91Ó°ÊÓ

To solve quadratic equations like our characteristic equation, we use the quadratic formula. This formula is essential for finding the roots of any polynomial of the form \( ax^2 + bx + c = 0 \). The roots can be found using:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, the characteristic equation is \( r^2 - ar - b = 0 \). Here, \( a = 1 \), \( b = -a \), and \( c = -b \). Plugging these into the quadratic formula gives us:
\[ r = \frac{a \pm \sqrt{a^2 + 4b}}{2} \]
This formula helps us find the roots \( r_1 \) and \( r_2 \) efficiently. Remember, the roots are the values of \( r \) that satisfy the quadratic equation. Using these roots, we solve recurrence relations and analyze long-term behavior.

A recurrence relation is a way of defining sequences based on previous terms. In this exercise, we are given the relation:
\[ x_{n+1} = a x_n + b x_{n-1} \]
To solve such a relation, we use the characteristic equation \( r^2 - ar - b = 0 \) to find the roots \( r_1 \) and \( r_2 \). The general solution of the recurrence relation is then:
\[ x_n = c_1 r_1^n + c_2 r_2^n \]
where \( c_1 \) and \( c_2 \) are constants determined by initial conditions. This form leverages the roots from our characteristic equation to build a formula for future terms of the sequence.

Asymptotic behavior examines how a sequence behaves as \( n \) approaches infinity. For the sequence \( x_n = c_1 r_1^n + c_2 r_2^n \), the most important factor is the value of the roots \( r_1 \) and \( r_2 \). If \( |r_1| > 1 \), the term \( r_1^n \) grows exponentially, dominating the behavior of the sequence.
Given that \( r_1 > r_2 \) and \( r_1 > 1 \), \( r_1^n \) will grow larger very quickly, causing \( x_n \) to approach infinity. Thus, we conclude:
\[ \lim_{n \to \infty} x_n = \infty \]
Understanding the asymptotic behavior is critical for predicting how sequences evolve over time, especially in fields like computer science and economics, where we often deal with iterative processes.