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Proving by mathematical induction that $3^{n-1}=\frac{3^n-1}{2}$

I am trying to show that
$$3^{n-1}=\frac{3^n-1}{2}$$
by mathematical induction.
How can I do this?
I tried using the fact that
$$3^2=9$$
$$3^3=27$$
$$3^4=81$$
but I couldn’t determine what should be the “next step”.

A:

The base case is: $3^{0-1}=\frac{3^0-1}{2}=\frac{1-1}{2}=\frac{0}{2}=0$.
The inductive step for the general statement is: $3^{n-1}=(3^{n-1})’+\frac{(3^{n-1})’}{2}=(3^{n-1})’+\frac{(3^{n})’-(3^{n-1})’}{2}=(3^{n-1})’+\frac{3^n-(3^{n-1})’}{2}=(3^{n-1})’+\frac{3^n-3^{n-1}}{2}=\frac{(3^n-1)-(3^{n-1})(1)}{2}=\frac{3^n-1}{2}$
Where $(3^n-1)=(3^{n-1})’+3^n-1=3