Question

In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13…. Starting with the third term, each term is the sum of the previous two. 2 = 1 + 1, 3 = 2 + 1, 5 = 2 + 3, and so on. Choose the correct options-
A. eighth term in Fibonacci series is 21
B. tenth term in Fibonacci series is 55
C. twelfth term in Fibonacci series is 144
D. twelfth term in Fibonacci series is 112

Answer 1

Hint: The knowledge of sequences and series will be applied here. We will begin by writing the general formula for the series, and then use that formula for the different values of n, which will give us the answers. Then we will match the values of the different terms with the sequence.

Complete step-by-step answer:
We know that any term in a Fibonacci sequence is equal to the sum of the previous two terms. So, the general term can be written as-
${{\text{a}}_{\text{n}}} = {{\text{a}}_{{\text{n}} - 1}} + {{\text{a}}_{{\text{n}} - 2}}$
From using the given general term, we will find all the terms upto the twelfth term, which can be done as-
$\begin{align}
 & {{\text{a}}_1} = 1 \\
 & {{\text{a}}_2} = 1 \\
 & {{\text{a}}_3} = {{\text{a}}_2} + {{\text{a}}_1} = 2 \\
 & {{\text{a}}_4} = {{\text{a}}_3} + {{\text{a}}_2} = 3 \\
 & {{\text{a}}_5} = {{\text{a}}_4} + {{\text{a}}_3} = 5 \\
 & {{\text{a}}_6} = {{\text{a}}_5} + {{\text{a}}_4} = 8 \\
 & {{\text{a}}_7} = {{\text{a}}_6} + {{\text{a}}_5} = 13 \\
 & {{\text{a}}_8} = {{\text{a}}_7} + {{\text{a}}_6} = 21 \\
 & {{\text{a}}_9} = {{\text{a}}_8} + {{\text{a}}_7} = 34 \\
 & {{\text{a}}_{10}} = {{\text{a}}_9} + {{\text{a}}_8} = 55 \\
 & {{\text{a}}_{11}} = {{\text{a}}_{10}} + {{\text{a}}_9} = 89 \\
 & {{\text{a}}_{12}} = {{\text{a}}_{11}} + {{\text{a}}_{10}} = 144 \\
\end{align} $

From the given data, we will now check each and every option to see which ones are correct. In option A, we can see that it is given that the eighth term is given as 21, which is true, as shown in the table.
In option B, the tenth term is given as 55, which is also true. In options C and D, we have been given the values of the twelfth term. From the data we can see that the twelfth term is 144, which matches with option C.
The correct options are A, B and C.

Note: Here, we can see that even if we have the general term, we need to find the value of each term individually, which may be confusing to some students. So, finding the term for very large numbers, that is, the twentieth term and so on is very calculative to do by hand. Also, it is important to note that the formula for the general term is not applicable for the first and the second terms.