# Recursive Def/Compounding Interest question

1. May 13, 2009

### rought

1. The problem statement, all variables and given/known data

When Mr. Howe retires at age 65 he expects to have a retirement account worth about $400,000. One month after he retires, and every month thereafter, he intends to withdraw$4000 from the account. The balance will be invested at 9% annual interest compounded monthly.

a) Let An represent the ammount in the account and n months after Mr. Howe's retirement. Give a recursive definition for An

b.) When will there be no money left in the bank account?

I know that the recursive formula is something like tn-1 but that's all I got =/...

2. May 13, 2009

### tiny-tim

Hi rought!
A recursive formula gives tn as a function of tn-1 (and sometimes also tn-2 etc)

In this case, An = … ?

3. May 13, 2009

### HallsofIvy

Staff Emeritus
A recursive equation can be written as either [/itex]t_n[/itex]= some function of $t_{n-1}$ or as $t_{n+1}$= some function of $t_n$. Either this amount is some function of last month's or next month's amount is some function of this month's.

Since you mention $t_{n-1}$, let's write it that way. This month's amount, $A_n$ is, first, last month's, $A_{n-1}$, minus any money taken out (how much money does he take out each month?) plus any money put in (interest earned. How much interest does the money earn each month?).

4. May 13, 2009

### rought

ah ok...

so heres the equation that i got: An=An-1(1+.09/12)-4000

Is this correct? I'm not sure how to do part b now though... =/

5. May 13, 2009

### HallsofIvy

Staff Emeritus
Okay, $A_n= 1.0075A_{n-1}- 4000$

A standard method for something like $A_n= rA_{n-1}$ is to try something like $A_n= n^x$ for some number. If that were true, then $A_{n-1}= (n^x)^r= n^{rx}$ and $A_n= 1.0075A_{n-1}$ becomes $n^r= 1.0075 (n^{r-1})$. Dividing both sides by $n^r$, $1= 1.0075r^{-1}$ so r= 1.0075. In fact, if we were to try $A_n= C(1.0075)^n$, for C any constant, we would have $A_n= C(1.0075)^n= 1.0075A_{n-1}= 1.0075C(1.0075)^{n-1}= C(1.0075)^n$ is true for all n because the "C"s cancel. $C(1.0075)^n$ is the general solution to the equation $A_n= 1.0075A_n$.

That's ignoring the "-4000" part but since that number is a constant, what if we try $A_n= A$, a constant? Now $A_n= 1.0075A_{n-1}- 4000$ becomes A= 1.0075A- 4000 or -.0075A= 4000.

Now the "theory" part: If $A_n$ is the general solution to the "homogeneous" equation and A is a single solution to the entire equation, then $A_n+ A$ is the general solution to the entire equation.

You should now be able to write out the general solution, use the fact that $A_1= 400000$ to find C and then determine when $A_n= 0$. (You may find that it is never 0. Just find when it is less than 1.)

6. May 13, 2009

### tiny-tim

Hi rought!
Yes, that's correct, except that you must also specify the initial condition …

An=An-1(1.0075) - 4000 and A0 = 400,000.

You really need to read up about recurrence relations (for example, in the PF Library) to find the general way of solving this.

To get you started, can you see what the solution would be for the simpler:

An=An-1(1.0075) and A0 = 400,000?

7. May 13, 2009

### rought

ok so I did part b here's what I got

A(n) = 400,000 * 1.0075^n - 4000 * (1.0075^n - 1) / .0075 = 0
400,000 * 1.0075^n = 4000 * (1.0075^n - 1) / .0075

400,000 = 4000 * (1 - 1.0075^-n) / .0075

100 = (1 - 1.0075^-n) / .0075
1 - 1.0075^-n = .75
1.0075^-n = .25
1.0075^n = 4
n * ln(1.0075) = ln(4)
n = ln(4) / ln(1.0075)
n = 185.5315 months

does this seem right?

Last edited: May 13, 2009