+26367 Find the remainder when r^13 + 1 is divided by r-1.
Geometric sequence:
\(\begin{array}{|rcll|} \hline \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} &=& \displaystyle\frac{r^{13}-1}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} &=& \displaystyle\frac{r^{13}}{r-1}- \frac{1}{r-1} \quad & | \quad + \frac{2}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} +\frac{2}{r-1} &=& \displaystyle\frac{r^{13}}{r-1}- \frac{1}{r-1} + \frac{2}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} +\frac{2}{r-1} &=& \displaystyle\frac{r^{13}}{r-1}+ \frac{1}{r-1} \\\\ \displaystyle 1+r+r^2+r^3+r^4+\ldots +r^{12} +\frac{\color{red}2}{r-1} &=& \displaystyle\frac{r^{13}+1}{r-1} \\ \hline \end{array}\)
The remainder is 2.
