The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set $A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$. Compute the sum $A+B$ in terms of $x$.
Source: Federal Mathematical Competition of Serbia and Montenegro 2004
Tags: Sequences
The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set $A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$. Compute the sum $A+B$ in terms of $x$.