The formula for sum of a geometric sequence can be derived from the formula for general terms of a geometric sequence.

Using the formula for the n^{th} term of a geometric sequence,

a_n = a_1r^{n-1}

in which a_1 is the first tirm, r is the common ratio.

a_1 = a_1r^0

a_2 = a_1r^1

a_3 = a_1r^2

\dots

\begin{aligned} S_n&=a_1+a_2+\dots+ a_n \\ &=a_1(1+r^1+r^2+\dots+r^{n-1}) \\ &= \dfrac{a_1(r-1)(1+r^1+r^2+\dots+r^{n-1})}{r-1} \\ &= \dfrac{a_1(r+r^2+\dots+r^n-1 -r-r^2-\dots-r^{n-1})}{r-1} \\ &= \dfrac{a_1(1-r^n)}{1-r} \end{aligned}

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}