Chapter 8.1, Problem 61E

(a)

To determine

The rule for sequence.

Answer to Problem 61E

  $a_n=2^n-1$

Explanation of Solution

Given: In a puzzle called the Tower of Hanoi, the object to use a series of moves to take the rings from one peg and stack them in order on another peg. A move consist of moving exactly one ring and no ring may be placed on top of smaller ring. The minimum number $a_n$ moves required to move $1$ rings is $1$ , $3$ for $2$ rings, $7$ for $3$ rings, $15$ for $4$ rings and $31$ for $5$ rings.

  

The pattern form of the series is

  $\begin{array}{l}{a}_1=1=2^1-1\\ a_2=3=2^2-1\\ a_3=7=2^3-1\\ a_4=15=2^4-1\\ a_5=31=2^5-1\\ \mathrm{..}\end{array}$

Thus the last term of the sequence is

  $a_n=2^n-1$

Hence, $a_n=2^n-1$

(b)

To determine

To Find: The minimum number of moves for $6$ rings, $7$ rings, $8$ rigs?

Answer to Problem 61E

  $63,127,255$

Explanation of Solution

Given: In a puzzle called the Tower of Hanoi, the object to use a series of moves to take the rings from one peg and stack them in order on another peg. A move consist of moving exactly one ring and no ring may be placed on top of smaller ring. The minimum number $a_n$ moves required to move $1$ rings is $1$ , $3$ for $2$ rings, $7$ for $3$ rings, $15$ for $4$ rings and $31$ for $5$ rings.

  

Thus the last term of the sequence is

  $a_n=2^n-1$

Substitute $n=6,7,8$

We get

  $\begin{array}{l}{a}_6=2^6-1=63\\ a_7=2^7-1=127\\ a_8=2^8-1=255\end{array}$