You deposit ​$40004000 in an account that pays 66​% interest compounded semiannually.  After 33 ​years, the interest rate is increased to 6.446.44​% compounded quarterly.  What will be the value of the account after a total of 66 ​years? 
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Part 1
The value of the account will be ​$ .
​(Round to the nearest dollar as​ needed.)

We can solve this problem using the compound interest formula:

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

Where:
- $A$ is the final amount.
- $P$ is the initial principal (deposit).
- $r$ is the annual interest rate (as a decimal).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the time the money is invested for (in years).

The process involves two stages:
1. **First 3 years:** The interest rate is 6% compounded semiannually.
2. **Next 3 years:** The interest rate increases to 6.44% compounded quarterly.

### Step 1: Calculate the value after the first 3 years
- $P = 4000$
- $r = 0.06$ (6% as a decimal)
- $n = 2$ (compounded semiannually)
- $t = 3$

The formula becomes:

\[
A_1 = 4000 \left(1 + \frac{0.06}{2}\right)^{2 \times 3}
\]

### Step 2: Calculate the value after the remaining 3 years
The value from Step 1 becomes the new principal, and the interest rate is increased to 6.44% compounded quarterly:
- New $P = A_1$
- $r = 0.0644$
- $n = 4$ (compounded quarterly)
- $t = 3$

The formula for the second period is:

\[
A_2 = A_1 \left(1 + \frac{0.0644}{4}\right)^{4 \times 3}
\]

Let me calculate the final value for you.The value of the account after 6 years will be approximately **$5,785**.

Would you like further clarification or details on any part of this calculation?

Here are five related questions for deeper exploration:
1. How does the compounding frequency affect the growth of an investment?
2. What happens to the total interest earned if the interest rate increases midway through an investment period?
3. How does semiannual compounding compare to quarterly compounding over the same period?
4. What is the difference between simple interest and compound interest in terms of long-term growth?
5. How does increasing the investment time affect the compound interest growth?

**Tip:** The more frequently interest is compounded, the more your investment grows, as you're earning interest on previously accumulated interest more often.

You deposit ​$4000 in an account that pays 6% interest compounded semiannually. After 3 years, the interest rate is increased to 6.44% compounded quarterly. What will be the value of the account after a total of 6 years?

Discover how to calculate the compound interest on a $4000 deposit with changing interest rates: 6% compounded semiannually for the first 3 years, and 6.44% compounded quarterly for the next 3 years. Follow the step-by-step solution to find the final account value after 6 years.

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