Question

Five annual deposits in the amounts of $10,000, $8,000, $6,000, $4,000, and $2,000, in that order, are made into a fund that pays interest at a rate of 8% compounded annually. Determine the amount in the fund immediately after the fifth deposit.

Answer 1

To find the total amount in a fund after a series of unequal annual deposits with annual compounding interest, calculate the future value of each deposit and sum them up. Each deposit will compound differently based on the time it has until the final value is calculated.

Step-by-step explanation:

The question involves calculating the future value of a series of unequal deposits made into a savings account with a given interest rate, which is a typical problem in financial mathematics. To find the amount in the fund immediately after the fifth deposit, we need to calculate the future value of each deposit separately, all compounded annually at an 8% interest rate, and then sum them up. The deposits are not all made at the same time, so each will have a different amount of time to compound.

Here's how the calculation is done for each deposit:

• The first deposit of $10,000 is made and then sits for 4 years at 8%, so the future value of that deposit is FV = $10,000 × (1 + 0.08)^4.
• The second deposit of $8,000 is made and then sits for 3 years at 8%, so the future value of that deposit is FV = $8,000 × (1 + 0.08)^3.
• The third deposit of $6,000 is made and then sits for 2 years at 8%, so the future value of that deposit is FV = $6,000 × (1 + 0.08)^2.
• The fourth deposit of $4,000 is made and then sits for 1 year at 8%, so the future value of that deposit is FV = $4,000 × (1 + 0.08)^1.
• The fifth deposit of $2,000 is made and does not have time to earn interest before the total is calculated, so its future value is simply FV = $2,000.

The total amount in the fund after the fifth deposit will be the sum of the future values of all these deposits.

Answer 2

Balance of fund just at the end of fifth year = $35,000.9856 + $2,000

= $37,000.9856

When rounded to nearest dollar amount = $37,001

Step-by-step explanation:

Provided, amounts are deposited annually,

Amount deposited assumed in the beginning of each year.

Thus, at the end of year 1 = $10,000 + 8% = $10,800

Amount deposited in second year = $8,000

Amount in the fund at end of second year = ($10,800 + $8,000) + 8% = $20,304

Amount added in third year = $6,000

Balance of fund at end of third year = ($20,304 + $6,000) + 8% = $28,408.32

Amount added in the 4th year = $4,000

Balance of fund at end of fourth year = ($28,408.32 + $4,000) + 8% = $35,000.9856

Amount added in fifth year = $2,000

Balance of fund just at the end of fifth year = $35,000.9856 + $2,000

= $37,000.9856

When rounded to nearest dollar amount = $37,001