Don't understand an induction proof for odometer principle Prove by induction that the Odometer Principle with base b does indeed give the representation $$\text{$x_{n-1}...x_1x_0$ for the natural number $N = x_{n-1}b^{n-1}+...+x_1b+x_0$} $$.![enter image description here]( So my question is, in the bracketed section of the image, how does one get from one line to the next?

Hint: This follows from the formula

$$ 1 + x + \dots + x^{n-1} = \frac{x^n - 1}{x - 1} $$

which holds for any positive integer $n$ and $x \
eq 1$.

You can prove this formula by doing the multiplication

\begin{align*} (1 + x + \dots + x^{n-1} )(x - 1) &= x^n + (x^{n-1} - x^{n-1}) + (x^{n-2} - x^{n-2}) + \dots + (x - x) - 1 \\\ & = x^n - 1. \end{align*}