To find the periodic payment $R$, we can use the future value of an ordinary annuity formula, which is as follows:

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

Where:

• $S$ is the future value (the amount accumulated),
• $R$ is the periodic payment,
• $r$ is the annual interest rate (in decimal form),
• $m$ is the number of compounding periods per year,
• $t$ is the total number of years,
• $mt$ is the total number of compounding periods over the life of the annuity.

We need to solve for $R$:

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

Given values:

• $S = 190,000$,
• $r = 5.5\% = 0.055$ (as a decimal),
• $t = 30$ years,
• $m = 6$ (compounded semi-annually).

Let’s calculate $R$.

$R = \frac{190,000 \times \frac{0.055}{6}}{(1 + \frac{0.055}{6})^{6 \times 30} - 1}$

Now, I will compute this step by step.The periodic payment $R$ required to accumulate a sum of $190,000$ dollars over 30 years, with interest earned at the rate of 5.5% per year compounded semi-annually, is $417.87.

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. How would the payment change if the interest rate were 6% instead of 5.5%?
2. What is the total amount of payments made over the 30 years?
3. How would the payment change if the compounding frequency was annual instead of semi-annual?
4. What is the future value if you only make payments for 20 years instead of 30?
5. How does increasing the compounding periods to quarterly affect the payment?

Tip: Always remember to convert percentages to decimals when using them in formulas.