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# Fibonacci Numbers Examples

A fibonacci number is a number in the sequence of numbers 1, 1, 3, 5, 8, 13, 21, 34, 55, ...... Each number in the sequence except the first two 1's, is got by adding the previous two terms.

Fibonacci sequence can therefore be defined as $a_{1}$ = $a_{2}$ = 1
$a_{n} = a_{n-1} + a_{n-2}$, where n is a positive integer which is greater than 2.

This is a recursive formula to find a term in the Fibonacci Sequence.

An important property of Fibonacci Numbers is that the ratio between two consecutive terms approximates very close to $\varphi$, the golden ratio after initial few terms. $\varphi$ is an irrational number which is equal to $\frac{\sqrt{5}+1}{2}$.

This property is used in finding a general term in fibonacci sequence.

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## Examples on Fibonacci Numbers

The formula to be used to find a general term in Fibonacci Sequence is as follows:

$a_{n}$ = $\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\sqrt{5}}$.

The value of the expression after substituting the value of n can be found using the calculator.

Verification Test for Fibonacci Number:
Suppose a number N is given. If the computed value of 5 $N^{2}$ + 4 or 5 $N^{2}$ - 4 is a perfect square, then N is a Fibonacci number.

### Solved Examples

Question 1: Find the 14th term in the fibonacci sequence.
Solution:
We need to find $a_{14}$ here.

$a_{14}$ = $\frac{(1+\sqrt{5})^{14}-(1-\sqrt{5})^{14}}{2^{14}\sqrt{5}}$

The value returned by the calculator for the above expression is 377
Hence, the 14th term in fibonacci sequence is 377.

The ratio between any two consecutive terms is $\varphi$. That is,

$\frac{a_{n}}{a_{n-1}}$ = $\frac{\sqrt{5}+1}{2}$

This property can be used to find the preceding or succeeding term of a given fibonacci number.

Question 2: 987 is a Fibonacci Number. Find the preceding and succeeding numbers in the sequence.
Solution:
Let us find the preceding number using the above relation. We have $a_{n}$ = 987 and let $a_{n-1}$ = x.

$\frac{987}{x}$ = $\frac{\sqrt{5}+1}{2}$

987 x 2 = x $(\sqrt{5}+1)$

x = $\frac{987\times 2}{\sqrt{5}+1}$

= 609.99954689.......
Rounding to the nearest integer, x = 610.

The number succeeding 987 is obtained by adding the number preceding to it.
Hence, the number that comes after 987 in the sequence is 987 + 610 = 1597.
We can also find the next number using the following proportion:

$\frac{y}{987}$ = $\frac{\sqrt{5}+1}{2}$

y = 987 x $\frac{\sqrt{5}+1}{2}$

y = $\frac{987(\sqrt{5}+1)}{2}$

= 1596.9995468......
Rounding this, we get the number 1597.

Question 3: Check whether the number 4181 is a Fibonacci number or not.
Solution:
Let us compute the two expressions given for the test.
5$N^{2}$ + 4 = $5(4181)^{2}$ + 4 = 87,403,809
$\sqrt{87,403,809}$ = 9349.000428
The computed value for 5$N^{2}$ + 4 is not a perfect square.
Similarly, 5$N^{2}$ - 4 = 87,403,801.
$\sqrt{87,403,801}$ = 9349
The computed value of 5$N^{2}$ - 4 is a perfect square.
Hence, 4181 is a Fibonacci number.

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