Proof by Induction for Natural Numbers Show that if the statement 
$$1 + 2 + 2^{2} + ... + 2^{n - 1} = 2^{n}$$
is assumed to be true for some $n,$ then it can be proved to be true for $n + 1.$ Is the statement true for all $n$?
Intuitively, then I don't think it holds for all $n.$

So, let $n = 3 \in \mathbb{N}$. Then we have, $$2^0+2^1+2^2=1+2+4=7 \neq 2^3$$ 
