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Sequence and Series - Arithmetic and Geometric Progressions

Introduction to Sequence and Series

Objective After going through this lesson, you shall be able to understand the following concepts. Sequence and Its Classification Series and Its Classification Introduction Consider a group of some numbers. These groups may or may not follow some pattern. However, we are mostly interested in the groups of numbers that follow or satisfy some pattern. In this chapter, we would study about such groups or lists of numbers. These groups of numbers form sequences. The study of patterns leads to generalisation, which is widely used in real life. Sequence and its Classification Let us consider the following pattern of numbers. (i) 5, 9, 13, 17, ... (ii) 15, 12, 9, 6, ... (iii) 4, 20, 100, 500,... In (i), except 5, every number is obtained by adding 4 to the previous number. In (ii), except 15, the other numbers are obtained by subtracting 3 from the previous number. In (iii), except 4, every number is obtained by multiplying the previous number by 5. So, we can see that each of these given groups of numbers follow a particular pattern or rule. An arrangement of numbers in a definite order following some rule is known as a sequence. In general, a sequence is denoted by {an} or < an>. This represents the sequence a1, a2, a3, ..., an. The numbers a1, a2, a3, …, an that occur in a sequence are called its terms. Here, the subscript denotes the position of the term in the sequence. The nth term or the general term of a sequence is denoted by an. This term enables us to find a certain term of the sequence by its number. This is an expression in terms of n such that on putting the value of n, we can find any term of the sequence. The first term is found by putting n = 1, the second term by putting n = 2, and so on. Note: In general, if a sequence is given, then we can find its nth term and if the  nth term of a sequence is given, then we can find the terms of the sequence. Consider the sequence 5, 9, 13, 17, .... Its nth term is an = 4n + 1 Now, by putting n = 1, 2, 3, 4,... in the expression 4n + 1, we can obtain the above sequence of numbers. There are two types of sequences. Finite and infinite sequences. Finite sequence: A sequence containing finite number of terms is called a finite sequence. For example: 5, 10, 15, 20, 25, 30, 35 is a finite sequence of 7 terms. Infinite sequence: A sequence containing infinite number of terms is called an infinite sequence. For example: Sequence of prime numbers (2, 3, 5, 7,...) and sequence of natural numbers (1, 2, 3, 4,...) are infinite sequences. Series and Its Classification Let a1, a2, a3,…, an be a sequence, then the expression a1 + a2 + a3 + …+ an is called the series associated with the sequence. The sum of the terms of a sequence is called the series of the corresponding sequence. For example: The series associated with the sequence 18, 36, 54, 72, 90,… is 18 + 36 + 54 + 72 + 90 + … In a compact form, the series associated with the sequence a1, a2, a3,…, an can be written in sigma notation as, where sigma ( ∑ ) denotes the sum. There are two types of series. Finite series: A series containing finite number of terms is called a finite series. For example: 5 + 10 + 15 + 20 + 25 + 30 + 35 is a finite series. Generally, we denote a finite series by the expression a1+a2+a3+...+an=∑k=1nak. Infinite series: A series containing infinite number of terms is called an infinite series. For example: Series of prime numbers (2 + 3 + 5 + 7 + 11 + ...) and series of natural numbers (1 + 2 + 3 + 4 +...) are infinite series. Generally, we denote an infinite series by the expression a1+a2+a3+...+an+...=∑k=1∞ak. Let us now consider some examples to understand the above concepts. Example 1: If nth term of a sequence is n2, then find its 6th term? Solution nth term of the sequence, an = n2 Putting n = 6, we get 6th term, a6 = 62 = 36 Example 2: Write the first four terms of the sequence that is defined by an=nn2+1. Solution nth term of the sequence, an=nn2+1 Putting n = 1, 2, 3, 4 in an=nn2+1, we get a1=112+1=12a2=222+1=25a3=332+1=310a4=442+1=417

Example 3: The nth term of a sequence is given by an =. Find the ratio of the 6th term of the sequence to its 4th term.

Solution The nth term of a sequence is given by an =.

Thus, the ratio of the 6th term of the sequence to its 4th term is given by

Hence, the required ratio is 129 : 17.

Example 4: Write the first five terms of the sequence whose nth term is given by. Solution It is given that the nth term of the sequence is given by.

∴ an =

On putting n = 1, 2, 3, 4, 5 successively in an, we get

a1 =

a2 =

a3 =

a4 =

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