Two algebraic forms for the sum of the first terms of a geometric progression are and

MATHEMATICS · Subjective · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Two algebraic forms for the sum of the first

n

terms of a geometric progression are

S_n = a\dfrac{r^n - 1}{r - 1}

and

S_n = a\dfrac{1 - r^n}{1 - r}

. Prove that these two expressions are identical for

r\neq 1