Question

A fractal tree starts with a single tree branch (the trunk). At each stage, each new branch from the previous stage grows two more branches, as shown.
a) List the number of new branches in each of the first stages. What type of sequence do these numbers form?

b) Write a function that describes the relationship between the number of branches B(n) in each step n.

c) What are the total number of branches growing out of the trunk in the 8th stage?​

Answer 1

A) New branch in:

2nd stage=2

3rd stage=4

4th stage=8

which can be listed as 2,4,8,...

Here,

2nd term/1st term=4/2=2

3rd term/2nd term=8/4=2

Since common ratio is 2, the given sequence is geometric sequence.

b) The required function is,

B(n)=2^(n-1)

{just put n=1,2,3.. here to check if it gives accurate no of branches in each step or not}

where n is stage number

c)Using above function to calculate total no of branches growing out in the 8th stage,

B(8)=2^(8-1)=2^7=128