Periodic Compound Interest: PN = P0 (1 + f )Nk
2. Continuous Compound Interest: PN = P 0 erN
3. Annuity Formula: PN = (\) : where PN is the balance after N years.
d(l-(1+ .!: )-Nk)
4. Payout Annuity: P0 = (f) : where P0 is the balance at the beginning. How much would you need to deposit in an account now in order to have $20: 000 in
the account in 10 years? Assume the account earns 5% interest compounded monthly
(Round the answer to two decimal place accuracy)

We are dealing with compound interest here, specifically with interest compounded monthly. To find out how much we need to deposit now to have $20,000 in 10 years at 5% interest compounded monthly, we will use the **Periodic Compound Interest** formula:

\[
PN = P_0 \left(1 + \frac{f}{k}\right)^{Nk}
\]

Where:
- $PN$ is the balance after $N$ years (in our case, $20,000),
- $P_0$ is the initial principal or the amount to deposit (what we are trying to find),
- $f$ is the annual interest rate (5% or 0.05),
- $k$ is the number of compounding periods per year (monthly compounding means $k = 12$),
- $N$ is the number of years (10 years).

Let's plug in the values and solve for $P_0$.

1. $PN = 20000$,
2. $f = 0.05$,
3. $k = 12$,
4. $N = 10$.

The formula becomes:

\[
20000 = P_0 \left(1 + \frac{0.05}{12}\right)^{10 \times 12}
\]

Now we can solve for $P_0$.The amount you would need to deposit now in order to have $20,000 in 10 years, assuming a 5% interest rate compounded monthly, is **$12,143.22** (rounded to two decimal places).

Do you want more details or have any other questions?

Here are 5 related questions to expand on this topic:
1. How would the deposit change if the interest was compounded quarterly instead of monthly?
2. What would happen if the interest rate was 6% instead of 5%?
3. How much total interest would you earn over the 10 years?
4. How does continuous compounding compare to monthly compounding in this scenario?
5. What if you wanted to reach $30,000 instead of $20,000 in 10 years?

**Tip:** Compounding more frequently generally results in more interest earned over time, as each compounding period builds on the last one.

How much would you need to deposit in an account now in order to have $20,000 in the account in 10 years? Assume the account earns 5% interest compounded monthly. (Round the answer to two decimal place accuracy)

Discover how much you need to deposit now to accumulate $20,000 in 10 years with 5% interest compounded monthly. This guide covers the use of the periodic compound interest formula, with step-by-step solutions and key insights into exponential growth and compounding frequency. Ideal for high school and college prep students.

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