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x_{n+2} = x_{n+1} + 2x_n, y_{n+2} = 2y_{n+1} + 3y_n

Source: panish Mathematical Olympiad 1975 P6

December 23, 2022
algebraSequencenumber theory

Problem Statement

Let {xn}\{x_n\} and {yn}\{y_n\} be two sequences of natural numbers defined as follow: x1=1,   x2=1,   xn+2=xn+1+2xnx_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n for n=1,2,3,...n = 1, 2, 3, ... y1=1,   y2=7,   yn+2=2yn+1+3yny_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n for n=1,2,3,...n = 1, 2, 3, ... Prove that, except for the case x1=y1=1x_1 = y_1 = 1, there is no natural value that occurs in the two sequences.
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