MATH1061/7861, Thu 3 Sep
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Question 1. Consider the sequence b0 = 0, b1 = 1, and bi = (bi−1 + bi−2 + 1)/2 + 3(i − 1) for each integer i ≥ 2. Calculate the first few terms of the sequence, guess an explicit formula, and prove your explicit formula correct.
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Question 2. Consider the sequence defined by d0 = 1 and di = di−1 / (1 + di−1) for each integer i ≥ 1. Calculate the first few terms of the sequence, guess an explicit formula, and prove your explicit formula correct.
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Challenge. Consider the sequence {ai} for n ≥ 0. Prove that if {ai} is both an arithmetic sequence and a geometric sequence, then it is a constant sequence (that is, a0 = a1 = a2 = …).
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Question 4. The Tower of Hanoi: Let Ti denote the smallest number of moves required for i discs.
◦ Compute T0, T1, T2 and T3.
◦ Write a recurrence relation for the sequence {Ti}. ◦ Prove your recurrence relation correct!
◦ Guess an explicit formula for the Ti.
◦ Prove your explicit formula correct!
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