6. Determine the regular payment for each annuity. a) An amount of 2500 is needed in 4 years by making regular.payments at the end of each month in an account that earns 4% per year, compounded monthly.

## Step 1: Convert the annual interest rate to a monthly rate.<br />### The annual interest rate is 4%, so the monthly interest rate is $\frac{4\%}{12} = \frac{0.04}{12} = 0.003333...$.<br /><br />## Step 2: Calculate the number of periods.<br />### The investment is for 4 years with monthly payments, so the number of periods is $4 \times 12 = 48$.<br /><br />## Step 3: Use the future value of an ordinary annuity formula.<br />### The future value of an ordinary annuity formula is $FV = PMT \times \frac{(1 + r)^n - 1}{r}$, where $FV$ is the future value, $PMT$ is the regular payment, $r$ is the interest rate per period, and $n$ is the number of periods. We want to find $PMT$, so we rearrange the formula: $PMT = FV \times \frac{r}{(1 + r)^n - 1}$.<br /><br />## Step 4: Substitute the values and calculate the regular payment.<br />### Substituting $FV = \$2500$, $r = 0.003333...$, and $n = 48$ into the formula, we get:<br />$PMT = 2500 \times \frac{0.003333...}{(1 + 0.003333...)^{48} - 1} \approx 2500 \times \frac{0.003333}{1.1730} \approx 2500 \times 0.002842 \approx 48.48$.