Sequences : An Overview

Sequences

Mathematics is the science of patterns and homogeneity. Sequences are all about them.

A sequence has a traditional meaning in mathematics similar to that in ordinary language.

A sequence is any regular pattern exhibiting some characteristic property throughout.

To be precise, any sequence is characterised by a property.

Let us take a simple example of a common most sequence of all, the series of natural numbers, 1, 2, 3 …n. This sequence is characterised by a property that every term is one more than the previous term. Therefore, the sequence is bounded by a property. We may find an infinite number of sequences. You may note that to generate a sequence we need two things. We begin with a starting term, and a rule defined to get the next term or a general term. This general term is often called generating term.

 There are of course sequences that are unbounded and may run infinitely on either side. These may be termed as infinite sequences. The series, which we are dealing with, are called finite sequences, for the apparent reason that they are bounded, i.e. they have the last term. The other types of sequences are called infinite series, which we would take afterwards.

We are now in a position to define a sequence, a finite series. We start with the bound of a series. The bound of a series are the values which the terms or members of a sequence can take. You may understand them as the limits of the series. For example, if we have specified that write all natural numbers less than 10, then we have clarified that our sequence has a limit of n varying from 1 to 9 since we have to find all natural numbers less than 10. A series may be generated by a rule alone along with the set or bound in which it is defined. We call this generating term as the general term of a sequence. You may now guess the general or generate the name of our series of natural numbers. Yes, it would be

t­n=n; here t­n is the symbol of the general term. A sequence is, therefore, a function of a general term. For each value of n, we get a particular term of the series. We are now in a position to define few finite sequences ourselves.

We start with a general term tn = 2n+1 ;

0 < n < 10

You can yourself get few values by putting n = 1, 2, 3 etc.

t1 = 2⋅1+1 = 3

t2 = 2⋅2+1 = 5

and t3 = 2⋅3+1 = 7 etc.

Note that this sequence is a contiguous finite sequence hence we would substitute only values in the range of 0 < n < 10; i.e. we have just nine terms in the series.

Till now we have an idea of a finite sequence. Discussion done till now summarises that a finite sequence has following properties.

Starting term or the initial term
The generating term or general term or the rule to get the next term

Please note that a finite sequence is always bounded, but the reverse is not always true. There may be a sequence which is bounded but may not be finite. Consider the sequence 1,.1,.01,.001… .Here the general term is given by tn =. You can see that all the values of this sequence lie in the range 0 and 1 (This sequence is a continuously decreasing sequence, and it will never reach zero as exponent are always positive, but they can reach as near to zero as possible. We call this phenomenon as converging which you will learn in higher grades). Hence this sequence is bounded in the semi-open set (0, 1], i.e. greater than zero and less than or equal to 1.

Let us take a few examples of finite sequences with general term given. You are required to find the first few terms.

tn = 3n+1 ; 0 < n < 3
tn= 2ⁿ-1 ; 0 < n < 13

Note that n can only take the value of positive integers, although it may have range.

You can quickly get terms of the sequences defined above, by putting the values of

n = 1, 2, 3, etc. You can check your answers from these:

The sequence is 4, 7, 10
The sequence is 1, 3, 7 … ,

Find more notes here.