Fibonacci Series: Properties and Ratio

Content Curator

Fibonacci series is named after Italian mathematician Leonardo Pisano Bigollo. Fibonacci numbers are represented in a series of numbers in which a number is the addition of the previous two numbers. This series of numbers start with 0 and 1. For example, let the first numbers be 0 and 1. So, the third number will be the sum of two numbers i.e 1 and so on. The recurrence relation of Fibonacci number series if given by x_n+2 = x_n + x_n+1. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ………. ∞.

Read Also: Isosceles Triangle theorems

Table of Content

Fibonacci Numbers
Properties of Fibonacci numbers
Fibonacci series Spiral and Golden ratio
Fibonacci series and Pascal’s triangle
Things to Remember
Sample Questions

Fibonacci Numbers

[Click Here for Sample Questions]

Fibonacci series formula in mathematics is used to find the missing terms in a Fibonacci number sequence. The formula to find the series can be used to express the (n+1)^th term in the sequence which is defined using the recursive formula, such that F₀ = 0, F₁ = 1 to give Fn.

The formula is given as follows -

F_n = F_n-1 + F_n-2

where n should be greater than 1.

Also Read:

Sequence and Series

Geometric Progression

Properties of Fibonacci Numbers

[Click Here for Sample Questions]

In Fibonacci series, if we take any three consecutive numbers from the series and add them, then we divide the result by 2, we will get the third number.
In Fibonacci series, if we take four consecutive numbers excluding 0, we multiply the outer number and the inner number. When we subtract these numbers, then the difference will always be equal to 1.
The numbers in the Fibonacci series are related to the golden ratio. We can calculate any number of fibonacci series using the formula,

Fn = \(\phi\)ⁿ - (1-\(\phi\))ⁿ / √5

where \(\phi\) is the golden ratio.

For finding 2 consecutive numbers of the series given as F_n+1 = F_n, the value of \(\phi\) can be calculated as limn->infinity F_n+1 / F_n.

Also Check:

Geometric Mean

Differentiation and Integration formula

Fibonacci Series and Golden Ratio

[Click Here for Sample Questions]

In mathematics, there is a close connection between the Fibonacci series and golden ratio. Any number of a Fibonacci series can be calculated using the below formula -

F_n = \(\phi\)ⁿ - (1-\(\phi\))ⁿ / √5

where ? is the golden ratio.

Golden ratio is expressed in terms of Fibonacci series as the limit of ratios of successive terms of Fibonacci series (or any other Fibonacci like sequence) as represented by Kepler in the given expression.

φ = lim_n->infinity F_n+1 / F_n

In simple words, we can say that if any number of a Fibonacci number is divided by its immediate predecessor in the given series, the quotient is approximately given by φ. As the value of n increases, the approximate value also increases i.e as n tends to infinity.

There are various operations of golden ratio in the field of architecture. For example, The Great Pyramid of Egypt and the Great Mosque of Kairouan are architectural miracles in which there is an involvement of the golden ratio concept.

Read More:

Trapezoid formula

Angle between two lines

Fibonacci Series and Pascal’s Triangle

[Click Here for Sample Questions]

Using Pascal’s Triangle, there is an interesting method to find numbers in a Fibonacci series. A Pascal’s triangle in mathematics is a triangular array which consists of binomial coefficients. Fibonacci numbers can be represented by calculating the sum of elements on rising diagonal lines in Pascal’s triangle in the Fibonacci series. We can observe in the figure that considering the first element in the Fibonacci series as ‘0’, the following terms can be calculated by adding the diagonal elements as shown.

Things to Remember

[Click Here for Sample Questions]

Fibonacci numbers are used in various fields of science like quantum mechanics, Cryptography etc.
Fibonacci retracement levels are widely used in technical analysis in financial market trading.
They are used in Coding fundamentals (Computer algorithms, distributed systems etc). For example, They are applied in the computational run-time analysis of Euclid’s Algorithm, and for determining the GCF of two integral values.
They are used in the grouping and arrangement of numbers and used to study other different special mathematical sequences.
Fibonacci series is important because of its relationship with the golden ratio and Pascal’s triangle.
The series of Fibonacci numbers also applies to numbers below zero.
From a financial perspective, the concept of Fibonacci numbers is applicable for whole numbers and decimal numbers.

