Question

The recurrence relation for the Tower of Hanoi sequence is mₖ = 2mₖ₋₁ + 1 for each integer k ≥ 2. Some of its initial values are m₁ = 1, m₂ = 3, and m₃ = 7. Use the recurrence relation and the given initial values to compute m₇ and m₈.

Answer 1

To find m_7 and m_8 in the Tower of Hanoi sequence, we apply the recurrence relation m_k = 2m_k-1 + 1 starting from the last known value, yielding m_7 = 127 and m_8 = 255.the correct answer is 255

Step-by-step explanation:

The student asked how to calculate m7 and m8 given the recurrence relation for the Tower of Hanoi sequence which is mk = 2mk-1 + 1 for each integer k ≥ 2, and the initial values m1 = 1, m2 = 3, and m3 = 7. To compute m7 and m8, we'll apply the recurrence relation successively, starting from the highest known value, m3.

• m4 = 2m3 + 1 = 2(7) + 1 = 15
• m5 = 2m4 + 1 = 2(15) + 1 = 31
• m6 = 2m5 + 1 = 2(31) + 1 = 63
• m7 = 2m6 + 1 = 2(63) + 1 = 127
• m8 = 2m7 + 1 = 2(127) + 1 = 255

Therefore, m7 = 127 and m8 = 255.