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(answered) – 1) Let Wn be the number of strings of length n formed fromDescriptionSolution downloadThe Question1) Let Wn be the number of strings of length n formed from letters A, B, C, and D that do not contain a substring AA, BA or CA. For example, for n = 2, all the strings with this property are AB, AC, AD, BB, BC, BD, CB, CC, CD, DA, DB, DC, DD and thus W2 = 13. (Note that W0 = 1, because the empty string satisfies the condition.)(a) Derive a recurrence relation for the numbers Wn. Justify it.(b) Find the formula for the numbers Wn by solving this recurrence. Show your work.CS/MATH111 ASSIGNMENT 3Problem 1: Let Wn be the number of strings of length n formed from letters A, B, C, and D that do notcontain a substring AA, BA or CA. For example, for n = 2, all the strings with this property areAB, AC, AD, BB, BC, BD, CB, CC, CD, DA, DB, DC, DDand thus W2 = 13. (Note that W0 = 1, because the empty string satis?es the condition.)(a) Derive a recurrence relation for the numbers Wn . Justify it.(b) Find the formula for the numbers Wn by solving this recurrence. Show your work.Problem 2: Solve the following recurrence equation:fn=13fn?2 + 12fn?3 + 2n + 1f0=0f1=1f2=1Show your work (all steps: the associated homogeneous equation, the characteristic polynomial and its roots,the general solution of the homogeneous equation, computing a particular solution, the general solution ofthe non-homogeneous equation, using the initial conditions to compute the ?nal solution.)Problem 3: Solve the following recurrence equation:tn=tn?1 + 2tn?2 + 3nt0=0t1=4Show your work (all steps: the associated homogeneous equation, the characteristic polynomial and its roots,the general solution of the homogeneous equation, computing a particular solution, the general solution ofthe non-homogeneous equation, using the initial conditions to compute the ?nal solution.)1

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