Intuition behind the finite geometric series formula?

You can visualize it by looking this identity as identity of polynomials: $$ (1-x)(1+x)=1-x^2 $$ $$ (1-x)(1+x+x^2)=1-x^3 $$ and in general $$ (1-x)(1+x+x^2+x^3+\ldots+x^{n-1})=1-x^n $$

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Intuition

Geometric Series

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