二阶与三阶线性递推

Introduction: From Geometric to Second-Order Recurrence

First-Order Linear Homogeneous Recurrence

What is the common ratio $q$ of the first-order recurrence $a_{k+1}+2a_k=0$?

Second-Order Characteristic Equation

What is the characteristic equation of the recurrence $a_{k+2}-5a_{k+1}+6a_k=0$?

General Terms for Distinct and Repeated Roots

What is the discriminant $\Delta$ of the characteristic equation $x^2-4x+4=0$?

Fibonacci and Binet's Formula

What is the characteristic equation of the Fibonacci sequence?

Worked Example: General Term of a Second-Order Recurrence

For the recurrence $a_{k+2}-5a_{k+1}+6a_k=0$ with $a_1=1$, $a_2=5$, find $a_3$. Enter an integer.

Find the root of the characteristic equation of the third-order recurrence $a_{k+3} - 6a_{k+2} + 12a_{k+1} - 8a_k = 0$ (triple root; enter an integer).

For the recurrence $a_{k+2}-4a_{k+1}+4a_k=0$ with $a_1=1$, $a_2=4$, find $a_3$. Enter an integer.

When the characteristic equation of a second-order linear recurrence has a repeated root, what is the form of the general term, and why is there an extra $n-1$ factor?

In your own words, explain why "trying a root" $a_n=x^{n-1}$ is reasonable for a linear homogeneous recurrence. Where does this idea come from?

Comprehensive Check

Common Pitfalls and Lesson Summary