Method of Proof (Computer Science)

Question

Prove that $1+r+r^{2}+...+r^{n-1}=\frac{r^{n}-1}{r-1}$, $r$ a positive integer, $r \neq 1$

Answer 1

$S=1+r+...r^{n-1}$
$rS= r+...r^{n}$
therefore $rS-S=r^n-1$ which means $S=\frac{r^n-1}{r-1}$

Answer 2

Note that for $r \neq 1$, $(r-1)(1+r+r^2+...+r^{n-1}) = r^n -1$, now just divide each side by $r-1$.

Answer 3

Hint: $$\frac{r^n-1}{r-1}+r^n=\frac{r^n-1+r^{n+1}-r^n}{r-1}=?$$

Answer 4

$\dfrac{r^n - 1}{r-1} = \dfrac{r^n - r^{n-1}}{r-1} + \dfrac{r^{n-1} - r^{n-2}}{r-1} + ...+ \dfrac{r - 1}{r-1} = r^{n-1} + r^{n-2} + ... + 1$