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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?

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    I **think** you are asking what $P$ is if $200000=P\left(1+\frac{0.03}{12}\right)^{12(55-24)}$. Nowhere near $6583.33$. The answer is more like $79,002$. If you divide this by $12$ you get roughly $6583.3$. That is probably what you did.2012-08-09

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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, $x=1.0025$.