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