Chapter 6.7, Problem 52E

(a)

To determine

The recursive rule for the length of the bones.

Answer to Problem 52E

The recursive rule is:

  $\begin{array}{l}{a}_1=9.5,\\ a_2=6,\\ a_n=a_{n-2}-a_{n-1}\end{array}$

Explanation of Solution

Given information:

The X-ray shows the lengths (in centimetres) of bones in a human hand.

  

Formula used:

Beginning with the third term, each term is the difference of the two previous terms.

Calculation:

Use a table to organize the terms & find the pattern:

  

The sequence does not have a common difference and not common ratio, so it is neither arithmetic nor geometric.

Find the difference of each pair of consecutive terms:

  $\begin{array}{l}{a}_1-a_2=9.5-6=3.5\leftarrow a_3\\ a_2-a_3=6-3.5=2.5\leftarrow a_4\\ a_3-a_4=3.5-2.5=1.5\end{array}$

Beginning with the third term, each term is the difference of the two previous terms.

Therefore, the recursive rule is:

  $\begin{array}{l}{a}_1=9.5,\\ a_2=6,\\ a_n=a_{n-2}-a_{n-1}\end{array}$

Conclusion:

The recursive rule is:

  $\begin{array}{l}{a}_1=9.5,\\ a_2=6,\\ a_n=a_{n-2}-a_{n-1}\end{array}$

(b)

To determine

The length of the different sections.

Answer to Problem 52E

The different sections of our hand do not have a pattern.

Explanation of Solution

Given information:

The X-ray shows the lengths (in centimetres) of bones in a human hand.

  

Formula used:

X-ray image is used.

Calculation:

No. The different sections of our hand do not have a pattern.

Conclusion:

The different sections of our hand do not have a pattern.