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* = 180,000, *r* = 5.5, *t* = 25, *m* = 6

$ 

To find the periodic payment $ R $ required to accumulate a sum $ S $, with interest compounded $ m $ times a year at an interest rate $ r \% $, we use the formula for the future value of an ordinary annuity:

\[
S = R \times \left( \frac{(1 + \frac{r}{100m})^{mt} - 1}{\frac{r}{100m}} \right)
\]

Where:
- $ S $ is the future value (total accumulated sum),
- $ R $ is the periodic payment,
- $ r $ is the annual interest rate (as a percentage),
- $ m $ is the number of compounding periods per year,
- $ t $ is the number of years.

Given:
- $ S = 180,000 $,
- $ r = 5.5\% $,
- $ t = 25 $,
- $ m = 6 $.

We need to solve for $ R $. Rearranging the formula to solve for $ R $:

\[
R = \frac{S \times \frac{r}{100m}}{(1 + \frac{r}{100m})^{mt} - 1}
\]

Now, substituting the values into the formula, we can calculate $ R $. Let's do that.The periodic payment $ R $ required to accumulate a sum of \$180,000 over 25 years, with an interest rate of 5.5% compounded 6 times per year, is approximately \$563.07.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How does increasing the interest rate affect the periodic payment?
2. What happens to the periodic payment if the compounding frequency changes?
3. How would the payment change if the accumulation period is shortened?
4. Can you calculate the total interest paid over the 25-year period?
5. How does the formula change if the payments are made continuously?

**Tip:** Increasing the frequency of compounding periods generally lowers the periodic payment needed for the same accumulated sum!

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 = 180,000, r = 5.5, t = 25, m = 6.

Learn how to calculate the periodic payment needed to accumulate $180,000 over 25 years with a 5.5% annual interest rate compounded 6 times per year. This problem uses the compound interest formula for annuities and is suitable for students in Grades 10-12.

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