Consider the sum
S=a+a w+a w^{2}+\cdot\cdot\cdot\cdot+a w^{n-1}=\sum_{i=1}^{n}a w^{i-1}of the first n terms in a geometric series. Show that the sum S in the above formula equal
\sum_{i=1}^{n}a w^{i-1}={\frac{a(1-w^{n})}{1-w}},provided that 𝑤 ≠ 1. (For 𝑤 = 1, we have trivially S = na.)