Using the recursive formula for a geometric sequence

a_n = a_{n-1}r

We get,

a_2 = a_1r

a_3 = a_2r

a_4 = a_3r

\dots

a_n = a_{n-1}r

Add the above equations,

a_2+a_3+a_4+\dots+a_n = (a_1+a_2+a_3+\dots+a_{n-1})r

S_n-a_1=(S_n-a_n)r

(1-r)S_n = a_1-a_nr

S_n = \dfrac{a_1-a_nr}{1-r}

\quad\space\space =\dfrac{a_1-a_1r^{n-1}r}{1-r}

\quad\space\space =\dfrac{a_1(1-r^n)}{1-r}