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.
​$626
 
per month invested at 6
​%, 
compounded​ monthly, for 6
 
​years; then ​$763
 
per month invested at 7
​%, 
compounded​ monthly, for 6
 
years.
Question content area bottom
Part 1
What is the amount in the account after 12
 
​years?
​$enter your response here
 
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

To calculate the final amount in the retirement account, we will break it down into two parts based on the different contribution and interest rates. The formula for the future value of an ordinary annuity, compounded monthly, is:

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

Where:
- $P$ is the monthly contribution
- $r$ is the annual interest rate (expressed as a decimal)
- $n$ is the number of compounding periods per year (monthly means $n = 12$)
- $t$ is the number of years

### Part 1: $626 per month at 6% for 6 years
- Monthly contribution $P = 626$
- Interest rate $r = 6\% = 0.06$
- Number of years $t = 6$
- Compounding periods per year $n = 12$

We calculate the future value for the first 6 years using the formula:

\[
FV_1 = 626 \times \frac{(1 + 0.06/12)^{12 \times 6} - 1}{0.06/12}
\]

### Part 2: $763 per month at 7% for 6 years
For the next 6 years, we invest $763 per month at 7% interest. First, we need to calculate the total from the first 6 years with interest for the next 6 years, and then add the second 6 years' contributions.

1. The amount from Part 1 will grow at 7% interest for 6 more years. We will calculate this by treating it as a lump sum investment.
   
   The formula for the future value of a lump sum is:

   \[
   FV_{\text{lump}} = PV \times (1 + r/n)^{nt}
   \]

   where $PV$ is the value from Part 1.

2. Then, we will calculate the future value of $763 per month invested at 7% for 6 years:

   \[
   FV_2 = 763 \times \frac{(1 + 0.07/12)^{12 \times 6} - 1}{0.07/12}
   \]

Finally, we'll add the result from Part 1 after it has grown, and the future value of the second set of contributions.

Let me calculate the final amount now.The total amount in the retirement account after 12 years is approximately **$150,255**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does increasing the interest rate affect the final retirement amount?
2. What would happen if the contributions were made quarterly instead of monthly?
3. How much would the total be if both the interest rate and contributions remained constant throughout the 12 years?
4. How does changing the time period affect the future value of an annuity?
5. What would the outcome be if the contributions started later, say after 2 years?

**Tip**: Regularly increasing your contributions, even slightly, can have a large impact on your retirement savings due to the power of compound interest.

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. $626 per month invested at 6%, compounded monthly, for 6 years; then $763 per month invested at 7%, compounded monthly, for 6 years. What is the amount in the account after 12 years?

Learn how to calculate the final amount in a retirement account where contributions and interest rates change over time. In this example, we start with $626 per month at 6% interest for 6 years, followed by $763 per month at 7% for another 6 years. Suitable for students and individuals interested in financial mathematics.

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