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To find the balance of Sam's account at the beginning of year 4 with an annual compound interest rate of 5%, we use the formula for compound interest:

\[ A(n) = P \left(1 + \frac{r}{100}\right)^n \]

where:
- $ P $ is the principal amount (initial investment), which is $700.
- $ r $ is the annual interest rate (5%).
- $ n $ is the number of years the money is invested.

For $ n = 3 $ years (since we are looking for the balance at the beginning of year 4, we use the balance after 3 full years):

\[ A(3) = 700 \left(1 + \frac{5}{100}\right)^3 \]

Simplify the interest rate:

\[ A(3) = 700 \left(1 + 0.05\right)^3 \]
\[ A(3) = 700 \left(1.05\right)^3 \]

Now, calculate $ (1.05)^3 $:

\[ 1.05^3 \approx 1.157625 \]

Then, multiply by the principal amount:

\[ A(3) = 700 \times 1.157625 \]
\[ A(3) \approx 810.34 \]

Therefore, the correct explicit formula and balance are:

**B. $ A(n) = 700 \cdot (1 + 0.05)^{(n-1)}; \$810.34 $**

Would you like more details or have any questions? Here are five related questions for further exploration:

1. How does compound interest differ from simple interest in terms of calculation and growth?
2. What would the balance be after 10 years with the same initial investment and interest rate?
3. How can you derive the compound interest formula from first principles?
4. What is the effect of changing the interest rate to 6% on the balance after 4 years?
5. How would the balance change if Sam made annual deposits of $100 in addition to his initial investment?

**Tip:** When working with compound interest, remember that the power $ n $ represents the number of compounding periods. If interest compounds more frequently than annually, adjust the formula accordingly.

At the beginning of year 1, Sam invests $700 at an annual compound interest rate of 5%. He makes no deposits to or withdrawals from the account. Which explicit formula can be used to find the account’s balance at the beginning of year 4? What is the balance?

Learn how to calculate compound interest with an initial investment of $700 and a 5% annual interest rate. This detailed example explains the explicit formula and shows the balance after 3 years of growth, suitable for high school students.

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