MathDB

Let

\left\{x_n\right\}

, with n \equal{} 1, 2, 3, \ldots, be a sequence defined by x_1 \equal{} 603, x_2 \equal{} 102 and x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}

\forall n \geq 1

. Show that: (1) The number

x_n

is a positive integer for every

n \geq 1

. (2) There are infinitely many positive integers

n

for which the decimal representation of

x_n

ends with 2003. (3) There exists no positive integer

n

for which the decimal representation of

x_n

ends with 2004.

Problem Statement