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 = 35,000, r = 4, t = 5, m = 4

To solve this, we use the formula for the **future value of an ordinary annuity**, which accumulates to a sum $S$ over time with periodic payments $R$, compounded $m$ times per year at an annual interest rate $r$.

The formula for the future value $S$ is:

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

We are given:
- $S = 35,000$
- $r = 4\%$
- $t = 5$ years
- $m = 4$ (quarterly)

We need to solve for $R$, the periodic payment.

---

### Step 1: Plugging in the known values
- $r = 4\%$, so $\frac{r}{100} = 0.04$.
- $m = 4$, so the interest rate per period is $\frac{0.04}{4} = 0.01$.
- $mt = 4 \times 5 = 20$.

The formula becomes:

\[
35,000 = R \times \left(\frac{(1 + 0.01)^{20} - 1}{0.01}\right)
\]

---

### Step 2: Simplifying the formula
- $1 + 0.01 = 1.01$
- $1.01^{20} \approx 1.219004$
- $1.219004 - 1 = 0.219004$
- $\frac{0.219004}{0.01} = 21.9004$

Now, the equation becomes:

\[
35,000 = R \times 21.9004
\]

---

### Step 3: Solving for $R$
\[
R = \frac{35,000}{21.9004} \approx 1,598.13
\]

---

### Final Answer
The required periodic payment $R$ is approximately:

\[
R \approx 1,598.13
\]

---

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

### Related Questions:
1. What happens to the payment if the interest rate increases?
2. How does changing the compounding frequency (m) affect the payment?
3. What would the payment be if the time period is extended to 10 years?
4. How is this formula derived from the future value concept?
5. What if payments are made at the beginning of each period instead?

**Tip:** When rounding financial values, always round to two decimal places to represent cents correctly.

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 = 35,000, r = 4, t = 5, m = 4

Find the periodic payment needed to accumulate $35,000 over 5 years with an interest rate of 4%, compounded quarterly. This solution explains the future value of ordinary annuities, compound interest concepts, and provides a step-by-step guide for students in Grades 10-12.

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