Fibonacci Sequence Examples

Problem 1: Calculate the 9th Fibonacci number if given golden ratio is 1.618034.

Solution: 

We can calculate the 9th Fibonacci number by using the formula:

x_n =  (φⁿ − (1−φ)ⁿ)/√5

x₉ = ((1.618034)⁹-(1-1.618034)⁹)/√5

x₉ = (76.0131604-(-0.0131556197))/√5 = 34.0000021 

x₉ = 34 

Problem 2: Find the next Fibonacci number of answers calculated in the above question.

Solution: 

Next Fibonacci number of 34 can be easily found by multiplying it by the Golden ratio that is 1.618034.

x₁₀ = 34×1.61803 = 55.01302

x₁₀ = 55(rounded off)

Problem 3: If the 5th and 6th terms of a Fibonacci sequence are 3 and 5 respectively, find the 7th term of the sequence.

Solution: 

With the use of the Fibonacci Sequence formula, we can easily calculate the 7th term of the Fibonacci sequence which is the sum of the 5th and 6th terms.

seventh term = 5th term + 6th term

= 3+5

= 8

The 7th term of the Fibonacci sequence is 8.

Problem 3: The first 4 numbers in the Fibonacci sequence are given as 1,1,2,3.

(a) What is the eighth term of the Fibonacci sequence? 

(b) What is the eleventh term of the Fibonacci sequence? 

Solution: 

By the use of the Fibonacci number formula, we can calculate the rest of the Fibonacci numbers like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.

(a) Therefore, the 8th term will be 21.

(b) 11th term will be 89.

Problem 4: Find the next 3 terms for each of the following Fibonacci-style sequences.  

(a) x, 4x, 5x, 9x,… 

(b) 3a, 3a+b, 6a+b, 9a+2b….  

Solution: 

With use of the Fibonacci Sequence formula, we can easily calculate the rest of the terms

(a)

Fifth term = 5x+9x = 14x, 

Sixth term = 9x+14x = 23x, 

Seventh term = 14x+23x = 37x

(b)

Fifth term = 6a+b+9a+2b = 15a+3b, 

Sixth term = 9a+2b+15a+3b = 24a+5b, 

Seventh term = 15a+3b+24a+5b = 39a+8b

Problem 5: John wants to generate a Fibonacci series with the first term as 3 and the second term as 4.

(a) Find the 3rd and 4th terms.

(b) He thinks that the sum of the first ten terms is equal to eleven times the seventh term of his sequence. Check if he is correct.

Solution: 

Using 3 and 4 as first and second terms, we can calculate the rest of the terms by simply adding the last two terms.

(a)

First term = 3

Second term = 4

Third Term = 3+4 = 7

Forth term = 4+7 = 11

(b)

On calculating the first ten terms of the series: 3,4,7,11,18,29,47,76,123,199.

Sum of first ten terms = 3+4+7+11+18+29+47+76+123+199 = 517

7th term = 47

Eleven times the 7th term = 11*47 = 517

As we can see that the sum of the first ten terms is equal to eleven times the seventh term of his sequence. Therefore, John was correct.

Problem 6: What is the first three-digit square number that appears on the list of Fibonacci numbers, if the first 4 terms are 0,1,1,2.

Solution: 

With the use of the Fibonacci Sequence formula, we can easily calculate the rest of the terms:

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

As we can see the first three-digit number which is a square that appears on the list of Fibonacci numbers is 144(square of 12).

Fibonacci Sequence Formula: Fibonacci sequence, the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers; that is, the nth Fibonacci number F_n = F_n − 1 + F_n − 2.

In the Fibonacci sequence, each number in the series is calculated by adding the two numbers before it. Generally, the first two terms of the Fibonacci series are 0 and 1. The Fibonacci sequence was known in India hundreds of years before Leonardo Pisano Bigollo knew about it. November 23rd is celebrated as Fibonacci Day, as it has the digits “1, 1, 2, 3” which is part of the sequence

In this article, we will learn about the Fibonacci Sequence, along with its formula, examples, golden ratio, etc.