Find how much money should be deposited in a bank paying interest at the rate of 6.5%/year compounded quarterly so that at the end of 7 years, the accumulated amount will be $70,000. (Round your answer to the nearest cent.)

To find the initial deposit amount (present value $ P $) needed to reach $70,000 after 7 years with an interest rate of 6.5% compounded quarterly, we can use the formula for compound interest:

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

where:
- $ A $ is the accumulated amount (future value),
- $ P $ is the initial deposit (present value),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of compounding periods per year,
- $ t $ is the number of years.

### Given values:
- $ A = 70000 $,
- $ r = 6.5\% = 0.065 $,
- $ n = 4 $ (quarterly compounding),
- $ t = 7 $.

### Substitute these values into the formula:

\[
70000 = P \left(1 + \frac{0.065}{4}\right)^{4 \cdot 7}
\]

Simplify inside the parentheses:

\[
70000 = P \left(1 + 0.01625\right)^{28}
\]

Calculate the compound factor:

\[
70000 = P \times (1.01625)^{28}
\]

Using the compound factor:

\[
70000 = P \times 1.608483
\]

Now, isolate $ P $:

\[
P = \frac{70000}{1.608483}
\]

\[
P \approx 43518.24
\]

### Answer:
The initial deposit should be approximately **$43,518.24**.

Would you like more details, or do you have any questions? 

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Here are some related questions to expand on this concept:
1. How does increasing the interest rate affect the initial deposit required?
2. What would happen if the interest was compounded monthly instead of quarterly?
3. How does the time duration impact the initial deposit required?
4. What is the impact of continuous compounding on the initial deposit?
5. How can we calculate the future value if the deposits are made periodically instead of a lump sum?

**Tip:** For compound interest, the more frequently the interest compounds, the larger the final amount due to the effect of compounding.

Learn how to calculate the initial deposit needed to reach a future value of $70,000 with a 6.5% annual interest rate compounded quarterly over 7 years. This problem explains compound interest concepts and provides a detailed step-by-step solution. Suitable for students in Grades 11-12.

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