I'm looking for a continuous function f(x) with the following properties. I've been playing with exponentials, but that doesn't seem to be the answer, although my high school mathematics is a bit rusty, I must admit.
- $f(1) = 2$
- $f(2) = 6 = 2+4$
- $f(3) = 14 = 2+4+8$
- $f(4) = 30 = 2+4+8+16$
- And so on
I'm looking for a continuous function, so something with a meaningful answer for $f(2.5)$, which is less then $6 + \frac{14 - 6}{2} = 10$.
My simple high school math made me look at something like $f(x) = a^x$, but that doesn't seem to be the answer.
Any better ideas?