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every integer occurs exactly once in the sequence.

Source: Indian Postal Coaching 2008 set 2 p1

May 25, 2020
SequencealgebraIntegerrecurrence relation

Problem Statement

Define a sequence <xn><x_n> by x0=0x_0 = 0 and xn={xn−1+3r−12if  n=3r−1(3k+1)xn−1−3r+12if  n=3r−1(3k+2)\large x_n = \left\{ \begin{array}{ll} x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\ & \\ x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\ \end{array} \right. where k,rk, r are integers. Prove that every integer occurs exactly once in the sequence.
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