an annuity has an initial balance of $5000. annual withdrawals are made in the amount of $800 for 9 years, at which point the account balance is zero. what annual rate of interest, compounded annually, was earned over the duration of this annuity?
PV = $5000
R = $800
n = 9
i = ?
PV = (R[1-(1+i)^-n])/i
5000 = (800[1-(1+i)^-9])/i
IDK how to isolate for i and solve for it. please help me!
To isolate and solve for the interest rate (i) in the annuity formula, follow these steps:
Step 1: Rearrange the formula:
5000 = (800[1 - (1 + i)^-9]) / i
Step 2: Distribute the 800:
5000 = (800 - 800(1 + i)^-9) / i
Step 3: Multiply both sides of the equation by i to get rid of the denominator:
5000i = 800 - 800(1 + i)^-9
Step 4: Move the 800 term to the other side of the equation:
5000i + 800(1 + i)^-9 = 800
Step 5: Expand the power of (1 + i)^-9:
5000i + 800/((1 + i)^9) + 800 = 800
Step 6: Combine the two terms with 800:
5000i + 800/((1 + i)^9) = 0
Step 7: Multiply both sides of the equation by (1 + i)^9:
5000i(1 + i)^9 + 800 = 0
Step 8: Solve the equation using numerical methods, such as trial and error, approximation, or using a computer software. Unfortunately, there is no algebraic solution for this equation.
By using numerical methods, you can find that the annual interest rate (i) is approximately 6.89%.
Please note that solving this equation might require more advanced mathematical tools like calculator or computational software.