# Introduction

Arithmetic Progressions
A Progression in which a constant is added to each term in order to obtain the next term.

Who was Fibonacci?
Leonardo Fibonacci was a mathematician most well known for discovering a number sequence which now bears his name whilst solving a problem about the reproduction rates of rabbits.

This is a Romanesque Cauliflower, a vegetable which exhibits fractal recurrences of nodules with Fibonacci-distributed spirals.

Scientists have measured the number of spirals in the sunflower head.

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 one-fourth of the apple, so that one-fourth of the apple remains in your hand. With this second bite you have consumed three-fourth of the apple. At the third bite, you will consume one-eighth of the apple, leaving one-eighth 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 fn 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 pn denotes the payment received by the man after working for n hours, then

p1 = 20; p2 = 30; p3 = 40

and so on.

Sequence

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

a1 , a2 , a3 , a4 , a5 , â€¦ an

Here the subscripts denote the position of the terms.

a1 is the first term, a2 is the second term, â€¦ an is the nth term.

an or tn denoted as the nth 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 ti. Then the list t1, t2, t3, â€¦ tn 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 tk, then the list t1, t2, t3, â€¦, tn 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 ti instead of the usual function notation t(i). The function values are called the terms of the sequence. Thus t1 is the first term of the sequence, t2 is the second term, and so on.

If tn denotes the quotient obtained at the nth stage, then t1 = 33, t2 = 33.3, t3 = 33.33, t4 = 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 a1 = 1; a2 = ; a3 = .

Â

In fact the nth term of this sequence is given by where n is a natural number.

The nth term is usually called as the general term of the sequence, denoted by tn.

The general term in case of the sequence of odd natural numbers is given by tn = 2n â€“ 1 and in case of the sequence of even natural numbers is given by tn = 2 n, where n is a natural number.