Let r be the common ratio of the series where | r | < 1, a₁ be the first term and a_n be the the nth term.

S = a₁ + a₂ + a₃ + ... + a_n

\( r = \frac{a_2}{a_1} = \frac{a_3}{a_2}\)

S = a₁ + r a₁ + r²a₁ + r³a₁+... + r^n-1a₁

\(\frac{S}{a_1} = 1 + r + r^2 +r^3 + ... + r^{n-1}\)

\(\frac{rS}{a_1} = r + r^2 + r^3 +r^4 + ... + r^{n-1}+ r^{n}\)

\(\frac{rS}{a_1} =( \frac{S}{a_1} - 1)+ r^{n}\)

\(\frac{rS}{a_1} - \frac{S}{a_1} = r^{n} - 1 \)

\(\frac{rS-S}{a_1} = r^{n}\)-1

\(\frac{S(r-1)}{a_1} = r^{n}\)-1

\( S = a_1 \frac{r^{n}-1}{r-1}\)

OR

\( S = a_1 \frac{1-r^{n}}{1-r}\)