Introduction
Arithmetic ProgressionsA Progression in which a constant is added to each term in order to obtain the next term.
They found, not only one set of short spirals going clockwise from the centre, but also another set of longer spirals going anti clockwise.
These two beautiful spirals of the sun flower head reveal the astonishing double connection with the Fibonacci series.
Arrangement of Petals in Flowers and Pine Cone
To get an idea of what a sequence is we begin with the following illustration.
Illustration 1
Suppose you are given an apple and asked to eat it in the following way. At each bite you have to consume half the apple in your hand. This means that at the very first bite you will consume half of the whole apple. (Do not waste your time thinking how you will take this first bite.) After this you will have half the apple in your hand. At the second bite, you will consume onefourth of the apple, so that onefourth of the apple remains in your hand. With this second bite you have consumed threefourth of the apple. At the third bite, you will consume oneeighth of the apple, leaving oneeighth in your hand. You continue eating the apple this way and simultaneously define f(n) as the part of the apple consumed after n bites, and g(n) as the part of the apple in your hand after you have taken n bites. Table gives the value of f(n) and g(n) for six values of n.
n 
f ( n) 
g (n) 
1 


2 


3 


4 


5 


6 


If you look carefully at the table, you will find that
f(n) = 1  and g(n) =
for n = 1, 2, 3, 4, 5, 6.
In fact, for each positive integer, f(n) is defined as above.
Note that the values of f and g depend on n. We call f and g as sequences. We usually denote the value of f at n by f_{n} instead of f(n).
We now look at another illustration of a sequence.
Illustration 2
Suppose a man is paid Rs. 20 for the first hour and Rs. 10 for every subsequent hour he works, how much money does he receive if he works for
a) one hour?
b) two hours?
c) three hours?
d) ten hours?
If the man works for just one hour he gets Rs. 20. If he works for two hours he gets Rs. 30. If he works for 3 hours he gets Rs. 40. If he works for 10 hours, he gets Rs. 110
If p_{n} denotes the payment received by the man after working for n hours, then
p_{1} = 20; p_{2} = 30; p_{3} = 40
and so on.
A sequence is an arrangement of numbers in a definite order according to some rule.
Add 2 to the previous number we get sequence.
The various numbers occurring in a sequence are called its terms.
We denote the terms of a sequence by
a_{1} , a_{2} , a_{3 ,} a_{4} , a_{5} , â€¦ a_{n}
Here the subscripts denote the position of the terms.
a_{1} is the first term, a_{2 }is the second term, â€¦ a_{n} is the n^{th }term.
a_{n} or t_{n} denoted as the n^{th} term is also called the general term of the sequence.
Suppose n is a positive integer and to each i in the set {1, 2, 3. â€¦, n} we associate a real number t_{i.} Then the list t_{1}, t_{2}, t_{3}, â€¦ t_{n} is called a finite sequence.
For example, consider the sequence,
, , , , ,
If to each k in the set {1, 2, 3, â€¦, n, â€¦} we associate a real number t_{k}, then the list t_{1}, t_{2}, t_{3}, â€¦, t_{n }â€¦ is called an infinite sequence.
For example,
If we divided 100 by 3, the successive quotients
33, 33.3, 33.33, 33.333, 33.3333, â€¦
form an infinite sequence.
Note that a finite sequence is a function with domain {1, 2, â€¦, n} and an infinite sequence is a function with domain {1, 2, 3, â€¦, n, â€¦}. If t is the name of a sequence, we write t_{i} instead of the usual function notation t(i). The function values are called the terms of the sequence. Thus t_{1} is the first term of the sequence, t_{2} is the second term, and so on.
If t_{n} denotes the quotient obtained at the n^{th} stage, then t_{1} = 33, t_{2} = 33.3, t_{3} = 33.33, t_{4} = 33.333 and so on.
Some other examples of sequences are
Sequence of odd natural numbers: 1, 3, 5, 7, â€¦
Sequence of even natural numbers: 2, 4, 6, 8, â€¦
Sequence of reciprocals:
TheÂ nthÂ term of a Sequence
A sequence is completely known if we know the rule(s) to write its various terms. Most of the time this rule is given in terms of an algebric formula. For instance, in case of sequences of reciprocals, we haveÂ a_{1} = 1; a_{2} = ; a_{3} = .Â
In fact the n^{th} term of this sequence is given by where n is a natural number.
The n^{th} term is usually called as the general term of the sequence, denoted by t_{n}.
The general term in case of the sequence of odd natural numbers is given by t_{n} = 2n â€“ 1 and in case of the sequence of even natural numbers is given by t_{n} = 2 n, where n is a natural number.