# How do I make an exponential function that doubles every fifth integer.

I don't understand exponential functions too well, so if you know the answer, please also show me the process I would use to come up with it.

I'm trying to figure out how to make an exponential function that doubles every five whole integers. That would mean that:

|input|output|
|:---:|:----:|
|f(1) |1     |
|f(6) |2     |
|f(11)|4     |
|f(16)|8     |


et cetera.

I know some people have suggested using a best-fit application, but I don't think those things are too accurate. Is there a tried and true process to creating a function that matches these outputs?

• How about $2^{(x-1)/5}=\left(\sqrt[5]2\right)^{x-1}$? Mar 24, 2021 at 21:37

You want an exponential function. This means it is of the form $$f(n)=a\cdot b^n$$ for some numbers $$a,b$$. You also want $$f(1)=1,f(6)=2$$. That's enough to solve your problem: $$\cases{a\cdot b^1=1\\ a\cdot b^6=2}$$ Dividing equation two by equation one, we get $$b^5=2$$. This is exactly the equation fifth roots are made to solve: $$b=\sqrt[5]2$$. Inserting this into equation one, we get $$a=\frac{1}{\sqrt[5]{2}}$$. Thus our function must be $$f(n)=\frac{\sqrt[5]2^n}{\sqrt[5]2}$$