Introduction to Fibonacci sequence Equation:
From this article, we may learn about the Fibonacci sequence equation in this article. The Fibonacci sequence Equation was introduced by the great Mathematician Leonardo Fibonacci .Fibonacci sequence is defined as set of numbers in each successive number is the sum of the two preceding numbers. Fibonacci sequence Equation can be programmed in computer languages using recursive equations. This sequence starts with the one or zero and followed by one and the next term is the sum of preceding Fibonacci sequence is also called as Lucas sequence.
Define Fibonacci Sequence Equation :
Fibonacci Sequence can be used in the fields of botany, computer science and in financial markets. The Fibonacci number sequence is 0,1,1,2,3,5,8……. ..
The Fibonacci number sequence always stats with a number 0 and 1 as first and second numbers. In recursive relation, the Fibonacci sequence is defined as Fn.
`F_n=F_(n-1) + F_(n-2)`
Where` F_0` ` =0 ` and `F_1 = F_2 = 1.`
Thus 1 +1= 2, 1 +2= 3, 2 + 3= 5, and so on.
The sequence of ratios of consecutive Fibonacci numbers:
`1/1,2/1,3/2,5/3,8/5`
Let` f_n ` be the nth term of the sequence. Then
`f_1= 1/1 ,f_2 = 2/1,............., f_n = (F(n+1))/(F(n))`
By using recursive relation,
`f_n = (F(n+1))/(F(n))`
`= (F(n)+F(n-1))/(F(n))`
`= 1 + (F(n-1))/(F(n))`
` = ` `1+ (1/(F(n))/(F(n-1)))`
`=` `1+(1/f_n-1)`
Let `f_n` and `f_n-1` having the same limits as
`lim_(n->oo)` ` f_n`= `f_(n-1)`
The real number f converges and satisfy the following equation:
`f=1+(1/f)`
`f^2-f-1=0`
By solving this quadratic equation we have roots as
`(1+- sqrt 5)/2`
The generating function for the Fibonacci number is
`G(x) = F_n f_n`
` = f/(1-f-f^2)`
` = f + f^2 +2f^3 +3f^4+`..........
Example Problem to Fibonacci Sequence Equation :
1.Complete the Fibonacci series . 1, 2 , 3,___,8, ___, 21,___….. using the Fibonacci equation
The solution is
`F_n = F_(n-1)+F_(n_2)`
= 2+3 =5
= 8+ 5 = 13
= 13+21 = 34
Answers are 5,13,34
2.Find the sum of the given Fibonacci sequence. 2,3,5,8,13,21,34,55,89,144
The solution is
To multiply the seventh term in the Fibonacci series by 11.
The seventh term is 34
34 * 11 =374
The Answer is 374
From this article, we may learn about the Fibonacci sequence equation in this article. The Fibonacci sequence Equation was introduced by the great Mathematician Leonardo Fibonacci .Fibonacci sequence is defined as set of numbers in each successive number is the sum of the two preceding numbers. Fibonacci sequence Equation can be programmed in computer languages using recursive equations. This sequence starts with the one or zero and followed by one and the next term is the sum of preceding Fibonacci sequence is also called as Lucas sequence.
Define Fibonacci Sequence Equation :
Fibonacci Sequence can be used in the fields of botany, computer science and in financial markets. The Fibonacci number sequence is 0,1,1,2,3,5,8……. ..
The Fibonacci number sequence always stats with a number 0 and 1 as first and second numbers. In recursive relation, the Fibonacci sequence is defined as Fn.
`F_n=F_(n-1) + F_(n-2)`
Where` F_0` ` =0 ` and `F_1 = F_2 = 1.`
Thus 1 +1= 2, 1 +2= 3, 2 + 3= 5, and so on.
The sequence of ratios of consecutive Fibonacci numbers:
`1/1,2/1,3/2,5/3,8/5`
Let` f_n ` be the nth term of the sequence. Then
`f_1= 1/1 ,f_2 = 2/1,............., f_n = (F(n+1))/(F(n))`
By using recursive relation,
`f_n = (F(n+1))/(F(n))`
`= (F(n)+F(n-1))/(F(n))`
`= 1 + (F(n-1))/(F(n))`
` = ` `1+ (1/(F(n))/(F(n-1)))`
`=` `1+(1/f_n-1)`
Let `f_n` and `f_n-1` having the same limits as
`lim_(n->oo)` ` f_n`= `f_(n-1)`
The real number f converges and satisfy the following equation:
`f=1+(1/f)`
`f^2-f-1=0`
By solving this quadratic equation we have roots as
`(1+- sqrt 5)/2`
The generating function for the Fibonacci number is
`G(x) = F_n f_n`
` = f/(1-f-f^2)`
` = f + f^2 +2f^3 +3f^4+`..........
Example Problem to Fibonacci Sequence Equation :
1.Complete the Fibonacci series . 1, 2 , 3,___,8, ___, 21,___….. using the Fibonacci equation
The solution is
`F_n = F_(n-1)+F_(n_2)`
= 2+3 =5
= 8+ 5 = 13
= 13+21 = 34
Answers are 5,13,34
2.Find the sum of the given Fibonacci sequence. 2,3,5,8,13,21,34,55,89,144
The solution is
To multiply the seventh term in the Fibonacci series by 11.
The seventh term is 34
34 * 11 =374
The Answer is 374
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