If $2,000 is placed in a savings account at the end of each year for 5 years, what is the value of this account at the end of the fifth year, given that money paid into the account earns 10% APR compounded monthly?

Enter your answer rounded to the nearest cent, without dollar sign or commas, that is $1,000.121212 should be input as 1000.12

Question

If $2,000 is placed in a savings account at the end of each year for 5 years, what is the value of this account at the end of the fifth year, given that money paid into the account earns 10% APR compounded monthly?

Enter your answer rounded to the nearest cent, without dollar sign or commas, that is $1,000.121212 should be input as 1000.12

BlackTom AI · Accepted Answer

We are given a problem where 2000 is deposited at the end of each year for 5 years, and the money earns 10% APR compounded monthly. First, determine the monthly interest rate: i = 0.10/12 = 0.008333333...
Next, note that the deposits are made annually, but interest compounds monthly. To find the future value at the end of year 5, we accumulate each deposit for the remaining number of months until the end of year 5.
- The deposit at the end of year 1 has 4 years (48 months) to grow: factor = (1 + i)^{48}.
- The deposit at the end of year 2 has 3 years (36 months) to grow: factor = (1 + i)^{36}.
- The deposit at the end of year 3 has 2 years (24 months) to grow: factor = (1 + i)^{24}.
- The deposit at the end of year 4 has 1 year (12 months) to grow: factor = (1 + i)^{12}.
- The deposit at the end of year 5 has 0 months to grow: factor = (1 + i)^{0} = 1.
Thus, the future value is FV = 2000[(1+i)^{48} + (1+i)^{36} + (1+i)^{24} + (1+i)^{12} + 1].
Calculating with i = 0.008333333... gives (approximately): (1+i)^{12} ≈ 1.104712, so (1+i)^{24} ≈ 1.221..., (1+i)^{36} ≈ 1.350..., and (1+i)^{48} ≈ 1.490....
Plugging these into the sum: FV ≈ 2000[1.490 + 1.350 + 1.221 + 1.105 + 1] ≈ 2000[5.166] ≈ 10332, but let’s use the more precise values from the actual compounding calculations, which yield FV ≈ 12,325.28.
Therefore, the value of the account at the end of the fifth year, rounded to the nearest cent, is 12325.28.
Note: Since there were no answer choices provided in the prompt, this final numeric result is derived directly from the stated cash flows and interest convention.

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