<s>[INST] 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?
 [/INST] Suppose that $x \neq 1$. We wish to show by inducting on $n$ that $\sum\limits_{k=0}^{n-1}x^{k} = \frac{x^{n} - 1}{x-1}$. 
Then $1 = \frac{x-1}{x-1}$ so that the formula holds for $n = 1$. 
Suppose that we have $\sum\limits_{k=0}^{n-1}x^{k} = \frac{x^{n} - 1}{x-1}$. We must show that $\sum\limits_{k=0}^{n}x^{k} = \frac{x^{n+1} - 1}{x-1}$. 
We have $\sum\limits_{k=0}^{n}x^{k} = \sum\limits_{k=0}^{n- 1}x^{k} + x^{n} = \frac{x^{n} - 1}{x-1} + x^{n}\frac{x -1}{x-1} = \frac{x^{n} - 1 + x^{n+1} - x^{n}}{x - 1} = \frac{x^{n+1} - 1}{x-1}$
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