JEE Advanced
JEE Main
BITSAT
View complete IIT JEE section
NTSE
KVPY
Olympiads
CBSE
ISCE
UAE
Saudi Arabia
Qatar
Kuwait
Oman
Bahrain
View complete NRI section
Physics
Chemistry
Maths
Revision notes
View complete study material
Buy IIT JEE/AIPMT study material
Buy CBSE Grade 8/9/10 study material
Progressions and Series What do we mean by a sequence? A sequence is basically a succession of numbers which is formed according to some definite rule. The most important thing in the concept of a sequence is that the domain is the set of natural numbers. Sequences may be finite or infinite depending on the number of terms they contain. If the range of the sequence is the set of real numbers then the sequence is said to be real valued. Similarly if the range is of complex numbers then the sequence is said to be complex valued. Eg: 1, 3, 5, 7 is a sequence which is formed by addition of 2 to every preceding term. What is a progression? If the terms of a sequence are written under some specific condition then the sequence is called a progression. What is a series? Series are formed with the terms of the sequence itself. For a given sequence {fn}, the corresponding series is given by f1 + f2 + …. + fn. Eg: For the sequence 1, 3, 5, 7, …. , the corresponding series is given by 1+3+5+7+……. . A series is generally expressed using the summation sign like the series (1+3+5+7) can be expressed as Σ (2n-1), where summation runs from 1 to 4. Types of Progressions: 1. Arithmetic ProgressionBasically there are three types of progressions: 2. Geometric Progression 3. Harmonic Progression All these progressions have been discussed in detail in the coming sections. Here we shall just discuss these topics without going into the intricacies: Arithmetic Progression: If the first term of the Arithmetic Progression which is generally called as A.P. is ‘a’ and the common difference is ‘d’ then the nth term is given by tn = a+(n-1)d.A sequence of numbers <an> is said to be in arithmetic progression if the difference between the terms is constant i.e. tn – tn-1 is a constant for all n ∈ N. the constant difference between the terms is called the common difference and is denoted by‘d’. Similarly, the sum of first ‘n’ terms of the A.P. is given by S_{n} = n/2 [2a + (n-1) d] = n/2 [a + l], where l is the last term of the A.P. Geometric Progression: If the first term of the geometric progression is ‘a’ and the common difference is ‘r’ then the nth term is given by tn = arn-1If every successive term is a fixed multiple of the preceding term then the series is said to be a geometric series. A progression of the form a, ar, ar2, ….. is a geometric progression where the common multiple r is called the fixed ratio. it follows from the definition itself that no term of the G.P can be zero because in such a case the series will get reduced to a zero series. The sum S_{n} of the first ‘n’ terms of the GP is given by S_{n} = a(r^{n} -1)/ (r-1) , if r ≠1 = na, if r =1. 1 + x + x^{2} +…. + = 1/(1-x)I^{n} case of an infinite G.P. the sum is given by Harmonic Progression: The nth term of H.P. is given by the sequence a1, a2, …. an, where ai ≠ 0 for every i is said to be in harmonic progression (H.P.) if the sequence 1/a1, 1/a2, …. ,1/an is in A.P. a_{n} =1 / [a+(n-1)d], where a = 1/a_{1} and d = 1/a_{2} -1/a_{1}. For harmonic progression problems and solutions, please refer the following sections. Progressions and Series is important from the perspective of scoring high in IIT JEE as it fetches 1-2 questions in most of the engineering examinations. There are few fixed pattern questions that are generally framed on this topic. You are expected to do all the questions based on this to remain competitive in IIT JEE examination. Related resources:
A sequence is basically a succession of numbers which is formed according to some definite rule. The most important thing in the concept of a sequence is that the domain is the set of natural numbers. Sequences may be finite or infinite depending on the number of terms they contain.
If the range of the sequence is the set of real numbers then the sequence is said to be real valued. Similarly if the range is of complex numbers then the sequence is said to be complex valued.
Eg: 1, 3, 5, 7 is a sequence which is formed by addition of 2 to every preceding term.
