Fibonacci Sequence part 2

This week is part two in a three-park series on the Fibonacci Sequence. However, before we get to that, I present the usual weekly logic puzzle.

Logic Puzzle

You are with two gods, one of which always speaks the truth and the other always lies. However, they both speak a foreign language in which the words for "yes" and "no" are, in no particular order, "ja" and "da." You may ask one of them one yes/no question, which must have a clear yes/no answer (so no paradoxes). Your task is to determine which god is which. What should you ask?

Fibonacci Sequence part 2

This week we continue our look at the Fibonacci Sequence. Before going further, let me define:

F_n= n^th number in Fibonacci Sequence.

As explained in last week’s newsletter:

F₁ = 1

F₂ = 2

F_n = F_n-1 + F_n-2 ,for n>2

So, the first ten numbers in the Fibonacci Sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

The question I pose is, as n approaches ∞, what is Fn Fn-1 ?

Let’s use the symbol Φ to represent this ratio of Fn Fn-1 as n approaches ∞.

Fn-2 Fn-1 the ratio of a term in the Fibonacci Sequence to the term ahead of it. As n approaches infinity, this will be the same as Fn-1 Fn = Φ

So, we have:

Since the Fibonacci is an increasing series, the only reasonable solution is 1 + √5 2 = ~ 1.61803398874989

This is known as the Golden Ratio and appears in mathematics all over the place.

One instance is if a rectangle has sides of a and b such that a/b = (a+b)/b, then a/b = Φ.

Another place you see the Golden Ratio is in a pentagram. In the image above, the ratios of the red to green, green to blue and blue to pink are all equal to the Golden Ratio.

Next week I will build on this lesson to show the formula for any given term in the Fibonacci Sequence.

Logic Puzzle Answer

Is "ja" is the word for "yes"? If you get a "ja" answer, you asked the truthful person. Otherwise, with a "da" answer, you asked the lying person.

The above is my short and simple answer. There are likely other possible answers.

Logic Puzzle Solution

Let's examine the four possible combinations, crossing you you are asking and the word for "yes."

Asking = truthful person, Yes = ja: Since ja means yes, he will answer truthfully and affirmatively with "ja."

Asking = truthful person, Yes = da: Since ja means no, he will answer turthfully and negatively with "ja."

Asking = lying person, Yes = ja: Since ja means yes, the correct answer is "yes" or "ja." However, you're asking the lying person so he will reverse the answer and say "da."

Asking = lying person, Yes = da: Since ja means no, the correct answer is "no" or "ja." However, you're asking the lying person so he will reverse the answer and say "da."

Note that you get a "ja" answer if you ask the truthful person, regardless of the word for "yes." Likewise, you get an answer of "da" if you ask the lying person regardless of the word for "yes."