To solve this, we can use the savings plan formula for periodic payments:

$A = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}$

Where:

• $A$ is the future value (target amount), which is $85,000 in this case.
• $P$ is the monthly deposit (this is what we're solving for).
• $r$ is the annual interest rate (APR), given as 7% or 0.07.
• $n$ is the number of times the interest is compounded per year. Since this is a monthly deposit, $n = 12$.
• $t$ is the time in years, which is 15 years.

Rearranging the formula to solve for $P$:

$P = \frac{A \times r/n}{(1 + r/n)^{nt} - 1}$

Substituting the values:

$P = \frac{85,000 \times 0.07 / 12}{(1 + 0.07/12)^{12 \times 15} - 1}$

Let me calculate that for you.

First, the monthly interest rate is $\frac{0.07}{12} = 0.0058333$, and the total number of months is $12 \times 15 = 180$.

Next, the term inside the parentheses becomes:

$(1 + 0.0058333)^{180} = 2.849$

Now, let's calculate $P$:

$P = \frac{85,000 \times 0.0058333}{2.849 - 1} = \frac{495.83}{1.849} \approx 268.1$

So, you should deposit approximately $268.10 each month to accumulate $85,000 in 15 years.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would the required monthly deposit change if the APR were 6% instead of 7%?
2. What if you could only deposit for 10 years instead of 15 years? How much would you need monthly?
3. What is the total amount deposited over the 15 years? How does that compare to the accumulated amount?
4. How would biweekly deposits affect the overall calculation instead of monthly?
5. What is the role of compound interest in achieving the target of $85,000?

Tip: Small changes in the interest rate can significantly impact your monthly deposit when saving over long periods.