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Problem on sequences

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

May 14, 2018
Sequences

Problem Statement

The sequence (an)(a_n) is given by a1=x∈Ra_1 = x \in \mathbb{R} and 3an+1=an+13a_{n+1} = a_n+1 for n≥1n \geq 1. Set
A=∑n=1∞[an−16]A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big], B=∑n=1∞[an+16]B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big].
Compute the sum A+BA+B in terms of xx.
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