Chapter 8.5, Problem 63E

a.

To determine

To list the number of new branches in each of the first seven stages and what type of sequence do these numbers form.

Answer to Problem 63E

  $a_7=13$ . Arithmetic sequence

Explanation of Solution

Given:

At each stage, each new branch from the previous stage grows two more branches.

Calculation:

The first tree has single branch so,

  $a_1=1$

Therefore the sequence $a_n$ is as follows-

  $\begin{array}{l}{a}_n=a_{n-1}+2\\ a_7=1+(7-1)(2)\\ a_7=1+12=13\end{array}$

Arithmetic sequence is formed.

b.

To determine

To write explicit rule and a recursive rule for the sequence in part (a).

Answer to Problem 63E

  $a_n=a_{n-1}+2$ , $a_n=2n-1$

Explanation of Solution

Given:

At each stage, each new branch from the previous stage grows two more branches.

Calculation:

The following is the recursive rule-

  $a_n=a_{n-1}+2$

And the following is the explicit rule-

  $\begin{array}{l}{a}_n=1+(n-1)2\\ a_n=1+2n-2\\ a_n=2n-1\end{array}$