Fibonacci Numbers: Overview, Questions, Preparation

A sequence is an ordered list of numbers that follow a specific pattern. A list of numbers that follow a sequence form series. If a series has countable numbers, it is known as finite series. If a series has uncountable numbers or has no end, it is known as infinite series.

Fibonacci Sequence:

A sequence in which the next number is obtained through the addition of the two previous numbers is known as a Fibonacci Sequence. This sequence follows a pattern of recurrence.

To obtain the Fibonacci sequence, we have a general formula:

F_m = F_m-1 + F_m-2

Where F_m represents the m^th term of a Fibonacci sequence. F_m-1 represents the (m-1)^th term of the Fibonacci sequence, Fm-2 represents the (m-2)th term of the Fibonacci Sequence.

Fibonacci Numbers:

To find the Fibonacci numbers, we use the general equation of the Fibonacci Sequence.

So, F_m = F_m-1 + F_m-2

Source: ciet.nic.in 

To begin this sequence, we consider the first two natural numbers, i.e. 0 and 1.
0, 1, F_m

The next term in the sequence is calculated by adding the previous two numbers.
F_m = 0 + 1= 1

So the sequence becomes- 0,1,1, F_m
Similarly, to find the next term of the sequence, add the earlier two numbers.
F_m = 1 + 1= 2

Continuing this pattern, we calculate the next few Fibonacci numbers.
F_m = 1 + 2= 3
F_m = 2 + 3= 5
F_m = 3 + 5= 8
F_m = 5 + 8= 13
F_m = 8 + 13= 21
F_m= 13 + 21= 34
F_m = 21 + 34= 55
F_m = 34 + 55= 89
F_m = 55 + 89= 144
F_m = 89 + 143= 233

And so on.

So, the Fibonnaci numbers are-
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,...

The topic Fibonacci Series is a part of the chapter Sequences and Series in class 11. This chapter holds a weightage of around 7-8 marks in the exams. Out of which 3-4 marks are assigned to this topic.

1. Find a_m+1 / am , for m= 1, 2, 3, 4, 5, if  a₁ = 1 =  a₂

Solution.

We know that a_m =  a_m-1  + a_m-2
So,  a₃ = 1+ 1 = 2,  a₄  = 2+ 1= 3, a₅ = 3+ 2= 5, a₆ = 5+ 3= 8
For m=1, a_m+1 / a_m = a₂/a₁= 1/1= 1
For m=2, a_m+1 / a_m = a₃/a₂= 2/1= 2
For m=3,a_m+1 / a_m = a₄/a₃= 3/2
For m=4, a_m+1 / a_m = a₅/a₄= 5/3
For m=5, a_m+1 / a_m = a₆/a₅= 8/5

2. Find a_m+1 -a_m , for m= 2, 3,4, if  a₁ = 0 and a₂= 1

Solution.
According to the formula of Fibonacci sequence, a_m =  a_m-1  + a_m-2
So,  a₃ = 1+ 0 = 1,  a₄  = 1+ 1= 2, a₅ =2 + 1= 3
For m=2, a_m+1 -a_m = a₃-a₂= 1-1= 0
For m=3, a_m+1 -a_m = a₄-a₃= 2-1 = 1
For m=4, a_m+1 -a_m= a₅-a₄= 3-2= 1

Q: What is the difference between sequence and series?

Q: Give the general equation of sequence and series.

Q: What are the types of sequences?

Q: Is there a repetition in the numbers of the Fibonacci sequence?

Q: What is the application of Fibonacci numbers?