# An example need to be turned into an integral

1. Nov 19, 2014

### vemvare

OK. We are producing something and then storing it. The first year, "1" is produced, the second year, "1,1" the third 1,12, so that the production capacity increases 10% per year. How do we convert this into a general formula for how much we have produced in total after year x?

It is obviously an integral, but all formulas I find for ∫ nx give nonsense when I apply them to this example, so I think I'm wrong in turing the formula into nx

How do I solve this?

2. Nov 19, 2014

### Mentallic

What you're looking for is called a geometric sum.

$$S=1+r+r^2+...+r^{n-1}=\frac{r^n-1}{r-1}$$

r is the ratio, in your case 1.1, n is the amount of years you're counting.

3. Nov 19, 2014

### vemvare

Ah, that gives reasonable numbers. So the formula resulting from the example is discrete? Is it the non-integer factor (1,1) that does that?

EDIT: No, I specified the increase as happening annually, if it isn't, if it is continous, the (r-1) is replaced with ln(r), which I found by experimenting, and it gives exactly the value my calculator does. Oddly I cannot seem to find any reference to this formula, though I'm most cenrtainly again screwing up annotations.

I need to refreshen my math-math, I've been in the world of molecules for too long...

Last edited: Nov 19, 2014
4. Nov 19, 2014

### SteamKing

Staff Emeritus
If the increase in production was constantly changing, every minute of every day saw an incremental increase, then you might use an integral to find the increase after a certain period. The integral is the limit of a certain process which can often be crudely represented as a summation.