Infinity infinity

Calculus Level 4

$\large S=\sum_{n=1}^{\infty} \dfrac{q_{n}}{3^{n}}$

Let the $$q_{n}$$ be a defined sequence for $$q_{1}=1$$ , $$q_{2}=2$$ , $$q_{3}=3$$ such that it satisfy the recurrence relation $$q_{n+3}=q_{n+2}-q_{n+1}+q_{n}$$ for positive integer $$n$$.

Compute $$S$$.

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