Find the final amount in the following retirement​ account, in which the rate of return on the account and the regular contribution change over time.
​$1400 per quarter invested at 6.6​%, compounded​ quarterly, for 13 ​years; then ​$1700 per quarter invested at 7.4​%, compounded​ quarterly, for 17 years.

To solve this problem, we need to calculate the future value of an annuity under two different periods with different interest rates and contributions. Here’s how we approach it:

### Step 1: Calculate the future value of the first phase
- **First Phase**:
  - Contribution: $1,400 per quarter
  - Interest rate: 6.6% annual rate compounded quarterly
  - Number of years: 13 years
  - Number of quarters per year: 4
  - Total number of quarters: $ 13 \times 4 = 52 $

#### Formula for the future value of an annuity:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- $ P $ is the quarterly contribution
- $ r $ is the quarterly interest rate
- $ n $ is the total number of quarters

For the first phase:
- $ P = 1,400 $
- Annual interest rate = 6.6%, so quarterly interest rate $ r = \frac{6.6\%}{4} = 0.0165 $
- $ n = 52 $ quarters

Now, calculating the future value for the first phase:
\[
FV_1 = 1,400 \times \frac{(1 + 0.0165)^{52} - 1}{0.0165}
\]
\[
FV_1 = 1,400 \times \frac{(1.0165)^{52} - 1}{0.0165}
\]

### Step 2: Calculate the future value of the second phase
After 13 years, the amount invested changes to $1,700 per quarter at a 7.4% annual interest rate for 17 years. We treat this period as a second annuity starting from the future value of the first phase.

- **Second Phase**:
  - Contribution: $1,700 per quarter
  - Interest rate: 7.4% annual rate compounded quarterly
  - Number of years: 17 years
  - Number of quarters: $ 17 \times 4 = 68 $

For the second phase:
- $ P = 1,700 $
- Annual interest rate = 7.4%, so quarterly interest rate $ r = \frac{7.4\%}{4} = 0.0185 $
- $ n = 68 $ quarters

Using the same formula, calculate the future value for the second phase, but first we need to grow the amount from the first phase by 17 years using the new interest rate. The new future value for the second phase would be:

\[
FV_1 \text{ (carried forward)} = FV_1 \times (1.0185)^{68}
\]

Next, calculate the future value of the new contributions during this period:
\[
FV_2 = 1,700 \times \frac{(1.0185)^{68} - 1}{0.0185}
\]

### Step 3: Add the results together
Finally, the total future value after both phases will be the sum of the two future values:
\[
\text{Total FV} = FV_1 \times (1.0185)^{68} + FV_2
\]

Let me compute these values.The final amount in the retirement account after both phases is approximately **$623,765.73**.

Would you like more details on the calculation process or any specific parts explained further?

Here are five related questions to expand on this topic:
1. How would the total change if the interest rates were compounded monthly instead of quarterly?
2. What would happen to the final amount if contributions were increased by 10% after 13 years?
3. How does changing the investment duration from 17 years to 20 years affect the final amount?
4. What is the impact of compounding frequency on long-term investment growth?
5. Can you calculate the future value of a lump-sum investment under similar conditions?

**Tip**: When saving for retirement, consistent contributions over time and taking advantage of compounding can significantly increase your savings.

Find the final amount in the following retirement​ account, in which the rate of return on the account and the regular contribution change over time. $1400 per quarter invested at 6.6​%, compounded​ quarterly, for 13 ​years; then ​$1700 per quarter invested at 7.4​%, compounded​ quarterly, for 17 years.

Learn how to calculate the future value of a retirement account where contributions and interest rates change over time. This problem involves $1,400 per quarter at 6.6% compounded quarterly for 13 years, followed by $1,700 per quarter at 7.4% compounded quarterly for 17 years. Perfect for high school students or financial math enthusiasts.

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