Question

The sequence \[1,1,2,3,5,8,13,21,34,...............\] is said to be in………………
A. Arithmetic Progression
B. Finite Sequence
C. Fibonacci Sequence
D. None of the above

Answer 1

Hint: According to given in the question we have to determine the given sequence which is \[1,1,2,3,5,8,13,21,34,...............\] said to be in Arithmetic Progression, Finite Sequence, Fibonacci Sequence or None of these. So, first of all if the given sequence is in Arithmetic Sequence so, we have to understand about the Arithmetic Sequence which is as explained below:
Arithmetic Sequence: A sequence is said to be in Arithmetic Sequence if the common difference between its terms are equal and common difference is the difference between the second and first term of the sequence and third and second term of the sequence and it the obtained differences are same to each-other than the sequence is said to be in Arithmetic Sequence.
Now, we have to check if the given sequence is a finite sequence but before that we have to understand about the finite sequence which is as explained below:
Finite Sequence: A finite sequence is the list of terms in a specific order. The sequence has the first term and the last term and the order of the terms of the finite sequence follows some type of mathematical pattern or any logical arrangement.
Now, we have to check for if the given sequence is in Fibonacci sequence but before that we have to understand about the Fibonacci sequence which is as explained below:
Fibonacci Sequence: A sequence is said to be in Fibonacci sequence if the third term of the sequence is the sum of its two last terms of the given sequence.

Complete answer:
Step 1: First, of all if the given sequence is in Arithmetic Sequence so, we have to understand about the Arithmetic Sequence which is as explained in the solution hint. Hence, we have to find the common difference (d) for the given sequence.
$ \Rightarrow $Difference between the second and first term of the given sequence,
$
   \Rightarrow d = 1 - 1 \\
   \Rightarrow d = 0 \\
 $
$ \Rightarrow $Difference between the third and second term of the given sequence,
$
   \Rightarrow d = 2 - 1 \\
   \Rightarrow d = 1 \\
 $
As we can see that, the obtained common differences are not the same hence, the given sequence is not in Arithmetic Progression.
Step 2: Now, we have to check if the given sequence is a finite sequence but before that we have to understand about the finite sequence which is as explained in the solution hint and as we can see that there is not any last number and any logical pattern hence, we can say that the given sequence is not a finite sequence.
Step 3: Now, we have to check for if the given sequence is in Fibonacci sequence but before that we have to understand about the Fibonacci sequence which is as explained in the solution hint so, we have to check if the third is the sum of its last two terms or not hence,
$ \Rightarrow $Third term = 2
$ \Rightarrow $Sum of two terms before the third terms$ \to 1 + 1 = 2$
Same as,
$ \Rightarrow $Fourth term = 3
$ \Rightarrow $Sum of two terms before the fourth terms$ \to 1 + 2 = 3$
It is clear that the given sequence is in Fibonacci.
Final solution: Hence, after understanding about Arithmetic Progression, Finite sequence, and Fibonacci sequence we have determined that the given sequence \[1,1,2,3,5,8,13,21,34,...............\]is said to be in Fibonacci sequence.

Therefore option (C) is correct.

Note: To determine the common difference we just have to find the difference between the second and the first term and third and the second term and same as for the next of the terms and id the obtained difference is same then we can say that the given sequence is as A.P.
The sequence has the first term and the last term and the order of the terms of the finite sequence follows some type of mathematical pattern or any logical arrangement.