**How much would you need to deposit in an account now in order to have $3000 in the account in 15 years? Assume the account earns 5% interest compounded daily.**

$ 

To determine how much you need to deposit now in order to have $3000 in an account in 15 years, with an interest rate of 5% compounded daily, we can use the formula for compound interest:

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

where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for in years.

Given:
- $ A = 3000 $
- $ r = 0.05 $
- $ n = 365 $ (since interest is compounded daily)
- $ t = 15 $

We need to find $ P $. Rearranging the formula to solve for $ P $:

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

Substituting the given values into the formula:

\[
P = \frac{3000}{\left(1 + \frac{0.05}{365}\right)^{365 \times 15}}
\]

Now, let's calculate this step-by-step.

1. Calculate $ \frac{0.05}{365} $:

\[
\frac{0.05}{365} \approx 0.000136986
\]

2. Add 1 to this value:

\[
1 + 0.000136986 \approx 1.000136986
\]

3. Raise this to the power of $ 365 \times 15 $:

\[
(1.000136986)^{5475}
\]

4. Calculate the power:

\[
(1.000136986)^{5475} \approx 2.114914
\]

5. Finally, divide 3000 by this result:

\[
P = \frac{3000}{2.114914} \approx 1418.53
\]

So, you would need to deposit approximately \$1418.53 now in order to have \$3000 in the account in 15 years, assuming the account earns 5% interest compounded daily.

Would you like any further details or have any questions?

---

1. What is compound interest, and how does it differ from simple interest?
2. Why is it important to know the frequency of compounding in interest calculations?
3. How does the number of compounding periods per year affect the final amount?
4. Can you explain how the compound interest formula is derived?
5. How would the required deposit change if the interest rate were different?
6. What if the interest were compounded monthly instead of daily?
7. What are some real-world applications of compound interest calculations?
8. How can understanding compound interest help in personal finance planning?

**Tip:** When dealing with compound interest problems, carefully consider the compounding frequency, as it significantly impacts the amount of interest earned over time.

Learn how to calculate compound interest to determine the initial deposit needed now to reach $3000 in 15 years at a 5% interest rate compounded daily. Step-by-step solution and formula explanation provided.

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