Chapter 7.5, Problem 2MRPS

Solution

i.

To state: The number of new branches in each of the first six stages if a fractal tree starts with a single branch (the trunk). At each stage, the new branches from the previous stage each grow two more branches as shown. The given diagram is:

  

The resultant terms are $1,2,4,8,16,32$ .

Given information:

A fractal tree starts with a single branch (the trunk). At each stage, the new branches from the previous stage each grow two more branches.

Explanation:

A fractal tree starts with a single branch (the trunk). It means the first term is 1.

At each stage, the new branches from the previous stage each grow two more branches. Then the second term will become $a_2=2\text{or}{a}_2=2^1$ .

The third term is: $a_3=4\text{or}{a}_3=2^2$

The fourth term is: $a_4=8\text{or}{a}_4=2^3$

Similarly the fifth and sixth terms are:

  $\begin{array}{l}{a}_5=16\text{or}{a}_5=2^4\\ a_6=32\text{or}{a}_6=2^5\end{array}$

Therefore, the first six terms are $1,2,4,8,16,32$ .

ii.

To state: Whether the sequence of numbers from part (a) is arithmetic, geometric or neither.

The sequence is a geometric sequence.

Given information:

The sequence from part (a) is $1,2,4,8,16,32$ .

Explanation:

From part (a) the sequence is:

  $1,2,4,8,16,32$

Check the sequence if it is arithmetic or not for doing so find the common difference. If the common difference is same then the sequence is considered as an arithmetic sequence.

  $\begin{array}{l}d=2-1,d=1\\ d=4-2,d=2\\ d=8-4,d=4\end{array}$

Now check the sequence if the sequence is geometric or not for doing so find the common ratio. If the common ratio is same then the sequence is considered as a geometric sequence.

  $\begin{array}{l}r=\frac{2}{1},r=2\\ r=\frac{4}{2},r=2\\ r=\frac{8}{4}r=2\end{array}$

Therefore, the sequence is geometric sequence.

iii.

To state: The explicit rule and a recursive rule for the sequence in part (a).

The explicit rule is $a_n=2^{n-1}$ and the recursive rule is $a_1=1,a_n=2a_{n-1}$ .

Given information:

The sequence from part (a) is $1,2,4,8,16,32$ .

Explanation:

Consider the given sequence: $1,2,4,8,16,32$

Find the first few terms of the sequence by substituting the values of n as:

  $1,2,3,4,5,6,7$

The terms can be written as:

  $\begin{array}{l}{a}_1=2^0,a_1=1\\ a_2=2^1,a_2=2\\ a_3=2^2,a_3=4\\ a_4=2^3,a_4=8\end{array}$

Thus, the explicit rule can be written as:

  $a_n=2^{n-1}$

The recursive rule can be obtained by writing the terms as:

  $\begin{array}{l}{a}_1=2^0,a_1=1\\ a_2=2.a_1,a_2=2\\ a_3=2.a_2,a_3=4\\ a_4=2.a_3,a_4=8\end{array}$

The recursive rule is: $a_1=1,a_n=2a_{n-1}$