Find the periodic payment R required to accumulate a sum of S dollars over t years with interest earned at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)
S = 50,000, r = 6, t = 9, m = 4

The periodic payment $ R $ required to accumulate a sum of $ S $ dollars over $ t $ years with interest rate $ r \% $ compounded $ m $ times a year is given by the future value of an ordinary annuity formula:

\[
S = R \times \frac{(1 + \frac{r}{m})^{mt} - 1}{\frac{r}{m}}
\]

We can rearrange this equation to solve for $ R $:

\[
R = \frac{S \times \frac{r}{m}}{(1 + \frac{r}{m})^{mt} - 1}
\]

### Given:
- $ S = 50,000 $
- $ r = 6\% = 0.06 $
- $ t = 9 $ years
- $ m = 4 $ (quarterly compounding)

Substitute these values into the formula to calculate $ R $:

\[
R = \frac{50,000 \times \frac{0.06}{4}}{(1 + \frac{0.06}{4})^{4 \times 9} - 1}
\]

Let me calculate that for you.The periodic payment $ R $ required to accumulate $ 50,000 $ dollars over 9 years with an interest rate of 6% compounded quarterly is $ 1,057.62 $ dollars.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. How would the result change if the compounding was done monthly instead of quarterly?
2. What if the interest rate was 5% instead of 6%?
3. How does increasing the accumulation period $ t $ affect the periodic payment?
4. What is the total amount of payments made over the 9 years?
5. How is the formula for an ordinary annuity different from that for a perpetuity?

**Tip**: Increasing the compounding frequency (e.g., from annually to quarterly) results in more frequent interest applications, which can slightly increase the future value.

Find the periodic payment R required to accumulate a sum of S dollars over t years with interest earned at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.) S = 50,000, r = 6, t = 9, m = 4

Learn how to find the periodic payment required to accumulate $50,000 over 9 years with an interest rate of 6%, compounded quarterly. This tutorial covers key financial mathematics concepts like compound interest and annuities, providing a detailed solution with formulas. Ideal for advanced high school and college students.

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