Answer:
[tex]1+r+r^{2}+r^{3}+.....+r^{n-1} = \frac{1-r^{n}}{1-r}[/tex]
Step-by-step explanation:
Taking the succession:
[tex]1+r+r^{2}+r^{3}+.....+r^{n-1}[/tex]
You can multiply and divide by 1-r without chaging the result:
[tex]\frac{1-r}{1-r} (1+r+r^{2}+r^{3}+.....+r^{n-1})[/tex]
Distributing the upper part of the fraction you have:
[tex]\frac{1}{1-r} (1-r+r-r^{2}+r^{2}-r^{3}+r^{3}-r^{4}+.....+r^{n-1}-r^{n})[/tex]
As can be seen all the intermediate members will be canceled by a same member with opposite sign, only [tex](1-r^{n})[/tex] will be left so:
[tex]1+r+r^{2}+r^{3}+.....+r^{n-1} = \frac{1-r^{n}}{1-r}[/tex]
