Questions: [OCR content]
Exam #3
Question 10 of 18 (1 point) Question Attempt 1 of 1
Time Remaining:
7 8 9 10 11 12 13 14 15 16
Find the future value of the annuity. Round your answer to the nearest cent. Do not round intermediate steps.
Payment Rate Compounded Time
4,000 1.66% Semiannually 3 years
The future value of the annuity is
Continue
Transcript text: [OCR content]
Exam #3
Question 10 of 18 (1 point) | Question Attempt 1 of 1
Time Remaining:
7 8 9 10 11 12 13 14 15 16
Find the future value of the annuity. Round your answer to the nearest cent. Do not round intermediate steps.
Payment Rate Compounded Time
$4,000 1.66% Semiannually 3 years
The future value of the annuity is $
Continue
Solution
Solution Steps
To find the future value of an annuity, we use the future value of an annuity formula:
FV=P×(r(1+r)n−1)
where:
P is the payment amount per period,
r is the interest rate per period,
n is the total number of periods.
Given that the interest is compounded semiannually, we need to adjust the interest rate and the number of periods accordingly.
Step 1: Given Values
We are given the following values for the annuity:
Payment per period, P=4000
Annual interest rate, r=0.0166
Compounding periods per year, m=2 (semiannually)
Total time in years, t=3
Step 2: Calculate Rate per Period and Total Periods
The interest rate per period is calculated as:
rp=mr=20.0166=0.0083
The total number of periods is:
n=m×t=2×3=6
Step 3: Calculate Future Value of the Annuity
Using the future value of an annuity formula:
FV=P×(rp(1+rp)n−1)
Substituting the values:
FV=4000×(0.0083(1+0.0083)6−1)
Calculating the expression gives:
FV=4000×(0.0083(1.0083)6−1)≈24503.55
Final Answer
The future value of the annuity is \\(\boxed{24503.55}\\).