8.1 Introduction

In mathematics, the word, “sequence” is used in much the same way as it is in ordinary English. 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. For example, population of human beings or bacteria at different times form a sequence. The amount of money deposited in a bank, over a number of years form a sequence. Depreciated values of certain commodity occur in a sequence. Sequences have important applications in several spheres of human activities.

Sequences, following specific patterns are called progressions. In previous class, we have studied about arithmetic progression (A.P). In this Chapter, besides discussing more about A.P.; arithmetic mean, geometric mean, relationship between A.M. and G.M., special series in forms of sum to $n$ terms of consecutive natural numbers, sum to $n$ terms of squares of natural numbers and sum to $n$ terms of cubes of natural numbers will also be studied.

8.2 Sequences

Let us consider the following examples:

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 $=\frac{300}{30}=10$

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

Consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division. In this process we get $3,3.3,3.33,3.333, \ldots$ and so on. These quotients also form a sequence. The various numbers occurring in a sequence are called its terms. We denote the terms of a sequence by $a_1, a_2, a_3, \ldots, a_n, \ldots$, etc., the subscripts denote the position of the term. The $n^{\text{th }}$ term is the number at the $n^{\text{th }}$ position of the sequence and is denoted by $a_n$. The $n^{\text{th }}$ term is also called 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, \ldots, a _{10}=1024 $

Similarly, in the example of successive quotients

$ a_1=3, a_2=3.3, a_3=3.33, \ldots, a_6=3.33333 \text{, etc. } $

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).

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.

Often, it is possible to express the rule, which yields the various terms of a sequence in terms of algebraic formula. Consider for instance, the sequence of even natural numbers $2,4,6, \ldots$

$ \begin{aligned} & \text{ Here } \quad a_1=2=2 \times 1 \quad a_2=4=2 \times 2 \\ & a_3=6=2 \times 3 \quad a_4=8=2 \times 4 \\ &\ldots & \ldots . & \ldots . & \ldots . & \ldots . & \ldots\\ & a _{23}=46=2 \times 23, a _{24}=48=2 \times 24 \text{, and so on. } \end{aligned} $

$ \begin{array}{lll} \text{Here } & a_1=2=2 \times 1 & a_2=4=2 \times 2 \\ & a_3=6=2 \times 3 & a_4=8=2 \times 4 \\ & \ldots \quad \dots \quad \dots & \ldots \quad \dots \quad \dots \\ & \ldots \quad \dots \quad \dots & \ldots \quad \dots \quad \dots \\ \end{array} \\ \quad \quad\quad a_{23}=46=2 \times 23, a_{24}=48=2 \times 24, \text{and so on} $

In fact, we see that the $n^{\text{th }}$ term of this sequence can be written as $a_n=2 n$, where $n$ is a natural number. Similarly, in the sequence of odd natural numbers $1,3,5, \ldots$, the $n^{\text{th }}$ term is given by the formula, $a_n=2 n-1$, where $n$ is a natural number.

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

$ \begin{aligned} & a_1=a_2=1 \\ & a_3=a_1+a_2 \\ & a_n=a _{n-2}+a _{n-1}, n>2 \end{aligned} $

This sequence is called Fibonacci sequence.

In the sequence of primes $2,3,5,7, \ldots$, we find that there is no formula for the $n^{\text{th }}$ prime. Such sequence can only be described by verbal description.

In every sequence, we should not expect that its terms will necessarily be given by a specific formula. However, we expect a theoretical scheme or a rule for generating the terms $a_1, a_2, a_3, \ldots, a_n, \ldots$ in succession.

In view of the above, a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it. Sometimes, we use the functional notation a(n) for $a_n$.

8.3 Series

Let $a_1, a_2, a_3, \ldots, a_n$, be a given sequence. Then, the expression

$ a_1+a_2+a_3+\ldots+a_n+\ldots $

is called the series associated with the given sequence. The series is finite or infinite according as the given sequence is finite or infinite. Series are often represented in compact form, called sigma notation, using the Greek letter $\sum$ (sigma) as means of indicating the summation involved. Thus, the series $a_1+a_2+a_3+\ldots+a_n$ is abbreviated

as $\sum_{k=1}^{n} a_k$.

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

We now consider some examples.

