Fibonacci ...

Just as a legend has its history, so too does mathematics. The history of this sequence dates back to its "tiling" in The Babylonian Tablet of Shamash (885–855 BC), which could be veneered with two orders of golden rectangles. The Book of Exodus (600–400 BC) describes the Ark of the Covenant, which follows a 3:5 golden ratio.

Each number is the total possible ways to arrange long (L) and short (S) syllables in a poetic meter of a specific length. There are 13 ways to arrange long and short syllables in meters of length 6.5. Five of these end with a long syllable, and eight end with a short syllable. This is because the number of patterns that end in L is given by the (n-2)th Fibonacci number and the number of patterns that end in S is given by the (n-1)th Fibonacci number. This is exactly what Virahanka’s Fibonacci sequence in Sanskrit prosody deals with.

The Fibonacci Sequence is connected to Sanskrit prosody, said Paramanand Singh in 1985. In the poetic culture of Sanskrit, there was an interest in enumerating all patterns of long (L) syllables (duration of 2 units) and short (S) syllables (duration of 1 unit). Counting various patterns of consecutive L and S syllables for a given total duration results in the Fibonacci sequence, where the number of patterns for a duration of ‘m’ units is F subscript (m + 1). The understanding of Fibonacci-like sequences existed earlier in the work of Pingala (450–200 BCE).

Paramanand Singh cited that Pingala’s formula ‘Misrau Cha’ (i.e. "the two are mixed") and its interpretation by scholars indicate that the number of patterns for ‘m’ beats F subscript (m + 1) is obtained by adding 1 (S) to Fₘ scenarios and 1 (L) to F subscript (m − 1) scenarios. Bharat Muni also describes the Fibonacci sequence in the Natya Shastra (100 BCE–350 CE). A clear explanation of the Fibonacci sequence appeared in Virahanka’s work (700 CE), which was lost but preserved in quotations by Gopala (1135 CE). Differences between two previous meters create variations. For example, for a cadence of 4, combining the variations of meters of 2 and 3 results in 5 (e.g., 8, 13, 21). This pattern applies to all Mātrā-vṛttas (prosodic combinations). Hema Chandra (1150 CE) is credited with recognizing this sequence, writing that the sum of the last term and the one before it gives the next Mātrā-vṛtta .

A survived copy of the book by Fibonacci (Leonardo of Pisa) called Liber Abaci (1202) was preserved in the Biblioteca Nazionale Centrale di Firenze. It showed the Fibonacci sequence with positions notated in Latin and Roman numerals and values in Hindu-Arabic numerals. It had this rabbit population growth hypothetical problem, with a newly born rabbit pair (a male and a female) kept in a field. Since rabbits can mate at 1 month, each female will surely produce a new pair starting from the end of their 2nd month. It was also assumed that rabbits never die. So now the question arises. Fibonacci wondered how many rabbit pairs have we got after a year passes by?

The total number of rabbit pairs for each month is calculated by dividing the number of pairs from the previous month (F(n-1)) by the number of new pairs (F(n-2)), which is equal to the pairs from two months prior. This is what we call recursive.

0+1=1

we get, ‘0,1,1’

1+1 =2

we get, ‘0,1,1,2’

1+2 = 3

we get, ‘0,1,1,2,3’

2+3 = 5

we get, ‘0,1,1,2,3,5’

3+5=8

we get, ‘0,1,1,2,3,5,8’

5+8=13

we get, ‘0,1,1,2,3,5,8,13’

8+13=21

we get, ‘0,1,1,2,3,5,8,13,21’

13+21=34

we get, ‘0,1,1,2,3,5,8,13,21,34’

21+34=55

We get, ‘0,1,1,2,3,5,8,13,21,34, 55’

If you don't necessarily know the preceding terms, better to have a general rule, an equation that governs the sequence :

One has to recall that finding the “summation” (Σ) of 2 previous numbers is recursive, where every term is just the sum of the two preceding numbers. The beauty of having a general rule is that it doesn’t matter if you don’t remember the sequence cause you can always reconstruct it. There’s also the perfect square method that helps verify if a number is Fibonacci. A number z is Fibonacci if and if only 5z^2 + 4 or 5z^2 – 4 apply.

** To what extent, can the sequence whose terms are the Sum of the ‘n’ previous terms be found out?

For example,

a)      “Can we find the ‘nth’ term and sum of ‘n’ terms for the sequence whose first term is 1, the 2nd term is 2 and the other terms are the sum of the previous 2 terms, hence the sequence under consideration being 1,2,3,5,8,13,21,34,….., with the same being applied for ?”

b)     “Can we find the ‘nth’ term and sum of ‘n’ terms for the sequence whose first term is 1, the 2nd term is 2, 3rd term is 3 and the other terms are the sum of the previous 3 terms, hence the sequence under consideration being 1,2,3,6,11,20,37,68,…?”

                                                     And so on….

                                                            OR

“Can we find the ‘nth’ term and sum of ‘n’ terms for the sequence whose first term is 1, the 2nd term is 2 and the other terms are the sum of the previous 2 terms, hence the sequence under consideration being 1,2,3,5,8,13,21,34,….., with the same being applied for Sum of the ‘n’ previous terms ?”                                                              

We know that the Exponential function has the variable in it’s exponent and constant in the base, due to which the trends portrayed by all the graphs lead to an assumption that they’re all modelled by exponential functions. Note that the above graphs are not Power functions, which will have variable at the base and a constant as the power.

*Graph in Desmos of “The Sequence whose terms are the Sum of the 2 Previous Terms” & Tn = 1.6^n

Sequences n Series : strikingly fascinating ? Why ? First off, series and sequence are different. The former is the sum of the terms of a sequence, while the latter is an ordered list of numbers. Do you like patterns and determining governing equations for it? Lateral thinking, visualization and intuition are super important.

So, we start with “The sequence whose terms are the sum of the 2 previous terms”.

The sequence is 1,2,3,5,8,13,21,34,55,….

In Desmos, after attempting to graph the equations such as, y = x^2, y = x^1.5 , y = 0.5^x , y = 1.5^x,………………………, the graph of y = 1.6^x has approximately matched the curve whose points are “The sequence whose terms are the sum of the 2 previous terms”. So, we can say that a constant raised to the power of a variable will be the nearest accurate function to mathematically model the sequence whose terms are the sum of the 2 previous terms. Since, the y -coordinate is ‘Tn’ and the x-coordinate is ‘n’, hence the equation that can model the sequence whose terms are the sum of the 2 previous terms is Tn = x^n, where x is a constant and n is the variable

T3 = T2 + T1 , which is generalized to be Tn = T subscript (n - 1) + T subscript (n - 2)  eq (1)

Substituting Tn = x^n in eq (1), we get,

This expression takes us ahead to solving further including homogeneous second order difference equation, Vieta’s and Cardano’s formula or Gaussian elimination, etc. We can feel that there’s a way to bring fun-loving math into this and to be able to do that might just be a cool process.

The name ‘Fibonacci’ was first used by 19th-century theorist Édouard Lucas, a number theorist. The sequence itself has popped up earlier as discussed above. For the most part, “invention” isn’t as convincing as “discovering”. What we do know is that it is a pattern visible in nature and anything creative.

· Flower pistils and petals

· Tree branches

· Shells

· Leaves

· Our faces

· Seed heads

· Fruits and vegetables

· Finger joints

· Animals, their body ratios, hair, horns, spinal and bone structure.

This language called Math was definitely supposed to create such a vast and infinite universe that we still have not explored entirely. MathEnigma, MatheMagic, MatheMiracle, and Mathematics...