How To Find The Terms In The Fibonacci Sequence

10.4: Fibonacci Numbers and the Golden Ratio

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A famous and important sequence is the Fibonacci sequence, named after the Italian mathematician known as Leonardo Pisano, whose nickname was Fibonacci, and who lived from 1170 to 1230. This sequence is:

\[\{1,1,2,3,5,8,xiii,21,34,55, \ldots \ldots \ldots\}\]

This sequence is defined recursively. This means each term is defined by the previous terms.

then on.

The Fibonacci sequence is defined past , for all , when and .

In other words, to get the next term in the sequence, add together the two previous terms.

\[\{i,1,2,3,5,viii,13,21,34,55,55+34=89,89+55=144, \cdots\}\]

The notation that we volition use to stand for the Fibonacci sequence is as follows:

\[f_{1}=i, f_{2}=i, f_{3}=2, f_{4}=3, f_{5}=5, f_{half-dozen}=8, f_{seven}=thirteen, f_{8}=21, f_{9}=34, f_{10}=55, f_{11}=89, f_{12}=144, \ldots\]

Example \(\PageIndex{one}\): Finding Fibonacci Numbers Recursively

Find the 13th, 14th, and 15th Fibonacci numbers using the above recursive definition for the Fibonacci sequence.

First, notice that in that location are already 12 Fibonacci numbers listed higher up, and so to detect the next three Fibonacci numbers, we simply add the two previous terms to get the next term as the definition states.

Therefore, the 13th, 14th, and 15th Fibonacci numbers are 233, 377, and 610 respectively.

Computing terms of the Fibonacci sequence can be tedious when using the recursive formula, peculiarly when finding terms with a large northward. Luckily, a mathematician named Leonhard Euler discovered a formula for calculating whatever Fibonacci number. This formula was lost for about 100 years and was rediscovered by another mathematician named Jacques Binet. The original formula, known as Binet's formula, is below.

Binet's Formula: The nth Fibonacci number is given by the following formula:

\[f_{north}=\frac{\left[\left(\frac{1+\sqrt{v}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\correct)^{n}\right]}{\sqrt{5}}\]

Binet's formula is an case of an explicitly defined sequence. This ways that terms of the sequence are not dependent on previous terms.

A somewhat more user-friendly, simplified version of Binet'south formula is sometimes used instead of the one in a higher place.

Binet'southward Simplified Formula: The nth Fibonacci number is given by the following formula:

Note: The symbol means "round to the nearest integer."

Example \(\PageIndex{2}\): Finding Explicitly

Find the value of using Binet's simplified formula.

Figure \(\PageIndex{1}\): Estimator Work for

Example \(\PageIndex{3}\): Finding Explicitly

Find the value of using Binet's simplified formula.

Figure \(\PageIndex{2}\): Calculator Piece of work for

Example \(\PageIndex{four}\): Finding Explicitly

Detect the value of using Binet'due south simplified formula.

Figure \(\PageIndex{3}\): Calculator Work for

All around us we can observe the Fibonacci numbers in nature. The number of branches on some copse or the number of petals of some daisies are often Fibonacci numbers

Figure \(\PageIndex{4}\): Fibonacci Numbers and Daisies

a. Daisy with xiii petals b. Daisy with 21 petals

a. Image result for daisy flower b.

(Daisies, n.d.)

Fibonacci numbers also appear in spiral growth patterns such as the number of spirals on a cactus or in sunflowers seed beds.

Figure \(\PageIndex{5}\): Fibonacci Numbers and Spiral Growth

a. Cactus with 13 clockwise spirals b. Sunflower with 34 clockwise spirals and 55 counterclockwise spirals

a. b.

(Cactus, n.d.) (Sunflower, northward.d.)

Another interesting fact arises when looking at the ratios of consecutive Fibonacci numbers.

It appears that these ratios are approaching a number. The number that these ratios are getting closer to is a special number called the Golden Ratio which is denoted by (the Greek letter of the alphabet phi). Yous have seen this number in Binet's formula.

The Gold Ratio:

\[\phi=\frac{1+\sqrt{5}}{2}\]

The Golden Ratio has the decimal approximation of \(\phi=1.6180339887\).

The Golden Ratio is a special number for a variety of reasons. It is also called the divine proportion and it appears in art and compages. Information technology is claimed past some to be the almost pleasing ratio to the centre. To observe this ratio, the Greeks cut a length into two parts, and permit the smaller piece equal i unit of measurement. The almost pleasing cut is when the ratio of the whole length to the long piece is the aforementioned as the ratio of the long piece to the short piece 1.

ane

cross-multiply to get

rearrange to become

solve this quadratic equation using the quadratic formula.

The Gilded Ratio is a solution to the quadratic equation meaning it has the property . This means that if you want to foursquare the Golden Ratio, just add 1 to it. To cheque this, just plug in .

Information technology worked!

Another interesting relationship betwixt the Golden Ratio and the Fibonacci sequence occurs when taking powers of .

And so on.

Find that the coefficients of and the numbers added to the term are Fibonacci numbers. This tin can be generalized to a formula known as the Golden Power Rule.

Gold Power Rule: \(\phi^{north}=f_{n} \phi+f_{n-1}\)

where\(f_{n}\) is the nth Fibonacci number and \(\phi\) is the Golden Ratio.

Example \(\PageIndex{5}\): Powers of the Golden Ratio

Find the following using the gilt power rule: a. and b.

Source: https://math.libretexts.org/Bookshelves/Applied_Mathematics/Book:_College_Mathematics_for_Everyday_Life_%28Inigo_et_al%29/10:_Geometric_Symmetry_and_the_Golden_Ratio/10.04:_Fibonacci_Numbers_and_the_Golden_Ratio

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