# Thread: Series and Exponential Growth

1. ## Series and Exponential Growth

If you invest $1000 at 5.7% interest compounded quarterly, and you deposit$1000 each year, in how many years will you have a total of $1,000,000? It's not difficult to set up the problem without the additional$1000 each year, but with it, I'm lost.

2. Assuming you $x_n$ dollars at the start of that year, let's try to find the amount we have at the end of the year:
at the end of each quarter, we multiply by 1.057,
hence, at the end of the year we have: $x_{n+1} = x_{n}*1.057^4+1000 = 1.248x_n+1000$. Our goal is to find n for which $x_n\geq 1,000,000$.
This is difference equation which looks like: $x_{n+1}-1.248x_{n}=1000$ whose solution is of the form $x_{n}=Ar^n+C$ (A, r and C are constants). To find these, solve first for $x_{n+1}-1.248x_{n}=0$. You get A and r. Then plug in what you got in the initial equation to find C.
Now that you have the closed form, you can solve for $x_{n}\geq 1,000,000$
hope this helps