John is 30 years old and wants to have 1 million dollars in savings by the time he retires at 65. He plans to open a savings account that pays 6% interest compounded quarterly and he will be making quarterly deposits of $200 into the account.

John will need to make an initial deposit of

We can calculate John's initial deposit using the future value formula for compound interest with regular deposits (also known as an annuity). The formula is:

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

Where:
- $FV$ = future value (\$1,000,000)
- $P$ = initial deposit (what we need to find)
- $D$ = regular quarterly deposit (\$200)
- $r$ = annual interest rate (6% or 0.06)
- $n$ = number of compounding periods per year (4 for quarterly)
- $t$ = time in years (35 years, from 30 to 65)

We'll isolate $P$ and solve for it.

Let's break this down step by step.

1. The total future value ($FV$) is $1,000,000.
2. Quarterly interest rate: $\frac{r}{n} = \frac{0.06}{4} = 0.015$
3. Total number of periods: $n \cdot t = 4 \times 35 = 140$

Let’s compute the initial deposit now.John will need to make an initial deposit of approximately **$112,706.09** to reach his goal of $1,000,000 by the time he retires at age 65, assuming he continues to make quarterly deposits of $200 and the account earns 6% interest compounded quarterly.

Would you like further details on how the calculation was done or any clarification?

Here are 5 related questions to explore further:
1. What if John increases his quarterly deposits to $300 instead of $200?
2. How does the interest rate affect John's initial deposit?
3. What would happen if John starts saving later, at age 40?
4. How would the scenario change if the interest were compounded monthly instead of quarterly?
5. What if John wants to achieve his goal with no initial deposit and just regular contributions?

**Tip:** The earlier you start saving, the more compound interest works in your favor, reducing the burden of large initial deposits.

John is 30 years old and wants to have 1 million dollars in savings by the time he retires at 65. He plans to open a savings account that pays 6% interest compounded quarterly and he will be making quarterly deposits of $200 into the account. John will need to make an initial deposit of...

Discover how to calculate the initial deposit John needs to make to reach $1 million in savings by age 65, assuming quarterly deposits of $200 and a 6% interest rate compounded quarterly. This problem involves compound interest and annuity calculations, suitable for advanced high school or college-level students.

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