# 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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