Questions: Use the appropriate formula to find the value of the annuity. Find the interest. Periodic Deposit Rate Time 100 at the end of every six months 4.5% compounded semiannually 20 years The value of the annuity is 110. (Do not round until the final answer. Then round to the nearest dollar as needed.)

Use the appropriate formula to find the value of the annuity.
Find the interest.
Periodic Deposit Rate Time
100 at the end of every six months 4.5% compounded semiannually 20 years
The value of the annuity is 110.
(Do not round until the final answer. Then round to the nearest dollar as needed.)

Transcript text: Use the appropriate formula to find the value of the annuity. Find the interest. Periodic Deposit & Rate & Time $100 at the end of every six months & 4.5% compounded semiannually & 20 years The value of the annuity is $110. (Do not round until the final answer. Then round to the nearest dollar as needed.)

Solution

Solution Steps

Step 1: Calculate the Future Value of the Annuity (FVA)

To calculate the future value of an annuity, we use the formula: \[FVA = PMT \times \left(\frac{(1 + \frac{r}{n})^{n \times t} - 1}{\frac{r}{n}}\right)\] Substituting the given values: $PMT = 100$, $r = 0.045$, $n = 2$, and $t = 20$, we get: \[FVA = 100 \times \left(\frac{(1 + \frac{0.045}{2})^{40} - 1}{\frac{0.045}{2}}\right) = 6379\]

Step 2: Calculate the Total Interest Earned

The total amount of deposits is calculated as: $Total Deposits = PMT \times n \times t = 4000$. Thus, the total interest earned is: $Total Interest = FVA - Total Deposits = 6379 - 4000 = 2379$.