Consider the following recurrence relation and initial conditions.
$t_k = 14t_{k-1} - 49t_{k-2}$, for each integer $k \ge 2$
$t_0 = 1$, $t_1 = 7$
(a) Suppose a sequence of the form $1, t, t^2, t^3, \dots, t^n, \dots$, where $t \ne 0$, satisfies the given recurrence relation (but not
necessarily the initial conditions). What is the characteristic equation of the recurrence relation?
What value of t is a solution to this equation?
$t = $
(b) Suppose a sequence $t_0, t_1, t_2, \dots$ satisfies the given initial conditions as well as the recurrence relation. Fill in the
blanks below to derive an explicit formula for $t_0, t_1, t_2, \dots$ in terms of $n$.
It follows from part (a) and the single roots theorem that for some constants $C$ and $D$, the terms of
$t_0, t_1, t_2, \dots$ satisfy the equation $t_n = $ for every integer $n \ge 0$.
Solve for $C$ and $D$ by setting up a system of two equations in two unknowns using the facts that $t_0 = 1$ and $t_1 = 7$.
The result is that $t_n = $ for every integer $n \ge 0$.
