Marlene deposits 67 000 NOK in the bank at the start of every year.
The annual interest rate is 2.1%. What is the value of her payments
at the end of the ninth year?

To calculate the value of Marlene's payments at the end of the ninth year, considering that she deposits 67,000 NOK at the start of each year with an annual interest rate of 2.1%, we'll use the future value of an annuity formula:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( FV \) is the future value of the annuity (the value of her payments at the end of the ninth year).
- \( P \) is the annual payment (67,000 NOK).
- \( r \) is the annual interest rate (2.1% or 0.021).
- \( n \) is the number of periods (9 years).

### Step-by-Step Calculation:

1. **Annual Interest Rate \( r \)**:  
   \[
   r = \frac{2.1}{100} = 0.021
   \]

2. **Future Value of the Annuity \( FV \)**:
   \[
   FV = 67,000 \times \frac{(1 + 0.021)^9 - 1}{0.021}
   \]
   Let's calculate it step by step.

3. **Calculate \((1 + r)^n\)**:
   \[
   (1 + 0.021)^9 = 1.021^9 \approx 1.199
   \]

4. **Calculate the future value \( FV \)**:
   \[
   FV = 67,000 \times \frac{1.199 - 1}{0.021} \approx 67,000 \times \frac{0.199}{0.021} \approx 67,000 \times 9.476
   \]
   \[
   FV \approx 67,000 \times 9.476 \approx 635,000.2 \text{ NOK}
   \]

So, the value of Marlene's payments at the end of the ninth year is approximately **635,000.2 NOK**.

Would you like further details or have any questions?

---

1. How does the future value change with a different interest rate?
2. What would happen if the interest were compounded more frequently?
3. How would the result change if she made monthly deposits instead of yearly?
4. What is the impact of extending the investment period by a few more years?
5. How does the formula differ for calculating the present value of an annuity?

**Tip:** Regular deposits combined with compound interest can significantly increase the future value of savings.

Learn how to calculate the future value of an annuity with compound interest using a detailed step-by-step solution. This problem covers financial concepts, including compound interest and the future value of an annuity formula. Suitable for advanced high school students.

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