Answer
The total value of the annuity after 8 years is approximately \$2763.36.
Solution
To find the total value of the annuity after 8 years, we can use the future value of an ordinary annuity formula:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period
- \( r \) = interest rate per period
- \( n \) = total number of payments
### Step 1: Identify the known values
- \( P = 74 \) (quarterly payment)
- Annual interest rate = \( 3.9\% \) or \( 0.039 \)
- Since the interest is compounded quarterly, we need to find the quarterly interest rate:
\[
r = \frac{0.039}{4} = 0.00975
\]
- The total number of payments over 8 years (with quarterly payments):
\[
n = 8 \times 4 = 32
\]
### Step 2: Substitute the values into the formula
Now we can substitute \( P \), \( r \), and \( n \) into the future value formula:
\[
FV = 74 \times \frac{(1 + 0.00975)^{32} - 1}{0.00975}
\]
### Step 3: Calculate the future value
Let's perform the calculation.
First, we calculate \( (1 + 0.00975)^{32} \):
\[
(1 + 0.00975)^{32}
\]
Then we can calculate the entire expression for \( FV \).
I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
\(\left(1+0.00975\right)^{32}\)
- step1: Add the numbers:
\(1.00975^{32}\)
- step2: Convert the expressions:
\(\left(\frac{4039}{4000}\right)^{32}\)
- step3: Use the properties of exponents:
\(\frac{4039^{32}}{4000^{32}}\)
Calculate or simplify the expression \( 74 * ((1 + 0.00975)^{32} - 1) / 0.00975 \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{74\left(\left(1+0.00975\right)^{32}-1\right)}{0.00975}\)
- step1: Add the numbers:
\(\frac{74\left(1.00975^{32}-1\right)}{0.00975}\)
- step2: Convert the expressions:
\(\frac{74\left(\left(\frac{4039}{4000}\right)^{32}-1\right)}{0.00975}\)
- step3: Subtract the numbers:
\(\frac{74\times \frac{4039^{32}-4000^{32}}{4000^{32}}}{0.00975}\)
- step4: Multiply the numbers:
\(\frac{\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{159}\times 125^{32}}}{0.00975}\)
- step5: Convert the expressions:
\(\frac{\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{159}\times 125^{32}}}{\frac{39}{4000}}\)
- step6: Multiply by the reciprocal:
\(\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{159}\times 125^{32}}\times \frac{4000}{39}\)
- step7: Rewrite the expression:
\(\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{159}\times 125^{32}}\times \frac{32\times 125}{39}\)
- step8: Rewrite the expression:
\(\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{159}\times 125^{32}}\times \frac{2^{5}\times 125}{39}\)
- step9: Reduce the numbers:
\(\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{154}\times 125^{31}}\times \frac{1}{39}\)
- step10: Multiply the fractions:
\(\frac{37\times 4039^{32}-37\times 4000^{32}}{2^{154}\times 125^{31}\times 39}\)
- step11: Multiply:
\(\frac{37\times 4039^{32}-37\times 4000^{32}}{39\times 2^{154}\times 125^{31}}\)
### Step 4: Calculate the values
1. **Calculate \( (1 + 0.00975)^{32} \)**:
\[
(1 + 0.00975)^{32} \approx 1.364092
\]
2. **Substituting into the future value formula**:
\[
FV = 74 \times \frac{1.364092 - 1}{0.00975}
\]
\[
FV = 74 \times \frac{0.364092}{0.00975}
\]
\[
FV \approx 74 \times 37.307
\]
\[
FV \approx 2763.362945
\]
### Step 5: Round the final answer
Rounding \( 2763.362945 \) to the nearest cent gives us:
\[
FV \approx 2763.36
\]
### Final Answer
The total value of the annuity in 8 years is approximately **\$2763.36**.
Answered by UpStudy AI and reviewed by a Professional Tutor
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