A long time ago when I was a high school students I've been taught the formula to sum the first $n$ terms of following geometric series
$$a_1,a_1q,a_1q^2,\ldots,a_1q^{n-1}$$
Is $S_n=\dfrac{a_1(1-q^n)}{1-q}$ . However when we asked the teacher why not writing it in easier form as $S_n=\dfrac{a_1(q^n-1)}{q-1}$, our teacher told us "because later we are going to learn about infinite sum of geometric series for $|q|<1$ and memorizing the formula in this way is easier to learn that" the formula for this case is $S_{\infty}=\dfrac{a_1}{1-q}$. And it is true it was better to learn the formula for $S_n$ as the former equation.
Here are my questions:
First: Is there another reason to encourage memorize formula as $S_n=\dfrac{a_1(1-q^n)}{1-q} ?$
Second: Now it is past about $5$ years since I learned about the formula for the first time. and I solved a lot of geometric series problem since that time( almost used the former formula every time). So can I use $S_n=\dfrac{a_1(q^n-1)}{q-1}$ in my head? doesn't that hurts? ( I want to use this because it is easier to calculate for example, $5^1+5^2+5^3+\ldots+5^{10}$ with $S=\dfrac{5(5^{10}-1)}{5-1}$ rather than having this unnecessary minus sign and see this as $S=\dfrac{5(1-5^{10})}{1-5}$