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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