Proof Of sum of geometric series goes wrong!!!

Question

I'm really struggling with the way this book proves the sum of geometric series. As I know the formula of sum of geometric series should be $$s_n= a\frac{1-r^n}{1-r}$$ But the book wrote it differently as $$s_n= a\frac{1-r^{n+1}}{1-r}$$ How is this formula derived?

Answer 1

As you have been proven in the text, $$s_n = \sum_{k=0}^n ar^k = a \frac{1-r^{n+1}}{1-r}$$ You just recalled an erroneous formula.

Answer 2

As I know the formula of sum of geometric series should be $s_n= a(1-r^n)/(1-r)$

Are you sure of this? What's your definition of $s_n$?