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Category: Number Theory

Fibonacci sequence the Golden Ratio

The Fibonacci sequence is the sequence defined by:

F0=0, F1=1, Fn=Fn1+Fn2,  for  n=2,3,4,....F_0 = 0,  F_1 = 1 ,  F_n = F_{n−1 }+ F_{n−2} \textrm{,  for }  n = 2, 3, 4, . . . .


First 15 sequence values:



The number

φ=1+52=1.61803... φ = \frac{1 + \sqrt{5}} {2} = 1.61803 . . .


is called the Golden Ratio. It satisfies the property

φ=1+1φ φ = 1 + \frac{1}{φ}


Can you show that the sequence of ratio of successive Fibonacci numbers

Fn+1Fn \frac{F_{n+1}}{F_n}


converges to the Golden Ratio?

limnFn+1Fn=φ \lim_{n\to\infty} \frac{F_{n+1}}{F_n} = φ


Observe that

F14/F13=377/2331.618025 F_{14} /F_{13} = 377/233 ≈ 1.618025


is already pretty close to φ.

What conditions must be met for this ratio to converge?

Fibonacci Spiral

Extension: Missing Squares

An elegant variation of this idea makes use of two rectangles of such proportion that they fit side by side to make a perfect 8 by 8 checkerboard. When the pieces are rearranged to make the larger rectangle, there is an apparent gain in area of one square unit.  Where did the extra square come from, and what does this have to do with Fibonacci?

fibonacci squares

Pythagorean Triples

Pythagorean triples are sets of three integers that are solutions to the associated identity for right triangles.

The question is this:

Can you find a formula or algorithm for generating Pythagorean Triples?

There may be more than one formula, and each might not be exhaustive. All we are after is a set of rules such that, given one number, can you find the other two that belong to the triplet?

pythagorean theorem

Probability of Coprimality

This one is rather easy to pose, and can be difficult to solve, even with a proof sitting right in front of you. The result is a bit surprising as well, and gives some insight into the structure of the integers and prime numbers.

Given two random integers, what is the probability that they have no common factors besides 1?

or, how it is usually posed:

What is the probability of two random integers being coprime?

Where “coprime” means “the greatest common divisor is one”.

coprime number distribution

Common Fractions & Decimal Equivalents

Step 1:  Use long division (no calculators please!) to convert the fractions (in this case, specifically reciprocal integers) to decimal representations.

Step 2:  Look at the results.  What do you notice?  Write down all the questions that come up.

Step 3:  As a group, we will make a list of all the questions.

Step 4: Think and write up your answers to the questions.

Step 5:  Discuss!


Note:  Work with a partner to divide up the work in Stages 1, 2, and 4.


1/2 = __________                         1/11 = __________

1/3 = __________                         1/12 = __________

1/4 = __________                         1/13 = __________

1/5 = __________                         1/14 = __________

1/6 = __________                         1/15 = __________

1/7 = __________                         1/16 = __________

1/8 = __________                         1/17 = __________

1/9 = __________                         1/18 = __________

1/10 = __________                        1/19 = __________

     1/20 = __________


Reciprocal Integers