My starting number is 7 and my second number is 5
For Mini Project #2 we’ll be exploring Math in Nature and looking at some interesting patterns of numbers!
The entire project needs to be TYPED UP and submitted as A SINGLE DOCUMENT.
Due Friday March 22nd (by 11:59 pm)
Recall the Lucas Sequence: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …
And the Fibonacci Sequence: 1,1,2,3,5,8,13,21,34, 55, 89,144, 233, 377, …
For this mini project: You are charged with making up the _____(insert your name here)____ Sequence of numbers that is different, yet similar to the sequences we explored in class.
When making up your own Fibonacci type sequence, start by picking any two numbers to start BUT !!!!! There is one MAJOR rule, you CANNOT start with the numbers 1 and 1, or 1 and 2, or 2 and 3, or 3 and 5, or 5 and 8, and so on from the Fibonacci Sequence. You also CANNOT start with the numbers 1 and 3, or 3 and 4, or 4 and 7, or 7 and 1, or 11 and 18, and so on from the Lucas Sequence.
Your starting numbers must be pre-approved.
NO TWO PEOPLE CAN HAVE THE SAME SEQUENCE – SO YOUR STARTING NUMBERS NEED TO BE PRE-APPROVED.
Once your starting numbers are approved, complete the following:
1. Name your sequences and clearly state the first two terms.
2. Generate a list of the first 30 numbers of your sequence. THEN
For each question/pattern below list show at least five (5) examples from your sequence to support that the relation holds true, OR five (5) counter examples that shows the relation does not hold true for your created sequence.
FOR full credit you MUST:
show all calculations,
write up an explanation for each problem and it must be in complete sentences explaining what you are seeing in your calculations
and explicitly state whether the conditions holds true or not for your sequence.
If the condition does not hold true, explain the pattern you are seeing with your numbers. Note: many WILL work, but it will just be a different number you will be seeing for your sequence.
You will be testing the following 3 patterns:
3. Test if the sum of every ten consecutive numbers in your sequence is divisible by 11. If it is not divisible by 11, does each sum have a different common factor?
4. The product of any two alternating numbers differs from the square of the middle number by 1. If the product and square does not differ by one, does each trial differ by the same number? Is this an important number in your sequence?
5. The sum of any number of (insert your sequence name) Numbers beginning with the first number of the sequence is always equal to 1 less the than the second number beyond the last one added. If the sum does not differ by one from the second number beyond, does each trial differ by the same number? Is this an important number in your sequence?
6. Attempt to estimate the golden ratio with 5 trials from your sequence. How close does your sequence approximate the golden ratio? Can you get to the tenth decimal place of the golden ratio (list at least 5 trials with your sequence numbers and their decimal approximation)? You will need to use a web-based calculator or google the ratio to the tenth decimal place, do not use your calculator for these values if the calculator cuts the number off at the ninth or tenth decimal place.
Finally research (i.e., google) the connection between the Fibonacci Sequence, the Golden Rectangle, and the Golden Ratio. Below are a few links that should help!
7. For the report you need to write up a (150 – 300 word) paragraph discussing the relation between the Fibonacci Sequence, the Golden Rectangle, and the Golden Ratio. In this write up make sure to fully define each of these separately and then explain how they are connected, and which can be observed in nature (and where!). Include in this write up at least five (5) examples of where we can see the golden ratio in nature.
Sources you can use:
www.livescience.com/37470-fibonacci-sequence.html
proofwiki.org/wiki/Properties_of_Fibonacci_Numbers
HERE ARE YOUR APPROVED STARTING NUMBERS: Note, two of you I had to change your numbers (Logan and Harrison)
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