Solved Examples of Fibonacci Sequence
1. Calculate the value of \(12^{\textrm{th}}\) and the \(13^{\textrm{th}}\) term of the Fibonacci sequence given that the \(9^{\textrm{th}}\) and \(10^{\textrm{th}}\) terms in the sequence are \(21\) and \(34\).
Solution: First find the \(11^{\textrm{th}}\) using the formula of Fibonacci sequence.
The formula of Fibonacci sequence is \(F_{n}=F_{n-1}+F_{n-2}\).
So \(F_{11}=F_{10}+F_{9}\)
Substituting the values of \(9^{\textrm{th}}\) and \(10^{\textrm{th}}\) terms in the formula, we get
\(F_{11}=34+21\) \(\Rightarrow\) \(F_{11}=55\)
Thus, the \(11^{\textrm{th}}\) of the Fibonacci sequence is \(55\).
Now find the \(12^{\textrm{th}}\) using the values of \(10^{\textrm{th}}\) and \(11^{\textrm{th}}\) terms.
So \(F_{12}=F_{11}+F_{10}\)
\(\Rightarrow\) \(F_{12}=55+34\)
\(\Rightarrow\) \(F_{12}=89\)
Thus, the \(12^{\textrm{th}}\) of the fibonacci sequence is \(89\).
Similarly, find the \(13^{\textrm{th}}\) using the values of \(11^{\textrm{th}}\) and \(12^{\textrm{th}}\) terms.
So \(F_{13}=F_{12}+F_{11}\)
\(\Rightarrow\) \(F_{13}=89+55\)
\(\Rightarrow\) \(F_{13}=144\)
Thus, the \(12^{\textrm{th}}\) of the fibonacci sequence is \(144\).
Therefore, the \(12^{\textrm{th}}\) and the \(13^{\textrm{th}}\) term of the fibonacci sequence are \(89\) and \(144\).
2. Find the fibonacci number when \(n=6\), using the formula of fibonacci sequence.
Solution: The formula of fibonacci sequence is \(F_{n}=F_{n-1}+F_{n-2}\).
Take \(F_{0}=0\) and \(F_{1}=1\)
Using the formula, we get
\(F_{2}=F_{1}+F_{0}=1+0=1\)
\(F_{3}=F_{2}+F_{1}=1+1=2\)
\(F_{4}=F_{3}+F_{2}=2+1=3\)
\(F_{5}=F_{4}+F_{3}=3+2=5\)
\(F_{6}=F_{5}+F_{4}=5+3=8\)
Therefore, the fibonacci number is \(8\) when \(n=6\).
3. The \(14^{\textrm{th}}\) term of the fibonacci sequence is \(377\). Find the next term
Solution: Given that \(F_{14}=377\).
We know that \(F_{15}=F_{14}\times\) (the golden ratio)
\(\Rightarrow\) \(F_{15}=F_{14}\times 1.618034\)
\(\Rightarrow\) \(F_{15}=609.99\)
\(\Rightarrow\) \(F_{15}=610\)
Therefore, the \(15^{\textrm{th}}\) of the fibonacci sequence is \(610\).
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