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\title{\textcolor{yellow}{Fibbonacci Series}}
\author{Amba Kulkarni}
\institute{ University of Hyderabad}
\date{22nd February 2018}

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\frametitle{Fibonacci}
The Fibonacci sequence: named after Leonardo of Pisa, who was known as Fibonacci. \\

Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics. \\

(By modern
convention, the sequence begins with F0 = 0. The Liber Abaci began the sequence with F1 = 1)\\

Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming
that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the
age of one month so that at the end of its second month a female can produce another pair of rabbits;
rabbits never die and a mating pair always produces one new pair (one male, one female) every month
from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one
year?
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\frametitle{Fibonacci}
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\item At the end of the first month, they mate, but there is still only 1 pair.
\item At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits
in the field.
\item  At the end of the third month, the original female produces a second pair, making 3 pairs in all in
the field.
\item At the end of the fourth month, the original female has produced yet another new pair, the female
born two months ago produces her first pair also, making 5 pairs.
\item At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which
is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth
Fibonacci number.
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\frametitle{M{\=a}tr{\=a} V\d{r}ttas}
{\=A}c{\=a}rya Virah{\=a}\.nka gave the rule for formation of numbers
of variations in m{\=a}tr{\=a} v\d{r}ttas for the first time.

Gop{\=a}la discusses Virah{\=a}\.nka's rule in detail as follows\\

Variations of two earlier meters [is the variation] is a m{\=a}tr{\=a}-v\d{r}tta.
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\frametitle{M{\=a}tr{\=a} V\d{r}ttas}
\begin{itemize}
\item For [a meter of] three [morae], variations of two earlier meters, one and two, being mixed three happens.
\item For [a meter] of four [morae], variations of meters of two morae [and] of three moare being mixed, five happens.
\item For [a meter] of five [morae], variations of meters of two earlier [meters] of three morae [and] of four morae, being mixed, eight is obtained.
\item In this way, for [a meter] of six morae, [variations] of four morae [and] of five morae being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, [variations of a meter] of seven morae [is] twenty-one.
\item In this way, the process shuould be followed in all m{\=a}tr{\=a}-v\d{r}ttas.
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\frametitle{M{\=a}tr{\=a} V\d{r}ttas}
\begin{itemize}
\item A meter of 3 m{\=a}tr{\=a}s = A meter of 1 m{\=a}tr{\=a} + G\\
       =  A meter of 2 m{\=a}tr{\=a} + L\\
      S[3] = S[1] + S[2]
\item A meter of 4 m{\=a}tr{\=a}s = A meter of 2 m{\=a}tr{\=a} + G\\
       =  A meter of 3 m{\=a}tr{\=a} + L\\
      S[4] = S[2] + S[3]
\item A meter of 5 m{\=a}tr{\=a}s = A meter of 3 m{\=a}tr{\=a} + G\\
       =  A meter of 4 m{\=a}tr{\=a} + L\\
      S[5] = S[3] + S[4]
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\frametitle{Fibonacci Sequence in Nature}
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\frametitle{Fibonacci Sequence in Nature}
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\frametitle{Fibonacci Sequence in Nature}
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\frametitle{Fibonacci Sequence in Nature}
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\frametitle{Fibonacci Sequence in Nature}
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\frametitle{Fibonacci Sequence in Nature}
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