Lemma 8.3.1
Let A and B be real numbers. A recurrence relation of the form

a_k = A× a_{k - 1} + B× a_{k - 2}

is satisfied by the sequence 1, t, t², t³, ..., tⁿ, ... where t is a nonzero real number, if, and only if, t satisfies the equation

t² - At - B = 0.

The equation t² - At - B = 0 is called the characteristic equation of the recurrence relation.

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Theorem 8.3.3 Distinct Roots Theorem
Suppose a sequence satisfies a recurrence relation

a_k = A× a_{k - 1} + B× a_{k - 2}

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

t² - At - B = 0

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

a_n = C× rⁿ + D× sⁿ,

where C and D are numbers whose values are determined by the values of a₀ and a₁.

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Theorem 8.3.5 Single-Root Theorem
Suppose a sequence satisfies a recurrence relation

a_k = A× a_{k - 1} + B× a_{k - 2}

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

t² - At - B = 0

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

a_n = C× rⁿ + D×n× rⁿ,

where C and D are numbers whose values are determined by the values of a₀ and any other known value of the sequence.