Question

What is the $ {{15}^{th}} $ term in the Fibonacci sequence of numbers?

Answer 1

Hint: We first explain the Fibonacci sequence of numbers. We use the theorem for $ {{n}^{th}} $ number as $ {{F}_{n}} $ , where we have $ {{F}_{n}}={{F}_{n-1}}+{{F}_{n-2}},n>2 $ . We also use $ {{F}_{1}}=0,{{F}_{2}}=1 $ . Then using the binary operation of addition, we find the $ {{15}^{th}} $ term.

Complete step by step solution:
In mathematics, the Fibonacci sequence commonly denoted as $ {{F}_{n}} $ is such that each number of the sequence is the sum of two preceding ones, starting from 0 and 1.
Therefore, if we denote the $ {{n}^{th}} $ number as $ {{F}_{n}} $ , then we have $ {{F}_{n}}={{F}_{n-1}}+{{F}_{n-2}},n>2 $ .
The starting numbers are $ {{F}_{1}}=0,{{F}_{2}}=1 $ .
There is no particular formula to find the $ {{n}^{th}} $ number. We have to find it manually by adding every number.
Therefore, we keep putting the number $ n=3,4,......15 $ in the equation of $ {{F}_{n}}={{F}_{n-1}}+{{F}_{n-2}} $ .
We put the value of $ n=3 $ to get $ {{F}_{3}}={{F}_{2}}+{{F}_{1}}=1+0=1 $ .
We put the value of $ n=4 $ to get $ {{F}_{4}}={{F}_{3}}+{{F}_{2}}=1+1=2 $ .
We put the value of $ n=5 $ to get $ {{F}_{5}}={{F}_{4}}+{{F}_{3}}=2+1=3 $ .
We put the value of $ n=6 $ to get \[{{F}_{6}}={{F}_{5}}+{{F}_{4}}=3+2=5\].
We put the value of $ n=7 $ to get $ {{F}_{7}}={{F}_{6}}+{{F}_{5}}=5+3=8 $ .
We put the value of $ n=8 $ to get $ {{F}_{8}}={{F}_{7}}+{{F}_{6}}=8+5=13 $ .
We put the value of $ n=9 $ to get $ {{F}_{9}}={{F}_{8}}+{{F}_{7}}=13+8=21 $ .
We put the value of $ n=10 $ to get $ {{F}_{10}}={{F}_{9}}+{{F}_{8}}=21+13=34 $ .
We put the value of $ n=11 $ to get $ {{F}_{11}}={{F}_{10}}+{{F}_{9}}=34+21=55 $ .
We put the value of $ n=12 $ to get $ {{F}_{12}}={{F}_{11}}+{{F}_{10}}=55+34=89 $ .
We put the value of $ n=13 $ to get $ {{F}_{13}}={{F}_{12}}+{{F}_{11}}=89+55=144 $ .
We put the value of $ n=14 $ to get $ {{F}_{14}}={{F}_{13}}+{{F}_{12}}=144+89=233 $ .
We put the value of $ n=15 $ to get $ {{F}_{15}}={{F}_{14}}+{{F}_{13}}=233+144=377 $ .
Therefore, the $ {{15}^{th}} $ term in the Fibonacci sequence of numbers is 377.
So, the correct answer is “377”.

Note: There is a specific formula to find $ {{n}^{th}} $ number of the series. Fibonacci number sequence can be used to create ratios or percentages that traders use. The ratio of the number is called as the golden ratio which appears in biological settings such as branching in trees, phyllotaxis, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts etc.