**Lemma 8.3.1**

Let A and B be real numbers. A recurrence relation of the form

*a _{k}* =

is satisfied by the sequence 1, *t*, *t*^{2}, *t*^{3},
..., *t ^{n}*, ... where

*t*^{2} - **A***t*
- **B** = 0.

The equation *t*^{2} - A*t* - B = 0 is called the *characteristic
equation* of the recurrence relation.

**Theorem 8.3.3 Distinct Roots Theorem**

Suppose a sequence satisfies a recurrence relation

*a _{k}* = A×

for real numbers A and B, B ¹ 0, and
all integers *k* ³ 2. If the
characteristic equation

*t*^{2} - **A***t*
- **B** = 0

has two distinct roots *r* and *s*, then the sequence satisfies
the explicit formula

*a _{n} = *C×

where C and D are numbers whose values are determined by the values of *a*_{0}
and *a*_{1}.

**Theorem 8.3.5 Single-Root Theorem**

Suppose a sequence satisfies a recurrence relation

*a _{k}* = A×

for real numbers A and B, B ¹ 0, and
all integers *k* ³ 2. If the
characteristic equation

*t*^{2} - **A***t*
- **B** = 0

has a single real root *r*, then the sequence satisfies the explicit
formula

*a _{n} = *C×

where C and D are numbers whose values are determined by the values of *a*_{0}
and *any other known value of the sequence*.