the variables have the following meaning:

A
= amount accumulated

P
= principal

r
= interest rate

n
= compoundings per period

t
= number of periods

Find the value of the function when  P=500,r=4%,n=12
, and  t=8
.

(Round your answer to the nearest whole number.)

The formula provided seems to have a minor typo. The correct formula for compound interest is:

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

Given the values:
- \( P = 500 \) (principal)
- \( r = 4\% = 0.04 \) (interest rate as a decimal)
- \( n = 12 \) (compoundings per period, monthly compounding)
- \( t = 8 \) (number of periods in years)

We can now plug these values into the formula:

\[
A = 500 \left(1 + \frac{0.04}{12}\right)^{12 \times 8}
\]

Let's calculate step-by-step:

1. **Calculate \(\frac{r}{n}\):**

\[
\frac{0.04}{12} = 0.003333\ldots
\]

2. **Calculate \(1 + \frac{r}{n}\):**

\[
1 + 0.003333\ldots = 1.003333\ldots
\]

3. **Calculate \(nt\):**

\[
12 \times 8 = 96
\]

4. **Calculate \((1 + \frac{r}{n})^{nt}\):**

\[
1.003333\ldots^{96}
\]

Let's compute the power:

\[
1.003333\ldots^{96} \approx 1.349353
\]

5. **Calculate the accumulated amount \(A\):**

\[
A = 500 \times 1.349353 \approx 674.6765
\]

Rounding to the nearest whole number:

\[
A \approx 675
\]

Thus, the accumulated amount after 8 years is **675**.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the accumulated amount change if the interest rate was 5% instead of 4%?
2. What would be the accumulated amount if the interest was compounded quarterly instead of monthly?
3. How does the value of \(A\) change if the number of periods \(t\) is doubled?
4. What would be the future value if the principal \(P\) was doubled?
5. How does continuous compounding affect the accumulated amount compared to monthly compounding?

**Tip:** To avoid confusion with interest rate percentages in formulas, always convert them into decimals by dividing by 100 before using them in calculations.

Learn how to calculate compound interest using the formula A(P, r, n, t) = P * (1 + r/n)^(nt). Given P = 500, r = 4%, n = 12 (monthly compounding), and t = 8 years, find the accumulated amount. Step-by-step solution included, rounded to the nearest whole number.

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