Renowned Sequences
The
world of mathematical sequences and series is quite fascinating and absorbing.
Such sequences are a great way of Mathematical Recreation. The sequences are
also found in many fields like Physics, Chemistry and Computer Science apart
from different branches of Mathematics. Only a few of the more famous
mathematical sequences are mentioned here:
(1) Fibonacci Series: Probably the most famous of all Mathematical
sequences; it goes like this----
1,1,2,3,5,8,13,21,34,55,89…
At first glance one may
wonder what makes this sequence of numbers so sacrosanct or important or famous.
However a quick inspection shows that it begins with two1 s and continues to
get succeeding terms by adding, each time, the last two numbers to get the next
number (i.e., 1 + 1 = 2, 1 + 2 = 3,
2 + 3 = 5, and so on).
By itself, this is not very remarkable. Yet there are
no numbers in all of mathematics as all-pervading as the fabulous Fibonacci
numbers. They pop up every now and then in nature, geometry, algebra, number theory,
Permutations and combinations and many other branches of mathematics. More
stunningly, they appear in nature abundantly; for example, the number of spirals
of bracts on a pinecone is always a Fibonacci number, and, similarly, the
number of spirals of bracts on a pineapple is also a Fibonacci number. The
appearances in nature seem boundless. The Fibonacci numbers can be found in
connection with the arrangement of branches on various species of trees, as
well as in the number of ancestors at every generation of the male bee on its
family tree. There is practically no end to where these numbers appear or be
sighted.
Fibonacci numbers are
very much connected to the famous ‘Golden Ratio’ or ‘Divine ratio’ whose value
is equal to 1.618…
The larger the Fibonacci
numbers, the closer their ratio of last two terms approaches the golden ratio. For
example, the quotient of the relatively small pair of consecutive Fibonacci
numbers:
13
--- = 1.625
8
Now, consider the
quotient of the somewhat larger pair of consecutive
Fibonacci numbers:
55
--- =
1.6176470588235294117 3
34
These increasingly
larger quotients seem to surround, the actual value of the Golden Ratio. When
we take much larger pairs of consecutive Fibonacci numbers, their quotients get
us ever closer to the actual value of the golden ratio.
4 181
----------
= 1.61803405…
2,584
There are many properties of Fibonacci series, only a few
are listed below:
i.
The sum of any ten consecutive Fibonacci numbers
is divisible by 11.
ii.
Two consecutive Fibonacci numbers do not have
any common factor, which means that they are Co-prime or relatively prime to
each other.
iii.
The Fibonacci numbers in the composite-number
(i.e., non-prime) positions are also composite numbers.
iv.
The sum of the first n Fibonacci numbers is equal
to the Fibonacci number two further along the sequence minus 1.Mathematically ,
F1 + F2+F3……..+Fn
= Fn+2 -1.
There are many counting
problems in combinatorics whose solution is given by the
Fibonacci Numbers.
For example a Question: In how many ways can a person climb a stair-case containing 10 steps if the person takes only one or two steps at a time?
No of Steps
|
The Ways a person can climb the stairs if he takes only one or two steps
at a time.
|
Total Number of ways
|
1
|
(1)
|
1
|
2
|
(1,1) (2)
|
2
|
3
|
(1,1,1), (1,2) (2,1)
|
3
|
4
|
(1,1,1,1) (2,2) (2,1,1) (1,2,1)
(1,1,2)
|
5
|
5
|
(1,1,1,1,1) (1,2,2) (212)(122) (1112)
(1121)(1211)(2111)
|
8
|
6
|
(111111)(222)(11112)(11121)(11211)(12111)(21111)
(2211)(2112)(2121)(1221)(1122)(1212)
|
13
|
One can
easily see how the no of steps are
increasing in Fibonacci Series . Thus if there are ten steps the person can
climb the stairs in 89 ways ; the tenth term of the sries being 89.
(2) Figurate Numbers series like square,
triangular, pentagonal, hexagonal no. series.
(a) Square Numbers Series: it is quite self
explanatory: 1, 4,9,16,25,36,49…
Pictorially, the square numbers can be represented as below:
(b) Triangular number Series: A triangular number or triangle number
counts the objects that can form
an equilateral triangle. The nth triangle
number is the number of dots or balls in a triangle with n dots on a side; it
is the sum of the n natural numbers from 1 to n.
Pictorially,
the triangular numbers can be represented as below:
(c) Pentagonal
number Series: A pentagonal number is a figurate
number that extends the concept of triangular
and square
numbers to the pentagon. A pentagonal is given by the formula:
for n ≥ 1. The first few pentagonal
numbers are:
Pictorially,
the Pentangular numbers can be can be represented as below:
(d) Hexagonal
Numbers: Similarly, pictorially, the hexagonal numbers can be represented
as below:
The
formula for the nth hexagonal number
The
first few hexagonal numbers are:
(e) The Lazy Caterer's Sequence: Formally also known as the central polygonal
numbers, it describes the maximum number of pieces (or bounded/unbounded
regions) of a circle
(a pancake
or pizza
is usually used to describe the situation) that can be made with a given number
of straight cuts. For example, three cuts across a pancake will produce six
pieces if the cuts all meet at a common point, but seven if they do not.
The
maximum number p of pieces that can be created with a given number of cuts n,
where n ≥ 0, is given by the formula
Using binomial
coefficients, the
formula can be expressed as
The constant sum in every row, column and diagonal is called the magic constant or magic sum, M. The magic constant of a normal magic square depends only on n and has the value
Thus the magic square series is like this: 15, 34, 65, 111,
175, 260…
There are many counting problems in
combinatorics
whose solution is given by the Catalan numbers.
1, 11, 21, 1211, 111221, 312211,
13112221, 1113213211, ...
To generate a term of the sequence from the
previous term, just ‘look and say or read’ the digits of the previous member.
For example:1 is read off as "one 1" or 11. ‘11’ is read off as
"two 1s" or 21. ‘21’ is read off as "one 2, then one 1" or
1211. ‘1211’ is read off as "one 1, then one 2, then two 1s" or
111221. ‘111221’ is read off as "three 1s, then two 2s, then one 1"
or 312211.
( The look-and-say sequence was introduced and
analyzed by John Conway in his paper "The Weird and Wonderful Chemistry of
Audioactive Decay" published in Eureka 46, 5–18 in 1986.However it has great
recreational value and it has appeared in several Management Entrance exams in
past.)
Finally a few special series are mentioned below from other branches than Mathematics:
(a)
1, 6, 30, 138, 606… It is about susceptibility
for the planar hexagonal lattice2 in Physics.
(b)
1, 1, 4, 8, 22, 51… It is about the numbers of
alkyl derivatives of benzene with n=6,7,…,carbon atoms.
(c)
3, 7, 46, 4436, 134281216… in Electrical Engineering
about Boolean functions of n variables.
(d)
0, 1, 3 , 5 , 9, 11, 14, 17, 25, 27 , . ..In Computer
Science about the number of comparisons needed to sort n elements by list
merging.
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