Summing up the Fibonacci Series

The Fibonacci Series is the most famous series of all time. It can be taught to kids, but Mathematicians are still unraveling the mysteries behind it.  

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

The series starts from 0 and 1, and the new term is the sum of its 2 preceding terms.

0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3 and so on.... The general term can be written as : - F(n) = F(n - 1) + F(n - 2), where F(0) = 0 and F(1) = 1

Another amazing feature of the series is The Fibonacci Spiral. To draw it:

Draw two squares of length 1, besides each other. Take the longer length of the rectangle, and draw a square on it, of length 1 + 1 = 2, and so on..... Then, draw a spiral, like in the figure(Area given inside box):

This figure is seen significantly in nature, in flowers, like sunflower. Amazing Fact : Africa looks like a Fibonacci Spiral.

The Fibonacci Spiral can be used to prove many results :

If you take the Area of the rectangle, till say, 13 x 13.

Area = 0² + 1² + 1² + 2² + 3² + 5² + 8² + 13²

The same thing can be written as the area of the rectangle, with side length 13 and 21.

Area = 13 x 21

This pattern can be continued due to the similar nature of the spiral, and the continuous blocks.

Therefore, it can be written as:

It is also known that the ratio of consecutive terms in the Fibonacci Series tends to a particular number, the Golden Ratio, which most people know because of The Da Vinci Code. We can calculate the value of Phi(Golden Ratio) :

As you can see that one value of phi is negative, and Fibonacci Numbers and increasing and positive, therefore, The Golden Ratio is taken as 1.618.

You might have noticed that the Golden Ratio derivation did not require the starting terms. So, if you take any series such that its any term is the sum of its two preceding terms, its terminal ratio will be the same - Golden Ratio.

Ex :- 5, 6, 11, 17, 28, 45, 73, ..... 

One fascinating thing is the way we can derive the nth term of the Fibonacci Series, using a Difference Equation.

We can represent the Fibonacci Series as a geometric series, as we know that it has a common ratio (Golden Ratio).

          

          

On solving for a and b by putting n = 0 and 1, we get :

        

The way the Fibonacci Series is defined, it starts from F(0) = 0, but if extend it towards the left, we get negative Fibonacci Numbers for negative indexes.

..... -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13,.....

F(-1) = 1

F(-2) = -1

You can see that in the negative direction, the mod of the numbers is the same, but it changes signs alternatively.

In the nth Fibonacci Formula, if you put negative numbers, you get the negative Fibonacci Numbers, as we have just listed out.

One thing, which will not make sense, but is still pretty cool, is Fractional Fibonacci Numbers. One problem which is faced is that most, almost all, FFN(Fractional Fibonacci Numbers) are Complex, as  is a negative number. 

Another thing, which is cool, is Imaginary Fibonacci Numbers(IFM), like of form F(a + bi)

Before that another formula :

From that, we can define .

Now, to find general formula for IFMs.

This is not a very neat formula, and no matter what, do not try to memorize it. The good thing about this formula is that, it gives you an expression with all positive constants, i.e. .

And that, is it.