Harmonic Mean and Infinite Sum

Algebra Level 4

k=113kak\large \sum_{k=1}^\infty \frac{1}{3^ka_k}

A sequence {an}\{a_n\} of real numbers is defined by a1=20,a2=14a_1=20,a_2=14 and for n3n\geq 3, ana_n is the harmonic mean of an1a_{n-1} and an2a_{n-2}.

If the infinite sum above can be represented in the form of pq \frac pq for some relatively prime positive integers pp and qq, find p+qp+q.

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