Reorganizing A Sum

Algebra Level 4

Consider the sum:

S=1+2(1x)+3(1x)(12x)+n(1x)(12x)...(1(n1)x) S = 1 + 2(1-x) + 3(1-x)(1-2x) + n(1-x)(1-2x)...(1-(n-1)x)

If S S can be written as:

S=1x[1(1a1x)(1a2x)(1a3x)(1anx)] S = \dfrac{1}{x} \left[1-(1-a_1x)(1-a_2x)(1-a_3x)\cdots(1-a_nx) \right]

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

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