Mathematics SEQUENCES AND SERIES - Definitions
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### Topics Covered

star Introduction
star Sequences
star Series

### Introduction

\color{red} ✍️ In mathematics, the word, color(blue)"“sequence”" is used in much the same way as it is in ordinary English.

\color{red} ✍️ When we say that a collection of objects is listed in a sequence, we usually mean that the collection is ordered in such a way that it has an identified first member, second member, third member and so on.

### Sequences

\color{fuchsia} \(★ mathbf\ul"finite sequence")

A sequence containing finite number of terms is called a finite sequence.

For example, sequence of ancestors is a finite sequence since it contains 10 terms (a fixed number).

\color{fuchsia} \(★ mathbf\ul"infinite sequence")

A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends.

color(red)"Let us consider the following examples:"

color(red)(=>) Assume that there is a generation gap of 30 years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over 300 years.

Here, the total number of generations = 300/30= 10

The number of person’s ancestors for the first, second, third, …, tenth generations are 2, 4, 8, 16, 32, …, 1024. These numbers form what we call color(blue)"a sequence."

color(red)(=>) Consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division.

In this process we get 3,3.3,3.33,3.333, ... and so on. These quotients also form a sequence.

\color{red} ✍️ The various numbers occurring in a sequence are called color(blue)"its terms." We denote the terms of a sequence by a_1, a_2, a_3, …, a_n, …, etc., the subscripts denote the position of the term.

\color{red} ✍️ The color(blue)(n^(th)) term is the number at the n^(th) position of the sequence and is denoted by color(blue)(a_n).

\color{red} ✍️ The n^(th) term is also called color(blue)"the general term of the sequence."

Thus, the terms of the sequence of person’s ancestors mentioned above are:

 \ \ \ \ \ \ \ \ \ \ \ \ a_1 = 2, a_2 = 4, a_3 = 8, …, a_10 = 1024.

Similarly, in the example of successive quotients

 \ \ \ \ \ \ \ \ \ \ \ \ a_1 = 3, a_2 = 3.3, a_3 = 3.33, …, a_6 = 3.33333, etc.

\color{fuchsia} \(★ mathbf\ul"Fibonacci sequence")

In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern, but the sequence is generated by the recurrence relation given by

 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_1 = a_2 = 1

 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_3 = a_1 + a_2

 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \a_n = a_(n – 2) + a_(n – 1), n > 2

This sequence is called color(blue)"Fibonacci sequence." we use the functional notation a(n) for a_n.

Often, it is possible to express the rule, which yields the various terms of a sequence in terms of algebraic formula.

color(red)(=>) Consider for instance, the sequence of even natural numbers 2, 4, 6, …

Here a_1 = 2 = 2 × 1 \ \ \ \ \ a_2 = 4 = 2 × 2

\ \ \ \ \ a_3 = 6 = 2 × 3 \ \ \ \ \ \ a_4 = 8 = 2 × 4
.... .... .... .... .... ....
.... .... .... .... .... ....
a_(23) = 46 = 2 × 23, \ \ \ \ a_(24) = 48 = 2 × 24,  and so on.

In fact, we see that the n^(th) term of this sequence can be written as color(red)(a_n = 2n), where n is a natural number.

color(red)(=>) Similarly, in the sequence of odd natural numbers color(red)(1,3,5, ......…,)

The n^(th) term is given by the formula, color(red)(a_n = 2n – 1), where n is a natural number.

a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3...k}. Sometimes, we use the functional notation a(n) for a_n.

### Series

\color{red} ✍️ Let color(red)(a_1, a_2, a_3,…..............,a_n,) be a given color(blue)(sequence).

\color{red} ✍️ Then, the expression

\ \ \ \ \ \ \ \ \ \ \ \ \color(red)( a_1 + a_2 + a_3 +,…......+ a_n + .........

is called the color(blue)"series" associated with the given sequence .

\color{red} ✍️ The series is finite or infinite according as the given sequence is finite or infinite.

\color{red} ✍️ Series are often represented in compact form, called color(red)"sigma notation," using the Greek letter color(blue)(Σ "(sigma)") as means of indicating the summation involved.

\color{red} ✍️ Thus, the series color(red)(a_1 + a_2 + a_3 +,…+ a_n) is abbreviated as color(blue)(Σ_(k=1)^(n) a_k)

color(red)"Key Concept" When the series is used, it refers to the indicated sum not to the sum itself.
For example, color(red)(1 + 3 + 5 + 7) is a finite series with four terms.
When we use the phrase color(blue)"“sum of a series,”" we will mean the number that results from adding the terms, the sum of the series is 16.

Q 3009112018

Write the first three terms in each of the following sequences defined by the following:

(i) a_n = 2n+5

(ii). a_n = (n-3)/4

Solution:

(i) Here a_n = 2n + 5

Substituting n = 1, 2, 3, we get a_1 = 2(1) + 5 = 7, a_2 = 9, a_3 = 11

Therefore, the required terms are 7, 9 and 11.

(ii). Here a_n = (n-3)/4 . Thus a_1 = (1-3)/4 = -1/2 , a_2 = -1/4 , a_3 = 0

Hence, the first three terms are -1/2 , -1/4 and 0
Q 3059212114

What is the 20th term of the sequence defined by a_n = (n-1)(2-n)(3+n) ?

Solution:

Putting n = 20 , we obtain
a_(20) = (20 – 1) (2 – 20) (3 + 20)
= 19 × (– 18) × (23) = – 7866.
Q 3019212119

Let the sequence an be defined as follows: a_1 = 1, a_n = a_(n – 1) + 2 for n ≥ 2. Find first five terms and write corresponding series.

Solution:

We have
a_1 = 1, a_2 = a_1 + 2 = 1 + 2 = 3, a_3 = a_2 + 2 = 3 + 2 = 5,
a_4 = a_3 + 2 = 5 + 2 = 7, a_5 = a_4 + 2 = 7 + 2 = 9.
Hence, the first five terms of the sequence are 1,3,5,7 and 9. The corresponding series
is 1 + 3 + 5 + 7 + 9 +...