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a. Use the appropriate formula to find the value of the annuity.
b. Find the interest.

\begin{tabular}{|l|l|l|}
\hline
Periodic Deposit & Rate & Time \\
\hline
[tex]$51,000$[/tex] at the end of each year & [tex]$7\%$[/tex] compounded annually & [tex]$30$[/tex] years \\
\hline
\end{tabular}

a. The value of the annuity is [tex]$\$[/tex][tex]$ $[/tex]\square[tex]$
(Do not round until the final answer. Then round to the nearest dollar as needed.)

b. The interest is $[/tex]\[tex]$[/tex] [tex]$\square$[/tex]
(Use the answer from part (a) to find this answer. Round to the nearest dollar as needed.)

Sagot :

To solve these questions, we'll follow step-by-step financial calculations.

### a. The value of the annuity

First, use the future value of an annuity formula compounded annually:

[tex]\[ FV = P \times \left( \frac{(1 + r)^t - 1}{r} \right) \][/tex]

where:
- [tex]\(FV\)[/tex] is the future value of the annuity
- [tex]\(P\)[/tex] is the periodic deposit (also called the payment per period)
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal)
- [tex]\(t\)[/tex] is the time in years

Given data:
- Periodic deposit ([tex]\(P\)[/tex]): [tex]$51,000 - Annual interest rate (\(r\)): 0.07 (7%) - Time (\(t\)): 30 years Plugging in the values into the formula: \[ FV = 51,000 \times \left( \frac{(1 + 0.07)^{30} - 1}{0.07} \right) \] Calculate the future value: \[ FV = 51,000 \times \left( \frac{(1.07)^{30} - 1}{0.07} \right) \] Using these specific parameters, the future value of the annuity, after making all the calculations, would be: \[ FV = \$[/tex]4,817,500 \]

Thus, the value of the annuity is \[tex]$4,817,500. ### b. The interest To find the interest earned, we need to calculate the total amount deposited and then subtract that from the future value. The total amount deposited over 30 years is: \[ \text{Total deposit} = \text{Periodic deposit} \times \text{Time} \] Given: - Periodic deposit: $[/tex]51,000
- Time: 30 years

[tex]\[ \text{Total deposit} = 51,000 \times 30 = 1,530,000 \][/tex]

Next, we need to subtract the total deposit from the future value to find the interest earned:

[tex]\[ \text{Interest earned} = FV - \text{Total deposit} \][/tex]

Given the future value ([tex]\(FV\)[/tex]) as:

[tex]\[ FV = 4,817,500 \][/tex]

[tex]\[ \text{Interest earned} = 4,817,500 - 1,530,000 = 3,287,500 \][/tex]

Thus, the interest earned is \[tex]$3,287,500. ### Summary a. The value of the annuity is \$[/tex]4,817,500.
b. The interest is \$3,287,500.