Application of Geometric Progression

Algebra Level 2

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

2 ( 1 2 ) 20 2-(\frac{1}{2})^{20} 2 + ( 1 2 ) 20 2+(\frac{1}{2})^{20} 2 + ( 1 2 ) 19 2+(\frac{1}{2})^{19} 2 ( 1 2 ) 19 2-(\frac{1}{2})^{19}

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