discrete assignment

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COT 3100 Intro. To Discrete Structures Homework #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 __ ___ for all integers k ≥ _____, where __ ___. 2. Given a recurrence relation of the form 𝑎𝑘 = 𝐴 𝑎𝑘−1 + 𝐵 𝑎𝑘−2 for all integers 𝑘 ≥ 2 , the characteristic equation of the relation is ___ _ _. 3. If a sequence 𝑎1 , 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 roots 𝑟 and 𝑠 (which could be complex numbers), then the sequence is given by an explicit formula of the form __ _ _ _. 4. If a sequence 𝑎1 , 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 root 𝑟 , then the sequence is given by an explicit formula of the form _ ____. 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 for all integers k ≥ 2 𝑎0 = 2, 𝑎1 = 2 Problem 3 [15 points] Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence. 𝑎𝑘 = 𝑎𝑘−1 + 6𝑎𝑘−2 for all integers k ≥ 2 𝑎0 = 0, 𝑎1 = 3 Problem 4 [15 points] Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence. 𝑎𝑘 = 4𝑎𝑘−2 for all integers k ≥ 2 𝑎0 = 1, 𝑎1 = −1 Problem 5 [15 points] Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence. 𝑟𝑘 = 2𝑟𝑘−1 − 𝑟𝑘−2 for all integers k ≥ 2 𝑟0 = 1, 𝑟1 = 4 Problem 6 [15 points] Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence. 𝑟𝑘 = 4𝑟𝑘−1 − 4𝑟𝑘−2 for all integers k ≥ 2 𝑟0 = 0, 𝑟1 = 3 Problem 7 [15 points] Suppose a sequence satisfies the below given recurrence relation and initial conditions. Find an explicit formula for the sequence. 𝑟𝑘 = 2𝑟𝑘−1 + 2 𝑟𝑘−2 for all integers k ≥ 2 𝑟0 = 1, 𝑟1 = 3 Submission Requirements The following requirements are for electronic submission via Canvas. ⚫ Your solutions must be in a single file with a file name yourname-hw7. ⚫ Upload the file by following the link where you download the homework description on Canvas. ⚫ Only submissions via the link on Canvas where this description is downloaded are graded. Submissions to any other locations on Canvas will be ignored.
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