What single lump sum deposited today would allow withdrawals of 2,000 dollars per year for 7 years at 5% interest compounded annually?

Introduction / Context:
This problem is about finding the present value of an ordinary annuity. Instead of knowing the deposit and computing future value, you are given a desired series of withdrawals and an interest rate, and you must find how much money must be set aside today to fund those withdrawals.

Given Data / Assumptions:

• Yearly withdrawal (payment) R = 2,000 dollars.
• Interest rate r = 5% per annum, compounded annually.
• Number of years n = 7.
• Withdrawals occur at the end of each year (ordinary annuity).
• No additional deposits are made after the initial lump sum.

Concept / Approach:
The present value of an ordinary annuity is given by:
PV = R * (1 - (1 + r)^(-n)) / r This formula discounts each future payment back to the present, summing the series into one equivalent lump sum today.

Step-by-Step Solution:
Step 1: Convert the rate to decimal form: r = 0.05. Step 2: Plug in values: PV = 2000 * (1 - (1.05)^(-7)) / 0.05. Step 3: Compute (1.05)^(-7). First find (1.05)^7 and then take its reciprocal. Step 4: After evaluation, the factor (1 - (1.05)^(-7)) / 0.05 is about 5.786. Step 5: Multiply by the payment: PV ≈ 2000 * 5.786 ≈ 11,572.75 dollars. So approximately 11,572.75 dollars is required today to support the withdrawals.

Verification / Alternative Check:
You can verify by taking the found present value and compounding it year by year while subtracting the 2,000 dollar payment at year end. After the seventh withdrawal the remaining balance should be very close to zero, confirming that the calculated present value is accurate.

Why Other Options Are Wrong:
Options B and D (12,000 and 13,000 dollars) ignore the effect of interest and are simple guesses based on total withdrawals.
Option C (10,500 dollars) is too low and would not generate enough interest to cover all payments.
Option E (9,800 dollars) is even lower and clearly insufficient.

Common Pitfalls:
One common error is to multiply the annual payment by the number of years and forget interest altogether. Another is to confuse present value with future value and apply the wrong formula. Always note whether payments are at the beginning or end of periods and match that with the correct annuity expression.

Final Answer:
The required lump sum deposit today is approximately 11,572.75 dollars.