Compound interest for retirement Determine how much you will have to save each month at $3$, $6$, $9$, and $12$ percent compounded monthly for you to accumulate a nest egg for retirement. The variables are current age, age of retirement, nest egg size, and interest rate. Show all work and cite as appropriate.
C = current age
A = Age of retirement
R = interest rate
N = nest egg size
P = Cash flow per period
T = Time
 $N=P(1+\frac{.03}{12})^{12}(55-24)$
$\$6583.33$ per month at $3$ percent
can someone please tell me if i did this right and my answer is accurate or not?
 A: One needs to specify the payment scheme a bit more accurately, because for exact computation we need those details.  Let $n$ be the number of months between current age and retirement age. I will assume that $n$ is an integer.
I will also assume that you make a payment $P$ every month, with first payment today, and the last payment done a month before retirement, for a total of $n$ payments. You will have to make minor adjustments if the payment scheme is slightly different. Let $N$ be the desired sum available at retirement.
Let $r$ be the nominal yearly rate. I assume we have monthly compounding. The monthly rate is $\frac{r}{12}$. Let $x=1+\frac{r}{12}$.
So in one month $1$ unit of currency grows to $x$ dollars. The payment $P$ you made a month before retirement has grown to $Px$ at retirement, not much! The one you made two months before retirement has grown to $Px^2$. The one three months before retirement has grown to $Px^3$.  And so on. Finally, the one you made $n$ months before retirement has grown to $Px^n$. (The payments made long before retirement have grown quite a bit.)  
We want to have accumulated a total of $N$, so
$$N=Px+Px^2+Px^3+\cdots +Px^n=Px(1+x+x^2+\cdots+x^{n-1}).$$
The sum of the geometric series $1+x+x^2+\cdots+x^{n-1}$ is $\frac{x^n-1}{x-1}$. 
(This is a standard formula, you can look it up on Wikipedia.)
So our equation becomes
$$N=Px\frac{x^n-1}{x-1}.$$
Now it is all ready for numerical calculation. With interest rate $3\%$, for example, $x=1.0025$.
