Assignment: Fibonacci Sequence Exploration
In this assignment, you will explore the Fibonacci sequence in greater depth by making and testing
hypotheses about its construction.
1. Write the first ten terms of the Fibonacci sequence
1,1,2,3,5,8,13,21,34,55
2. Find the sum of the first ten terms of the Fibonacci sequence that you generated in problem 1.
Divide the sum by 11. What do you notice?
sum is 143 , divide it by 11 =13 and 13 is in the sequence
3. Choose two numbers other than 1 and 1. The numbers do not have to be equal. Generate a
Fibonacci-like sequence, beginning with these two numbers. Write the first ten terms of your
sequence.
For example, you could choose the numbers 5 and 6 and the first two numbers. The third term
would be 11 (5 + 6), the fourth term would be 17 (6 + 11), and so on. Do NOT use 5 and 6 as
your first two terms.
3,5,8,13,21,34,55,89,144,233
4. Find the sum of the sequence you created in problem 3. Divide the sum by 11. What do you
notice?
the sum is 605 , divide it but 11 it is 55, 55 is in the sequence even after using two different starting
numbers
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5. Make a hypothesis. What is special about the sum on the first ten terms of any Fibonacci-like
sequence?
It's a known property of fibonacci numbers that the quotient of adjacent terms F_n, F_(n+1)
approaches the golden ratio as n approaches infinity. http://en.wikipedia.org/wiki/
Fibonacci_number#limit_of_consecutive_quotients
6. Create another Fibonacci-like sequence beginning with two different numbers than you chose in
problem 3. Do NOT use 5 and 6 as your first two terms. Write the first ten terms of your
sequence. Does the hypothesis you wrote in problem 5 apply to this sequence?
2,8,10,18,28,46,74,120,194,314
yes , because when you add 2+8=10 then 10+8=18 and so on
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