Answer
The formula was proved for $n=1$
The formula is correct if $n$ is changed by $n+1$
Work Step by Step
Let's prove the formula for $n=1$:
$2^1-1=2-1=1$
It is correct!
Now, suppose that the formula is correct, that is:
$1+2+2^2+2^3+...+2^{n-1}=2^n-1$
Now, let's prove the formula for $n+1$:
$1+2+2^2+2^3+...+2^{n-1}+2^{(n+1)-1}=(1+2+2^2+2^3+...+2^{n-1})+2^n=2^n-1+2^n=2(2^n)-1=2^1(2^n)-1=2^{n+1}-1$
That is exactly the given formula if $n$ is changed by $n+1$
