Problem #PRU-61458

Problems Algebra and arithmetic Sequences Recurrent relations Linear recurrent relations Methods Mathematical induction

Problem

Definition. The sequence of numbers $a_{0}, a_{1}, \dots, a_{n}, \dots$ , which, with the given $p$ and $q$ , satisfies the relation $a_{n + 2} = p a_{n + 1} + q a_{n}$ ( $n = 0, 1, 2, \dots$ ) is called a linear recurrent sequence of the second order.

The equation $x^{2} - p x - q = 0$ is called a characteristic equation of the sequence ${a_{n}}$ .

Prove that, if the numbers $a_{0}$ , $a_{1}$ are fixed, then all of the other terms of the sequence ${a_{n}}$ are uniquely determined.

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