8.4 Geometric Progression (G. P.)

Let us consider the following sequences:

(i) $2,4,8,16, \ldots$,

(ii) $\frac{1}{9}, \frac{-1}{27}, \frac{1}{81}, \frac{-1}{243}$

(iii) $.01, .0001, .000001, \ldots$

In each of these sequences, how their terms progress? We note that each term, except the first progresses in a definite order.

In (i), we have $a_1=2, \frac{a_2}{a_1}=2, \frac{a_3}{a_2}=2, \frac{a_4}{a_3}=2$ and so on.

In (ii), we observe, $a_1=\frac{1}{9}, \frac{a_2}{a_1}=\frac{1}{3}, \frac{a_3}{a_2}=\frac{1}{3}, \frac{a_4}{a_3}=\frac{1}{3}$ and so on.

Similarly, state how do the terms in (iii) progress? It is observed that in each case, every term except the first term bears a constant ratio to the term immediately preceding it. In (i), this constant ratio is 2; in (ii), it is $-\frac{1}{3}$ and in (iii), the constant ratio is 0.01 . Such sequences are called geometric sequence or geometric progression abbreviated as G.P.

A sequence $a_1, a_2, a_3, \ldots, a_n, \ldots$ is called geometric progression, if each term is non-zero and $\frac{a_k+1}{a_k}=r$ (constant), for $k \geq 1$.

By letting $a_1=a$, we obtain a geometric progression, $a, a r, a r^{2}, a r^{3}, \ldots$, where $a$ is called the first term and $r$ is called the common ratio of the G.P. Common ratio in geometric progression (i), (ii) and (iii) above are $2,-\frac{1}{3}$ and 0.01 , respectively.

As in case of arithmetic progression, the problem of finding the $n^{\text{th }}$ term or sum of $n$ terms of a geometric progression containing a large number of terms would be difficult without the use of the formulae which we shall develop in the next Section. We shall use the following notations with these formulae:

$ \begin{aligned} & a=\text{ the first term, } r=\text{ the common ratio, } l=\text{ the last term, } \\ & n=\text{ the numbers of terms, } \\ & S_n=\text{ the sum of first } n \text{ terms. } \end{aligned} $

8.4.1 General term of $a$ G.P.

Let us consider a G.P. with first non-zero term ’ $a$ ’ and common ratio ’ $r$ ‘. Write a few terms of it. The second term is obtained by multiplying $a$ by $r$, thus $a_2=a r$. Similarly, third term is obtained by multiplying $a_2$ by $r$. Thus, $a_3=a_2 r=a r^{2}$, and so on.

We write below these and few more terms.

$1^{\text{st }}$ term $=a_1=a=a r^{1-1}, 2^{\text{nd }}$ term $=a_2=a r=a r^{2-1}, 3^{\text{rd }}$ term $=a_3=a r^{2}=a r^{3-1}$

$4^{\text{th }}$ term $=a_4=a r^{3}=a r^{4-1}, 5^{\text{th }}$ term $=a_5=a r^{4}=a r^{5-1}$

Do you see a pattern? What will be $16^{\text{th }}$ term?

$ a _{16}=a r^{16-1}=a r^{15} $

Therefore, the pattern suggests that the $n^{\text{th }}$ term of a G.P. is given by $a_n=a r^{n-1}$.

Thus, $a$, G.P. can be written as $a, a r, a r^{2}, a r^{3}, \ldots a r^{n-1} ; a, a r, a r^{2}, \ldots, a r^{n-1} \ldots ;$ according as G.P. is finite or infinite, respectively.

The series $a+a r+a r^{2}+\ldots+a r^{n-1}$ or $a+a r+a r^{2}+\ldots+a r^{n-1}+\ldots$ are called finite or infinite geometric series, respectively.

