The Fibonacci Sequence

The Fibonacci sequence is given by

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . .

The nth Fibonacci number is denoted by fn. For example,

f1 = 1, f2 = 1, f3 = 2, f4 = 3, f5 = 5 . . .

Also, many authors assume that f0 = 0.

Solution. The first 20 Fibonacci numbers are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765.

The table below has three columns. The first column simply lists some positive integers n. The second column shows the nth Fibonacci number to the (n - 1)th (or the previous) Fibonacci number.

If we plot the values in the third column as a line graph, we obtain the following:

So as n gets larger, the values in the third column tend to converge at a specific value 1.61803. Technically, we say "fn/fn-1 approaches 1.61803 as n approaches infinity."

Mathematicians have worked on this ratio for hundreds of years and they found out that its value, when rounded off to 20 decimal places is

1.6180339887498948482

and with exact value of

This value is called the golden ratio and is usually denoted by the Greek letter ϕ. Like the other important mathematical constants π and e, the golden ratio ϕ is also an irrational number. This means that it has a non-repeating and non-terminating decimal expansion.

The golden ratio is actually more interesting that how much we will be using it for in this lecture. The ancient Greeks believed that it is the "standard of beauty". Since antiquity, the golden ratio is believed to be present in nature and is applied by many artists and architects in their work. For more about this, you can start with the following video.

In this lecture, we will only be using the golden ratio for a specific purpose -- Binet's formula. Binet's formula is a special formula designed to give us the nth Fibonacci number without actually generating the first (n - 1) numbers in the sequence. Without this formula, if we are asked for, say f20, we would need to generate the first 19 Fibonacci numbers before reaching the 20th. This can be particularly tedious.

Binet's formula is given by

where

Again, ϕ is called the golden ratio while ϕ' is usually referred to as the complement of the golden ratio.

Solution. Using Binet's formula:

Observe that the formula works better if we will be more precise with our value for ϕ and ϕ'. This means that the more decimal places we use for both constants, the more accurate our answer would be.

[1] The Story of Fibonacci. From https://www.fibonicci.com/fibonacci/.

[2] Encyclopedia Britanncca. Fibonacci. From https://www.britannica.com/biography/Fibonacci

1. What is the Rabbit Problem and how is it related to the Fibonacci sequence?

2. Explain how to generate the Fibonacci sequence.

3. What is the value of the golden ratio and how was it obtained from the Fibonacci sequence?

4. Is it possible to obtain the nth Fibonacci number without actually generating the first few terms of the Fibonacci sequence? How?