Introduction
The Sequence and Series chapter of Class 11 Mathematics is the study of patterns found in sequences and series of numbers. This chapter introduces the student to the concept of mathematical sequences and series, and how to work with them. It also covers topics such as arithmetic and geometric progressions, infinite series, and the summation of finite and infinite series.
The aim of this chapter is to equip the student with the knowledge and skills to identify, analyse and solve problems involving sequences and series. By the end of the chapter, the student should be able to determine the sum of a given sequence, find the nth term of a sequence, and be able to find the sum of a finite or infinite series. They should also be able to use the properties of arithmetic and geometric progressions to solve equations.
Arithmetic Sequence
A sequence of numbers in which a certain number can be added to (or subtracted from) any term to get the next term is called an arithmetic sequence (or arithmetic progression). In other words, a sequence in which the difference between successive terms always has the same value is called an arithmetic sequence. The arithmetic sequence is also known as the arithmetic progression. As an abbreviation, the arithmetic sequence and the arithmetic progression are respectively written as A.S. and A.P. respectively. For examples,
(a) 1,3,5,7,9,…
(b) 9,-6,-3,0,3,…
Geometric Sequences
A geometric sequence is a sequence of numbers in which the ratio of any term to its preceding term is always the same The geometric sequence is also known as the geometric progression. Geometric sequence and geometric progression are written as G.S., and G.P respectively. Examples of such sequences are;
(a) 1,2,4,8, 16,…..
(b) 3,-9,27,-81,…….
Harmonic Sequences
A sequence is said to be a harmonic sequence if the reciprocals of its terms form an arithmetic sequence. The harmonic sequence is also known as the harmonic progression. Harmonic sequence and harmonic progression are also written as H.S. and H.P. respectively. Some examples of harmonic sequence (harmonic progression) are;
(a) 1,1/2,1/3, 1/4 ,……
(b) 1,1/3,1/5, 1/7 ,…..
Exercise – 4.1
Introduction
Class 11 Mathematics fourth chapter named ‘The Sequence and Series’ is the study of patterns found in sequences and series of numbers. We already have learned about sequence and series in grade 9 and 10 but this chapter was included in grade 11 also inorder to introduce the student the concept of mathematical sequences and series, and how to work with them. It also covers topics such as arithmetic progress, geometric progressions, harmonic progression, infinite series, and the summation of finite and infinite series.
The main motive of this chapter is to equip the student with the knowledge and skills to identify, analyse and solve problems involving sequences and series. By the end of the chapter, the student should be able to determine the sum of a given infinite as well as finite geometric sequence, find the nth term of a sequence, and be able to find common ratio of geometric sequence. Students should also be able to use the properties of arithmetic progression and geometric progressions to solve the given equations.
Geometric Sequences
A geometric sequence is a sequence of numbers in which the ratio of any term to its preceding term is always the same The geometric sequence is also known as the geometric progression. Geometric sequence and geometric progression are written as G.S., and G.P respectively. Examples of such sequences are;
(a) 1,2,4,8, 16,…..
(b) 3,-9,27,-81,…….
Sum of Infinite Geometric Series
As I already mentioned, in this exercise we are going to learn, how to find sum of infinite geometric series. While finding the sum of infinite geometric series we are going to use a formula which is, S = a/1 – r , where a = first term and r = common ratio of the given series.
