Future Value of Money

[mathdad]

Suppose that you make deposits of 500 dollars at the start of every year into an Individual Retirement Account (IRA) earning interest r. At the start of the first year, the value of the account will be 500 dollars; at the start of the second year, the value on the account, will be 500r + 1000.

A. Verify that the value of the account at the start of the third year is T(r) = 500r^2 + 1500r + 1500.

B. The account value at the start of the fourth year is F(r) = 500r^3 + 2000r^2 + 3000r + 2000. If the annual rate of interest is 5 percent, what will be the value of the account at the start of the fourth year?

NOTE: I seek solution steps for parts A and B. I want to try this on my own (following your steps) before posting my work here.

 

[ksdhart2]

I'm not sure I understand the scenario correctly. Is the interest applied before or after the new deposit is made? If we assume the interest is calculated before the deposit (say, at the end of each year), then the account balance at the end of the first year would be $500r$. It then follows that the account balance at the start of the second year would be $500r + 500$. On the other hand if the interest, is calculated after the deposit is made, then the account balance after the second year's deposit would be $500 + 500 = 1000$. It then follows that the account balance at the start of the second year would be $1000r$.

However, in neither case is the account balance what the problem text states it should be. So clearly there's an error somewhere.

 

[MarkFL]

I would make a table:

Year	Start Amount	End Amount
1	500	500+500r
2	1000+500r	1000+500r+(1000+500r)r=500r^2+1500r+1000
3	500r^2+1500r+1500	500r^2+1500r+1500+(500r^2+1500r+1500)r=500r^3+2000r^2+3000r+1500
4	500r^3+2000r^2+3000r+2000	...

 

[mathdad]

I would make a table:

Year	Start Amount	End Amount
1	500	500+500r
2	1000+500r	1000+500r+(1000+500r)r=500r^2+1500r+1000
3	500r^2+1500r+1500	500r^2+1500r+1500+(500r^2+1500r+1500)r=500r^3+2000r^2+3000r+1500
4	500r^3+2000r^2+3000r+2000	...

Mark, I seek the solution steps as stated in the question. Nice table but it does not lead me to start.

 

[MarkFL]

Mark, I seek the solution steps as stated in the question. Nice table but it does not lead me to start.

I was hoping you would see that for the end amount, I am adding r times the start amount to the start amount. Then for the next row, the start amount is the end amount from the previous row with 500 added to it. See it now?

 

[mathdad]

I was hoping you would see that for the end amount, I am adding r times the start amount to the start amount. Then for the next row, the start amount is the end amount from the previous row with 500 added to it. See it now?

I'll get back to you.