If the terms of a sequence are written under some specific condition then the sequence is called a progression.
Series are formed with the terms of the sequence itself. For a given sequence {fn}, the corresponding series is given by f1 + f2 + …. + fn.
Eg: For the sequence 1, 3, 5, 7, …. , the corresponding series is given by
1+3+5+7+……. .
A series is generally expressed using the summation sign like the series (1+3+5+7) can be expressed as Σ (2n-1), where summation runs from 1 to 4.
1. Arithmetic ProgressionBasically there are three types of progressions:
2. Geometric Progression
3. Harmonic Progression
All these progressions have been discussed in detail in the coming sections. Here we shall just discuss these topics without going into the intricacies:
If the first term of the Arithmetic Progression which is generally called as A.P. is ‘a’ and the common difference is ‘d’ then the nth term is given by tn = a+(n-1)d.A sequence of numbers <an> is said to be in arithmetic progression if the difference between the terms is constant i.e. tn – tn-1 is a constant for all n ∈ N. the constant difference between the terms is called the common difference and is denoted by‘d’.
Similarly, the sum of first ‘n’ terms of the A.P. is given by
S_{n} = n/2 [2a + (n-1) d]
= n/2 [a + l], where l is the last term of the A.P.
Geometric Progression: If the first term of the geometric progression is ‘a’ and the common difference is ‘r’ then the nth term is given by tn = arn-1If every successive term is a fixed multiple of the preceding term then the series is said to be a geometric series. A progression of the form a, ar, ar2, ….. is a geometric progression where the common multiple r is called the fixed ratio. it follows from the definition itself that no term of the G.P can be zero because in such a case the series will get reduced to a zero series.
The sum S_{n} of the first ‘n’ terms of the GP is given by
S_{n} = a(r^{n} -1)/ (r-1) , if r ≠1
= na, if r =1.
1 + x + x^{2} +…. + = 1/(1-x)I^{n} case of an infinite G.P. the sum is given by
Harmonic Progression: The nth term of H.P. is given by the sequence a1, a2, …. an, where ai ≠ 0 for every i is said to be in harmonic progression (H.P.) if the sequence 1/a1, 1/a2, …. ,1/an is in A.P.
a_{n} =1 / [a+(n-1)d], where a = 1/a_{1} and d = 1/a_{2} -1/a_{1}.
For harmonic progression problems and solutions, please refer the following sections.
Progressions and Series is important from the perspective of scoring high in IIT JEE as it fetches 1-2 questions in most of the engineering examinations. There are few fixed pattern questions that are generally framed on this topic. You are expected to do all the questions based on this to remain competitive in IIT JEE examination.
Related resources:
Click here for the Complete Syllabus of IIT JEE Mathematics.
Look into the Past Year Papers with Solutions to get a hint of the kind of questions asked in the exam.
You can get to know the Useful Books of Mathematics here.
Refer the Revision Notes on Arithmetic, Geometric and Harmonic Progression for a quick revision.
Geometric Mean Geometric mean between two numbers...
Harmonic Mean Harmonic Mean is one of the several...
Arithmetic Mean of mth Power The concept of...
Examples Based on Relationship between AM, GM and...
Arithmetic mean between two numbers When three...
Relation between A.M., G.M. and H.M. Let there are...
Geometric Progression A sequence is said to be in...
Arithmetic Progression (A.P.) A sequence is said...
Sum of first n terms of AP A sequence of numbers...
Method of Differences Suppose a 1 , a 2 , a 3 ,...
Arithmeticogeometric Progression Solved Examples...
Arithmetic-Geometric Progression Suppose a 1 , a 2...
Arithmetic Mean Examples What exactly do you mean...
Weighted Means Let a 1 , a 2 , ……, a...
Basic Principle of AP: Progressions and series...
Solved Examples based on GP Solved examples :...
Solved Examples Based on Harmonic mean...
Geometric Progression Geometric Progression as the...
Basic Concepts of Sequences and Pattern Sequence A...
Harmonic Progression Harmonic progression is an...
Download IIT JEE Solved Examples Based on...