Application of Geometric Progression

Algebra Level pending

Let \(\{a_n\}\) be a geometric progression with \(a_1=2\) and common ratio \(-\frac{1}{2}.\) For all \(n\in \mathbb{N}\), let us define two sets of coordinates \(P_n\) and \(Q_n\) as follows: \[P_n=(n, a_n), Q_n=(n,0).\] What is the value of \(\displaystyle \sum_{n=1}^{20}A_n\) when \(A_n\) is the area of the triangle \(P_nQ_nQ_{n+1}?\)

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