Proof by induction (exponents) 
Use proof by induction and show that the formula holds for all positive integers:$$1+3+3^2+\dots+3^{n-1}=\frac{3^n -1}2$$

The confusing step in my opinion is the first expression: $3^{n-1}$, when I have to show for $k+1$. Any solutions?

For n+1 it is 
$1+3+3^2+\ldots+3^{n-1}+3^n=\frac{3^{n+1}-1}{2}$
$\left( 1+3+3^2+\ldots +3^{n-1}\right) +3^n=\frac{3\cdot 3^{n}-1}{2}$
$\left( 1+3+3^2+\ldots +3^{n-1}\right) +3^n=\frac{(1+2)\cdot 3^{n}-1}{2}$
$\left( 1+3+3^2+\ldots +3^{n-1}\right) +3^n=\frac{1\cdot 3^{n}-1}{2}+\frac{2\cdot 3^{n}}{2}$
$\left( 1+3+3^2+\ldots +3^{n-1}\right) +3^n=\left( \frac{1\cdot 3^{n}-1}{2}\right) + 3^n$