8.4.2. Sum to $n$ terms of $a$ G.P.

Let the first term of a G.P. be $a$ and the common ratio be $r$. Let us denote by $S_n$ the sum to first $n$ terms of G.P. Then

$ S_n=a+a^{n}+a r^{2}+\ldots+a r^{n-1} \quad \quad \quad \quad \quad \ldots (1) $

Case 1 If $r=1$, we have $S_n=a+a+a+\ldots+a(n$ terms $)=n a$

Case 2 If $r \neq 1$, multiplying (1) by $r$, we have

$ r S_n=a r+a r^{2}+a r^{3}+\ldots+a r^{n} \quad \quad \quad \quad \quad \ldots (2) $

Subtracting (2) from (1), we get $(1-r) S_n=a-a r^{n}=a(1-r^{n})$

This gives

$ \text{ or } \quad S_n=\frac{a(r^{n}-1)}{r-1} $

8.4.3 Geometric Mean (G.M.)

The geometric mean of two positive numbers $a$ and $b$ is the number $\sqrt{a b}$. Therefore, the geometric mean of 2 and 8 is 4 . We observe that the three numbers $2,4,8$ are consecutive terms of a G.P. This leads to a generalisation of the concept of geometric means of two numbers.

Given any two positive numbers $a$ and $b$, we can insert as many numbers as we like between them to make the resulting sequence in a G.P.

Let $G_1, G_2, \ldots, G_n$ be $n$ numbers between positive numbers $a$ and $b$ such that $a, G_1, G_2, G_3, \ldots, G_n, b$ is a G.P. Thus, $b$ being the $(n+2)^{\text{th }}$ term, we have

$ b=a r^{n+1}, \quad \text{ or } \quad r=(\frac{b}{a})^{\frac{1}{n+1}} \text{. } $

Hence $G_1=a r=a(\frac{b}{a})^{\frac{1}{n+1}}, G_2=a r^{2}=a(\frac{b}{a})^{\frac{2}{n+1}}, G_3=a r^{3}=a(\frac{b}{a})^{\frac{3}{n+1}}$,

$ G_n=a r^{n}=a(\frac{b}{a})^{\frac{n}{n+1}} $

8.5 Relationship Between A.M. and G.M.

Let $A$ and $G$ be A.M. and G.M. of two given positive real numbers $a$ and $b$, respectively. Then

$ A=\frac{a+b}{2} \text{ and } G=\sqrt{a b} $

Thus, we have

$ \begin{aligned} A-G & =\frac{a+b}{2}-\sqrt{a b}=\frac{a+b-2 \sqrt{a b}}{2} \\ & =\frac{(\sqrt{a}-\sqrt{b})^{2}}{2} \geq 0 \quad \quad \quad \quad \quad \quad \quad \ldots (1) \end{aligned} $

From (1), we obtain the relationship $A \geq G$.

Summary

By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type ${1,2,3, \ldots . k}$. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.

Let $a_1, a_2, a_3, \ldots$ be the sequence, then the sum expressed as $a_1+a_2+a_3+\ldots$ is called series. A series is called finite series if it has got finite number of terms.

A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by $a$ and its common ratio by $r$. The general or the $n^{\text{th }}$ term of G.P. is given by $a_n=a r^{n-1}$. The sum $S_n$ of the first $n$ terms of G.P. is given by

$ S_n=\frac{a(r^{n}-1)}{r-1} \text{ or } \frac{a(1-r^{n})}{1-r} \text{, if } r \neq 1 $

The geometric mean (G.M.) of any two positive numbers $a$ and $b$ is given by $\sqrt{a b}$ i.e., the sequence $a, G, b$ is G.P.

Historical Note

Evidence is found that Babylonians, some 4000 years ago, knew of arithmetic and geometric sequences. According to Boethius (510), arithmetic and geometric sequences were known to early Greek writers. Among the Indian mathematician, Aryabhatta (476) was the first to give the formula for the sum of squares and cubes of natural numbers in his famous work Aryabhatiyam, written around 499. He also gave the formula for finding the sum to $n$ terms of an arithmetic sequence starting with $p^{\text{th }}$ term. Noted Indian mathematicians Brahmgupta

(598), Mahavira (850) and Bhaskara (1114-1185) also considered the sum of squares and cubes. Another specific type of sequence having important applications in mathematics, called Fibonacci sequence, was discovered by Italian mathematician Leonardo Fibonacci (1170-1250). Seventeenth century witnessed the classification of series into specific forms. In 1671 James Gregory used the term infinite series in connection with infinite sequence. It was only through the rigorous development of algebraic and set theoretic tools that the concepts related to sequence and series could be formulated suitably.