Math / Algebra

Questiona. 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 $\$ 2000$ at the end of each year & $7 \%$ compounded annually & 15 years \\ \hline \end{tabular}
Click the icon to view some finance formulas. a. The value of the annuity is $\$$ $\square$ (Do not round until the final answer. Then round to the nearest dollar as needed.)

Studdy Solution

STEP 1

1. We are dealing with an ordinary annuity where deposits are made at the end of each period.
2. The periodic deposit is \$2000.
3. The interest rate is 7% compounded annually.
4. The time period is 15 years.

STEP 2

1. Identify the formula for the future value of an ordinary annuity.
2. Substitute the given values into the formula.
3. Calculate the future value of the annuity.
4. Calculate the total amount deposited.
5. Find the interest earned by subtracting the total deposits from the future value of the annuity.

STEP 3

Identify the formula for the future value of an ordinary annuity:
The future value $FV$ of an ordinary annuity can be calculated using the formula:
$FV = P \frac{(1 + r)^n - 1}{r}$
where: - $P$ is the periodic deposit, - $r$ is the interest rate per period, - $n$ is the number of periods.

STEP 4

Substitute the given values into the formula:
Given $P = 2000$ , $r = 0.07$ , and $n = 15$ , substitute these into the formula:
$FV = 2000 \frac{(1 + 0.07)^{15} - 1}{0.07}$

STEP 5

Calculate the future value of the annuity:
First, calculate $(1 + 0.07)^{15}$ :
$(1 + 0.07)^{15} = 2.75903$
Now substitute back into the formula:
$FV = 2000 \frac{2.75903 - 1}{0.07}$ $FV = 2000 \frac{1.75903}{0.07}$ $FV = 2000 \times 25.129$ $FV = 50258$
The future value of the annuity is approximately $\$50,258$ .

STEP 6

Calculate the total amount deposited:
The total amount deposited over 15 years is:
$\text{Total Deposits} = 2000 \times 15 = 30000$

STEP 7

Find the interest earned:
The interest earned is the difference between the future value and the total deposits:
$\text{Interest} = 50258 - 30000 = 20258$
The value of the annuity is $\$50,258$ and the interest earned is $\$20,258$ .

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Studdy Solution

STEP 1

1. We are dealing with an ordinary annuity where deposits are made at the end of each period.
2. The periodic deposit is \$2000.
3. The interest rate is 7% compounded annually.
4. The time period is 15 years.

STEP 2

STEP 3

Identify the formula for the future value of an ordinary annuity:
The future value $FV$ of an ordinary annuity can be calculated using the formula:
$FV = P \frac{(1 + r)^n - 1}{r}$
where: - $P$ is the periodic deposit, - $r$ is the interest rate per period, - $n$ is the number of periods.

STEP 4

Substitute the given values into the formula:
Given $P = 2000$ , $r = 0.07$ , and $n = 15$ , substitute these into the formula:
$FV = 2000 \frac{(1 + 0.07)^{15} - 1}{0.07}$

STEP 5

STEP 6

Calculate the total amount deposited:
The total amount deposited over 15 years is:
$\text{Total Deposits} = 2000 \times 15 = 30000$

STEP 7

Find the interest earned:
The interest earned is the difference between the future value and the total deposits:
$\text{Interest} = 50258 - 30000 = 20258$
The value of the annuity is $\$50,258$ and the interest earned is $\$20,258$ .

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Studdy Solution

STEP 1

1. We are dealing with an ordinary annuity where deposits are made at the end of each period.2. The periodic deposit is \$2000.3. The interest rate is 7% compounded annually.4. The time period is 15 years.

STEP 2

STEP 3

STEP 4

Substitute the given values into the formula:Given P=2000 P = 2000 P=2000, r=0.07 r = 0.07 r=0.07, and n=15 n = 15 n=15, substitute these into the formula:FV=2000(1+0.07)15−10.07 FV = 2000 \frac{(1 + 0.07)^{15} - 1}{0.07} FV=20000.07(1+0.07)15−1​

STEP 5

STEP 6

Calculate the total amount deposited:The total amount deposited over 15 years is:Total Deposits=2000×15=30000 \text{Total Deposits} = 2000 \times 15 = 30000 Total Deposits=2000×15=30000

STEP 7

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1. We are dealing with an ordinary annuity where deposits are made at the end of each period.
2. The periodic deposit is \$2000.
3. The interest rate is 7% compounded annually.
4. The time period is 15 years.

Substitute the given values into the formula:
Given $P = 2000$ , $r = 0.07$ , and $n = 15$ , substitute these into the formula:
$FV = 2000 \frac{(1 + 0.07)^{15} - 1}{0.07}$

Calculate the total amount deposited:
The total amount deposited over 15 years is:
$\text{Total Deposits} = 2000 \times 15 = 30000$