Function For Graphing Exponentially Slowing Growth

Question

I need the following type of function for a project at uni:

I'm trying to study growth patterns that slow by a constant in the following manner (you could use anything in place of 0.5, say 0.3, etc...):

x, y:

0, 0

1, (0.5^0)

2, (0.5^0+0.5^1)

2, (0.5^0+0.5^1+0.5^2)

3, (0.5^0+0.5^1+0.5^2+0.5^3)

And so on......

I would like to find a reasonably simple function that can be used in and manipulated by means of an online graphing calculator. So far, I have had no luck; the closest thing I've found thing are log functions but they don't do, as far as I can see, what I'm talking about.

I know I can write something in js or some such to do what I want but I'd rather avoid that if I can.

Thanks for your time.

Answer 1

You are facing a geometric pregression, that is to say that $$y_n=\frac{1-x^{n+1}}{1-x}$$