Read More:

First Order Differential Equation

Sum of n terms

Differential Equations

Sample Questions

Ques. Find the 10^th term of the fibonacci series if the 8^th and 9^th terms are 13 and 21. (2 Marks)

Ans. Using the Fibonacci series recursive formula, the 10^th term is the sum of 8^th and 9^th terms.

10^th term = 8^th term + 9^th term

10^th term = 13 + 21

10^th term = 34

Ques. Using the fibonacci series formula, find the value of the 21^st and 22^nd term of fibonacci series given that the 19^th and 20^th terms are 2584 and 4181. (2 Marks)

Ans. Using the fibonacci formula, we can say that the 21^th term is the sum of the 19^th and 20^th term.

21st term = 19^th term + 20^th term = 2584 + 4181 = 6765

Now, 22^nd term = 20^th term + 21^st term = 6765 + 4181 = 10946

Ques. What are the examples of fibonacci series in nature and its uses? (3 Marks)

Ans. The fibonacci series can be seen in biological settings around us in different forms. It can be found in spirals in petals of some flowers such as in flower heads of sunflowers. It can also be found in branches of trees.

Fibonacci series have various applications in our day-to-day lives. They are used to study patterns of numbers and data in mathematical analysis in various fields.

Ques. Calculate the sum of the first 15 fibonacci numbers in the series. (2 Marks)

Ans. As from the series, the first 15 fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.

Then to find the sum, we have to add all these numbers in the given sequence.

0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144 + 233 + 377

= 986

Ques. Find a fibonacci number when n = 4 using recursive relation. (3 Marks)

Ans. As we know the formula for fibonacci number,

F_n = F_n-1 + F_n-2

Where F_n is n^th term

F_n-1 is (n-1)^th term

F_n-2 is (n-2)^th term

Since the first and second terms are 0 and 1, so F₀ = 0, F₁ = 1.

Hence third term is F₂ = F₀ + F₁ = 0 + 1 = 1

Fourth term is F₃ = F₂ + F₁ = 1 + 1 = 2

Fifth term is F_zw = F₃ + F₂ = 1 + 2 = 3

So, 3 is the fibonacci number when n = 4 using recurrence relation

Ques. Find the fibonacci number using the golden ratio when n = 6. (3 Marks)

Ans. The formula to calculate fibonacci number using golden ratio is

Fn = \(\phi\)ⁿ - (1-\(\phi\))ⁿ / √5

In which \(\phi\) is golden ratio whose value is approximately equal to 1.618 n where n is n^th term of the series.

In the question, n = 6 given

Substituting both values in the equation we get,

F⁶ = 1.618⁶ - (1-1.618)6 / √5

F⁶ = 17.942 - (0.618)6 / √5

F⁶ = 17.942 - 0.056 / 2.236

F⁶ = 17.886 / 2.236

F⁶ = 7.999

F⁶ = 8 (rounded value)

So, the fibonacci number is 8 when n = 6.

Ques. Find the sum of GP.: 1, 2,4,8, ... up to the 10^th term. (2 Marks)

Ans. Using the formula for summation of n terms of a GP:

Sn = a(r^n-¹)/(r-1)

Putting the value of a=1, r =2 , n=10

We get the solution :

2^¹0 - 1

Ques. Find the sum of an infinite GP 3,1,1/3,….? (2 Marks)

Ans. The sum of an infinite GP is given by,

Sn = a / 1 - r

a = 3, r=1/3.

Putting the values in the equation :

3 / (1-1/3)= 9/2

Mathematics Related Links:

Properties of Determinants	Quadrilateral Formula	Integers as Exponents?
Harmonic Mean	Summation Formula	Sequence and Series

Fibonacci Series: Properties and Ratio

Fibonacci Numbers

Properties of Fibonacci Numbers

Fibonacci Series and Golden Ratio

Fibonacci Series and Pascal’s Triangle

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

2.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

Comments

SUBSCRIBE TO OUR NEWS LETTER

Fibonacci Series: Properties and Ratio

Fibonacci Numbers

Properties of Fibonacci Numbers

Fibonacci Series and Golden Ratio

Fibonacci Series and Pascal’s Triangle

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

2.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

4. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

2.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]