Diana invested $1,700 at the beginning of every 6 months in an RRSP for 11 years. For the first 8 years it earned interest at a rate of 4.30% compounded semi-annually and for the next 3 years it earned interest at a rate of 5.50% compounded semi-annually.
a. Calculate the accumulated value of his investment at the end of the first 8 years.
b. Calculate the accumulated value of her investment at the end of 11 years.

To solve this problem, we need to break it into two parts, corresponding to the different interest rates over two periods.

### Part A: Accumulated Value at the End of 8 Years

Diana invests $1,700 at the beginning of every 6 months for 8 years at an interest rate of 4.30% compounded semi-annually.

1. **Interest rate per period**: Since the interest is compounded semi-annually, we divide the annual interest rate by 2:
   \[
   i = \frac{4.30\%}{2} = 0.0215
   \]

2. **Number of periods**: Since she invests every 6 months for 8 years, the number of periods is:
   \[
   n = 8 \times 2 = 16
   \]

3. **Future Value of Annuity Due Formula**: Since the investments are made at the beginning of each period, we use the future value of an annuity due formula:
   \[
   FV_{\text{due}} = P \left( \frac{(1 + i)^n - 1}{i} \right) (1 + i)
   \]
   where $ P $ is the payment per period.

4. **Substitute the values**:
   \[
   FV_{\text{due}} = 1700 \left( \frac{(1 + 0.0215)^{16} - 1}{0.0215} \right) (1 + 0.0215)
   \]

Let's calculate this step-by-step.

#### Step-by-Step Calculation:

\[
(1 + 0.0215)^{16} = 1.0215^{16} \approx 1.399808
\]

\[
\frac{1.399808 - 1}{0.0215} \approx \frac{0.399808}{0.0215} \approx 18.5892
\]

\[
FV_{\text{due}} = 1700 \times 18.5892 \times 1.0215 \approx 1700 \times 18.5892 \times 1.0215 \approx 32158.36
\]

So, the accumulated value at the end of 8 years is approximately $32,158.36.

### Part B: Accumulated Value at the End of 11 Years

For the next 3 years, the interest rate changes to 5.50% compounded semi-annually.

1. **New interest rate per period**: 
   \[
   i = \frac{5.50\%}{2} = 0.0275
   \]

2. **Number of periods**: 
   \[
   n = 3 \times 2 = 6
   \]

3. **Calculate the future value of the amount accumulated at the end of 8 years over the next 3 years**:
   \[
   FV = 32158.36 (1 + 0.0275)^6
   \]

4. **Calculate the future value of new annuity payments made during the next 3 years**:
   \[
   FV_{\text{due}} = 1700 \left( \frac{(1 + 0.0275)^6 - 1}{0.0275} \right) (1 + 0.0275)
   \]

#### Step-by-Step Calculation:

\[
(1 + 0.0275)^6 = 1.0275^6 \approx 1.171659
\]

\[
FV = 32158.36 \times 1.171659 \approx 37708.72
\]

Now, for the new annuity payments:

\[
\frac{1.171659 - 1}{0.0275} \approx \frac{0.171659}{0.0275} \approx 6.2436
\]

\[
FV_{\text{due}} = 1700 \times 6.2436 \times 1.0275 \approx 10934.77
\]

Finally, add the two future values to get the total accumulated value at the end of 11 years:

\[
FV_{\text{total}} = 37708.72 + 10934.77 \approx 48643.49
\]

So, the accumulated value of Diana's investment at the end of 11 years is approximately $48,643.49.

---

Would you like more details or have any questions?

Here are some related questions you might want to explore:
1. What is the present value of Diana's investment at the end of 11 years?
2. How would the accumulated value change if the interest rate was compounded quarterly instead?
3. What if Diana invested at the end of each 6-month period instead of the beginning?
4. How much interest did Diana earn in total over the 11 years?
5. How would changes in the payment amount affect the future value of her investments?

**Tip:** When working with compound interest, always make sure to align the compounding periods with the payment periods to accurately calculate the future value.

Discover how to calculate the accumulated value of an investment with changing interest rates over time. This problem involves investing $1,700 every 6 months for 11 years, with different interest rates for different periods. Learn to use the Future Value of Annuity Due and Future Value formulas to solve similar investment problems.

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