Hey, I was wondering if someone could help me with a question I have.
Prove by induction
x^n-1 + x^n-2 + .... + x + 1 = (x^n)-1 / x - 1
Thank you
It is very clear that this is the case if $n=1$
So suppose that $J>1~\&~\dfrac{{{x^J} - 1}}{{x - 1}} = \sum\limits_{k = 0}^{J - 1} {{x^k}}$
$ \begin{align*}\frac{{{x^{J + 1}} - 1}}{{x - 1}} &= {x^{J}} + \frac{{{x^J} - 1}}{{x - 1}}\\&= {x^{J + 1}} + \sum\limits_{k = 0}^J {{x^k}}\\&= \sum\limits_{k = 0}^{J + 1} {{x^k}} \end{align*}$
$\displaystyle \begin{align*}x^n + x^{n-1} + x^{n-2} + \ldots + 1 &= x^n + (x^{n-1} + x^{n-2} + \ldots + 1) \\ &=x^n + \frac{x^n - 1}{x-1} &\text{by the induction hypothesis} \\ &= \frac{x-1}{x-1}x^n + \frac{x^n - 1}{x-1}\end{align*}$
I'm sure you can finish from there.