Harmonic Mean and Infinite Sum

Algebra Level 4

\[\large \sum_{k=1}^\infty \frac{1}{3^ka_k}\]

A sequence \(\{a_n\}\) of real numbers is defined by \(a_1=20,a_2=14\) and for \(n\geq 3\), \(a_n\) is the harmonic mean of \(a_{n-1}\) and \(a_{n-2}\).

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

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