You are saving for retirement. To live​ comfortably, you decide you will need to save ​$1,600,000 by the time you are 66.  Today is your 25th ​birthday, and you​ decide, starting today and continuing on every birthday up to and including your 66th ​birthday, that you will put the same amount into a savings account. If the interest rate is 11 %​, how much must you set aside each year to make sure that you will have ​$1,600,000 in the account on your 66th ​birthday?
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Part 1
The amount to deposit each year is ​$
  
enter your response here. ​ (Round to the nearest​ cent.)

To solve this problem, we need to use the formula for the future value of an annuity, as you are making regular contributions and earning interest over time. The formula for the future value $ FV $ of an annuity is:

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

Where:
- $ FV $ is the future value, which is $1,600,000 in this case.
- $ P $ is the amount deposited each year (this is what we're solving for).
- $ r $ is the interest rate per period (in decimal form, so 11% becomes 0.11).
- $ n $ is the total number of deposits (from age 25 to 66, inclusive, so 66 - 25 + 1 = 42 years).

We will rearrange the formula to solve for $ P $:

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

Let's now calculate it.The amount you must set aside each year to ensure you have $1,600,000 by your 66th birthday is approximately **$2,225.38**.

Would you like more details or have any further questions?

Here are some related questions to expand on this topic:

1. How does increasing the interest rate affect the annual deposit?
2. What happens if you decide to retire at 60 instead of 66?
3. How much total interest will you have earned by age 66?
4. What would be the effect of starting the deposits at a later age, such as 30?
5. What if the interest is compounded monthly instead of annually?

**Tip:** Regular contributions, even in small amounts, can grow significantly over time due to compound interest, which is why starting early is beneficial!

You are saving for retirement. To live comfortably, you decide you will need to save $1,600,000 by the time you are 66. Today is your 25th birthday, and you decide, starting today and continuing on every birthday up to and including your 66th birthday, that you will put the same amount into a savings account. If the interest rate is 11%, how much must you set aside each year to make sure that you will have $1,600,000 in the account on your 66th birthday?

Find out how much you need to save annually starting at age 25 to reach $1,600,000 by your 66th birthday, with an 11% interest rate. This solution uses the future value of an annuity formula and demonstrates the importance of compound interest in retirement planning. Suitable for college-level finance or math learners.

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