Sum of n terms of a G.P. confusion in formulas

Question

If we have to find the sum of n terms of a G.P. then we have two formulas for it (1) $a(1-r^n)/(1-r)$ and (2) $a(r^n-1)/(r-1)$. Now I know how the (1) has been derived but dont know about the (2)(is it obtained by multiplying denominator and numerator of (1) by -1?). I am also confused when two use them and why there exists 2 formulas for the same objective? pls explain it.

Answer 1

As you have mentioned in your post and as Dr. Sonnhard Graubner has mentioned in his answer, you can get one expression from the other by multiplying by $-1$ on the numerator and denominator. Which one you use is just a matter of preference. In particular, when working with geometric series where $r > 1$, I would guess that people prefer to work with (2), since $r^n - 1$ and $r-1$ are positive quantities, so the resulting fraction is more aesthetic, but, for the same reason, (1) will tend to be used when $r < 1$.

Answer 2

Multiply numerator and denominator by $(-1)$ $$\frac{a(r^n-1)(-1)}{(r-1)(-1)}$$