Resolve a recursive series

Question

Consider the following recursive function:

$f(n) = 1 + \sum_{i=0}^{n-1} f(i)$ with $f(0)=1$

I need to derive a non recursive form. By simply trying values, I have inferred that it must be $f(n)=2^n$. However, I don't know how to derive this formally. Any ideas?

Answer 1

By the definition given $$f(n)=1+\sum_{i=0}^{n-1}f(i)$$ and $$f(n+1)=1+\sum_{i=0}^{n}f(i).$$ Therefore $$f(n+1)-f(n)=f(n).$$ This reduces to $$f(n+1)=2f(n).$$