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x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2, perfext square, matrix

Source: Romania BMO TST 1992 p2

February 19, 2020

linear algebramatrixPerfect SquareSequence

Problem Statement

For a positive integer

a

, define the sequence (

x_n

) by

x_1 = x_2 = 1

and

x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2

, for n

\ge 1

. Show that

x_n

is a perfect square and that for

n > 2

its square root equals the first entry in the matrix

\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}