To solve this problem, we can use the future value of an annuity formula, which is applicable when you're making regular payments into an account that earns interest over time.

The formula for the future value of an annuity compounded quarterly is:

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

Where:

• $FV$ is the future value (the amount you want to save, $21,000).
• $P$ is the quarterly deposit (what we're solving for).
• $r$ is the annual interest rate (4.3% or 0.043).
• $n$ is the number of compounding periods per year (quarterly means $n = 4$).
• $t$ is the number of years (6 years in this case).

Step-by-step solution:

1. Identify known values:

   • $FV = 21,000$
   • $r = 0.043$
   • $n = 4$
   • $t = 6$

2. Rearrange the formula to solve for $P$: $P = \frac{FV \times \frac{r}{n}}{(1 + \frac{r}{n})^{nt} - 1}$

3. Plug in the values: $P = \frac{21,000 \times \frac{0.043}{4}}{(1 + \frac{0.043}{4})^{4 \times 6} - 1}$

4. Calculate: $P = \frac{21,000 \times 0.01075}{(1.01075)^{24} - 1}$

Let's calculate this next:

$P = \frac{21,000 \times 0.01075}{(1.01075)^{24} - 1} \approx \frac{21,000 \times 0.01075}{0.2944}$

$P \approx \frac{225.75}{0.2944} \approx 766.72$

Final Answer:

You must deposit $766.72 quarterly to reach your goal of $21,000 in 6 years at 4.3% interest compounded quarterly.

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Would you like further details on how I calculated that?

Here are some related questions:

1. How does compound interest affect the growth of savings over time?
2. How would the result change if the interest rate were higher or lower?
3. How would you calculate the future value if the deposits were made monthly instead of quarterly?
4. What is the effect of increasing the number of years on the required quarterly deposit?
5. How can you adjust this formula to account for annual or monthly compounding instead of quarterly?

Tip: Make sure to always check if the interest is compounded periodically (quarterly, monthly, etc.) and adjust the formula accordingly.