A Professor Fernando's tree grows according to the following rule:

- in the first week, the tree starts growing with just one branch;
- after growing for two weeks, this branch gives rise to a new branch per week;
- each new branch generated continues to grow, and after growing for two weeks, it gives rise to a new branch per week.

The figure below illustrates Professor Fernando's tree after five weeks from the start of its growth.

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-08.jpg?height=954&width=554&top_left_y=834&top_left_x=657)

Note that after three weeks, there were two branches; after four weeks, there were three branches; and after five weeks, there were five branches.

a) How many branches will there be after six weeks?

b) How many branches will there be after seven weeks?

c) How many branches will there be after thirteen weeks?

#
Solution

a) Following the growth rules of Professor Fernando's tree, we can continue the drawing shown in the problem statement to obtain an illustration of this tree after the sixth week of growth:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-33.jpg?height=717&width=754&top_left_y=515&top_left_x=754)

This figure shows that after the sixth week of growth, we have a total of eight branches.

b) Note that the sequence determining the number of branches per week of growth is as follows:

$$
1, 1, 2, 3, 5, 8, \ldots
$$

This sequence satisfies the following rule: starting from its third term, each term is the sum of the two preceding terms. For example, the third term is 2, which is equal to the sum of the first term 1 and the second term 1. Also, the fourth term is 3, which is equal to the sum of the third term 2 and the second term 1.

The number of branches present after the seventh week of growth will be given by the seventh term of the sequence. To generate this seventh term, we need to add the sixth term (8) to the fifth term (5). That is, the seventh term of the sequence is equal to \(8 + 5 = 13\).

c) The number of branches after 13 weeks of growth will be given by the thirteenth term of the sequence that appears in the solution to part \(b\). Remembering that each term in that sequence is given by the sum of the two preceding terms, we can generate the remaining terms as follows:

- sixth term: 8;
- seventh term: 13;
- eighth term: \(8 + 13 = 21\);
- ninth term: \(13 + 21 = 34\);
- tenth term: \(21 + 34 = 55\);
- eleventh term: \(34 + 55 = 89\);
- twelfth term: \(55 + 89 = 144\);
- thirteenth term: \(89 + 144 = 233\);

Thus, the number of branches after 13 weeks is equal to 233.

Note: The sequence obtained is:

$$
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, \ldots
$$

This is the famous Fibonacci sequence!