Reorganizing A Sum

Algebra Level 4

Consider the sum:

\[ S = 1 + 2(1-x) + 3(1-x)(1-2x) + n(1-x)(1-2x)...(1-(n-1)x) \]

If \( S \) can be written as:

\[ S = \dfrac{1}{x} \left[1-(1-a_1x)(1-a_2x)(1-a_3x)\cdots(1-a_nx) \right] \]

Find : \[ \displaystyle \sum_{i=1}^n a_i \]

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