Chapter ISG, Problem 7.10.10P

a.

To determine

To discuss: The reason why the given recursive formula makes sense.

Answer to Problem 7.10.10P

Recursive formula makes sense due to the bifurcating node structure of the method used to solve the problem.

Explanation of Solution

Given Information: Problem is regarding ‘Tower of Hanoi’.

Calculation:

By solving the problem for some initial values manually, sequence is obtained as,

  $\text{1,3,7,15}\mathrm{.....}$

Analyzing the pattern it can be inferred that explicit formula is,

  $a_n=2^n-1$$-(i)$

  $a_{n-1}=2^{n-1}-1$$-(ii)$

From $(i)$ and $(ii)$ ,

  $\begin{array}{l}{a}_{n-1}=\frac{2^n}{2}-1=\frac{2^n-2}{2}=\frac{2^n-1}{2}-\frac{1}{2}\\ =\frac{a_n-1}{2}\\ \Rightarrow a_n=2a_{n-1}+1\end{array}$

  $\Rightarrow a_n=a_{n-1}+1+a_{n-1}$

Thus, recursive formula makes sense due to the bifurcating node structure of the method to solve the problem.

b.

To determine

To simplify: The recursive formula and write what is $a_1$ and state the reason behind answer.

Answer to Problem 7.10.10P

Simplified form is $a_n=2a_{n-1}+1$ and $a_1=1$.

Explanation of Solution

Given Information: Problem is regarding ‘Tower of Hanoi’, explicit formula from previous part.

Calculation:

  $a_n=a_{n-1}+1+a_{n-1}$

  $a_n=2a_{n-1}+1$

  $a_1=1$ , because only one move is required to place a single disk from the leftmost peg to the rightmost peg.

Thus, simplified form is $a_n=2a_{n-1}+1$ and $a_1=1$.

c.

To determine

To find: The number of moves it takes for a game of $8$ disks.

Answer to Problem 7.10.10P

For $8$ disks, $255$ moves are required.

Explanation of Solution

Given Information: Problem is regarding ‘Tower of Hanoi’, spreadsheet should be used.

Calculation:

Using formula feature of spreadsheet software,

  

Thus, for a game of $8$ disks, $255$ moves are required.

d.

To determine

To write: An explicit formula for $a_n$.

Answer to Problem 7.10.10P

The explicit formula is $a_n=2^n-1$.

Explanation of Solution

Given Information: Problem is regarding ‘Tower of Hanoi’.

Calculation:

Using formula feature of spreadsheet software,

  

The sequence of moves required is $1,3,7,15,31,\mathrm{63.....}$

Careful examination of the sequence unveils the pattern,

Explicit formula: $a_n=2^n-1$

Thus, the explicit formula is $a_n=2^n-1$.