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 $n.$

If we assume that $1+2+\cdots+2^{n-1}=2^n$, we can easily prove that $1+2+\cdots+2^n=2^{n+1}$.  For
$$1+2+\cdots+2^n=(1+2+\cdots+2^{n-1})+2^n=2^n+2^n=2^{n+1}.$$
But of course it is not true that $1+2+\cdots+2^{n-1}=2^n$.  The base case $n=1$ does not hold. For then $1+2+\cdots+2^{n-1}=1\ne 2^1$.
The whole point of this exercise is that in order to prove a result by induction, we must do the induction step (which worked) and we must verify the base case (which failed).
In fact, $1+2+\cdots+2^{n-1}=2^n-1$.  The proof is straightforward.
