# Some Special Sequences, Objectives, Define a Series, Fibonacci Series

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Suppose you are asked to collect pebbles every day in such a way that on the first day if you collect one pebble, second day you collect double of the pebbles that you have collected on the first day, third day you collect double of the pebbles that you have collected on the second day, and so on.

Then you write the number of pebbles collected day wise, you will have a sequence,

From a sequence we derive a series. The series corresponding to the above sequence is

One well known series is Fibonacci series

In this lesson we shall study some special types of series in detail.

## Objectives

After studying this lesson, you will be able to:

## Series

An expression of the form is called a series, where is a sequence of numbers. The above series is denoted by . If is finite then the series is a finite series, otherwise the series is infinite. Thus we find that a series is associated to a sequence.

Thus a series is a sum of terms arranged in order, according to some definite law.

Consider the following sets of numbers:

(a) (b)

(c) (d)

(a), (b), (c), (d) form sequences, since they are connected by a definite law. The series associated with them are :

Example:

Write the term of each of the following series:

(a)

(b)

(c)

(d)

Solution:

(a) The series is

Here the odd terms are negative and the even terms are positive. The above series is obtained by multiplying the series. by

(b) The series is

(c) The series is

The above series can be written as

(d) The series is

i.e.