COT 3100 Intro. To Discrete StructuresHomework #7 Problem 1 Fill in blanks [10 points] 1. A second-order linear homogeneous recurrence relation with constant coefficients is a recurrence relation of the form 𝒂𝒌 = 𝑨𝒂𝒌−𝟏 + 𝑩𝒂𝒌−𝟐for all integers k ≥ 2, where A and B are fixed real number, B≠0. 2. Given a recurrence relation of the form𝑎𝑘 = 𝐴 𝑎𝑘−1 + 𝐵 𝑎𝑘−2 for all integers𝑘 ≥ 2, the characteristic equation of the relation is 𝒓𝟐 − 𝑨𝒓 − 𝑩 = 𝟎. 3. If a sequence𝑎1 , 𝑎2 , 𝑎3 , ⋯is defined by a second-order linear homogeneous recurrence relation with constant coefficients and the characteristic equation for the relation has two distinct roots𝑟and𝑠(which could be complex numbers), then the sequence is given by an explicit formula of the form 𝒏𝟐 = 𝑮(𝒓)𝒏 + 𝒄𝟐 (𝒔)𝒏 , 𝒏 ≥ 𝟎, 𝑮, 𝒄𝟐 ∈ 𝑹 4. If a sequence𝑎1 , 𝑎2 , 𝑎3 , ⋯is defined by a second-order linear homogeneous recurrence relation with constant coefficients and the characteristic equation for the relation has only a single root𝑟, then the sequence is given by an explicit formula of the form 𝒏𝟐 = (𝑮 + 𝒄𝟐 𝒏)𝒓𝒏 , 𝒏 ≥ 𝟎, 𝑮, 𝒄𝟐 ∈ 𝑹 Problem 2 [15 points] Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence. 𝑎𝑘 = 7𝑎𝑘−1 − 10𝑎𝑘−2 ...
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