Sure, let's address each part of this question step by step:
a. Conjecture about each of the Fibonacci numbers after the first two.
The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones. The sequence usually starts with 1 and 1, and then is followed by the sum of these two numbers, and so forth.
Mathematically, the [tex]\(n\)[/tex]-th Fibonacci number can be defined as:
[tex]\[ F(n) = F(n-1) + F(n-2) \][/tex]
with seed values:
[tex]\[ F(1) = 1, \, F(2) = 1 \][/tex]
b. Write the next three numbers in the pattern.
Starting from the given sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34
To find the next number in the pattern, we add the last two numbers in the sequence:
[tex]\[ 21 + 34 = 55 \][/tex]
So, the next number is 55.
Next, we add the last two numbers again:
[tex]\[ 34 + 55 = 89 \][/tex]
So, the next number is 89.
Finally, we add the last two numbers one more time:
[tex]\[ 55 + 89 = 144 \][/tex]
So, the next number is 144.
Thus, the next three numbers in the Fibonacci sequence are:
[tex]\[ 55, 89, 144 \][/tex]
c. Research to find a real-world example of this pattern.
The Fibonacci sequence frequently appears in nature, showing a surprising connection between mathematics and the natural world. One prominent real-world example is the arrangement of leaves on a stem, known as phyllotaxis. This pattern optimizes sunlight exposure for leaves.
Additionally, the sequence is evident in the branching patterns of trees, the arrangement of pine cones and pineapples, and even in the flowering of artichoke and sunflower petals. It is a remarkable pattern demonstrating efficiency and aesthetic beauty in nature.
In summary:
- a. Each Fibonacci number after the first two is the sum of the two preceding numbers.
- b. The next three numbers in the pattern are 55, 89, and 144.
- c. A real-world example of the Fibonacci pattern is the arrangement of leaves on a stem and the branching patterns in trees.
